Volumes of hyperbolic Haken manifolds, I

Marc Culler^{1} and Peter B. Shalen^{1}

University of Illinois at Chicago

Introduction

In [14] a program was initiated for using the topological theory of 3-manifolds to obtain lower bounds for volumes of hyperbolic 3-manifolds. In [1], by a combination of new geometric ideas with relatively standard (but specifically 3-dimensional) topological techniques, we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is at least 3 has volume exceeding 0.92. By contrast, the best known lower bound [10,5] for the volume of an arbitrary closed hyperbolic 3-manifold is approximately .0012.

In [3] we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is 2 has volume exceeding 0.34. The proof depended on supplementing the results and techniques of [1] with ingenious elementary arguments due to Zagier [11] and numerical computations.

In the present paper we shall show that if one excludes certain special manifolds, such as fiber bundles over S^{1}, then the lower bound of 0.34 also holds for hyperbolic 3-manifolds with Betti number 1. The proof depends heavily on the results of [1] and [3], but it involves much deeper topological ideas than these papers. The new topological results needed for the proof occupy most of the present paper. To some extent these results have the flavor of general topology, but the proofs make use of such specifically low-dimensional techniques as the characteristic submanifold theory [9,8], the interaction between trees and incompressible surfaces, and Scott’s theorem [12] that surface groups are locally extended residually finite.

Before giving a precise statement of our main result we must review a few elementary notions from 3-manifold theory.

Let M be a connected, closed, orientable topological 3-manifold. By an incompressible surface in M we mean a closed, connected, orientable, locally flat 2-manifold S ⊂ M such that genus S > 0 and π_{1}(S) → π_{1}(M) is injective. Recall from [6, p. 62] that M contains a non-separating incompressible surface if and only if H_{1}(M; Q) ≠ 0.

A topological 3-manifold-with-boundary N is said to be boundary-irreducible if for every component C of ∂M the natural homomorphism π_{1}(C) → π_{1}(M), induced by inclusion, is injective. It is plain that if S is an incompressible surface in a connected, closed, orientable 3-manifold M, then each component of the manifold obtained by splitting M along S is boundary-irreducible.

Throughout this paper we will let I denote the unit interval [0, 1] ⊂ R. An I-bundle over a topological 2-manifold B is a locally trivial fiber bundle over B with fiber I.

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1Supported by an NSF grant

Definition. Let N be a compact, connected, orientable topological 3-manifold with boundary. We shall say that N is a book of I-bundles if it has the form N = E ∪ V, where

(i) E is an I-bundle over a non-empty compact 2-manifold-with-boundary B; (ii) each component of V is homeomorphic to D^{2} × S^{1}; (iii) the set A = E ∩ V is the inverse image of ∂B under the bundle projection E → B;

and (iv) each component of A is an annulus in ∂V which is homotopically non-trivial in V.

It follows from the above conditions that int E ∩ int V = ∅.

(The reason for calling N a “book of I-bundles” is that we may regard N as the regular neighborhood of a complex X ⊂ N which, up to homeomorphism, is obtained from the disjoint union of the 2-manifold B with a 1-manifold C (the core of V) by attaching ∂B to C via a covering map. Thus X is a “book of surfaces”: B consists of its “pages” and C is its “binding.” One obtains N from X by thickening up the pages to form I-bundles and the binding to form solid tori.)

Note that in the definition of a book of I-bundles we do not require B to be orientable or connected. Note also that we do allow V to be empty, in which case N is an I-bundle over B. It is not hard to show that a book of I-bundles is boundary-reducible if and only if there is a component V_{0} of V which intersects E in a single annulus and this annulus carries the fundamental group of V_{0}. In particular, books of I-bundles of this exceptional type do not arise from splitting closed manifolds along incompressible surfaces.

Definition. Let S be an incompressible surface in a closed, orientable topological 3manifold M. Let N denote the manifold-with-boundary obtained by splitting M along S. We shall say that S is a fibroid if each component of N is a book of I-bundles.

Note that in particular S is a fibroid if each component of N is an I-bundle. In this case S is either a fiber in a fibration of M over S^{1}, or the common boundary of two twisted I-bundles whose union is M. Incompressible surfaces of these types have been extensively studied and are known to appear as exceptions in the statements or proofs of many theorems about incompressible surfaces. Fibroids constitute a somewhat larger but still very special class of incompressible surfaces. It is this class that appears as exceptional in the main theorem of this paper.

Main Theorem. Let M be a closed, orientable hyperbolic 3-manifold. Suppose that M contains a non-separating incompressible surface which is not a fibroid. Then vol M > 0.34.

There is no evidence that the conclusion of the above theorem is false when the given surface is a fibroid. However, if it remains true it will require a different proof in this case.

In the sequel to this paper we will investigate the case where M contains a separating incompressible surface which is not a fibroid.

The above theorem will be proved in Section 11 by combining the results of [3] with Theorem A below, a result which is of independent interest.

By a closed curve in a topological space X we mean a map α : S^{1} → X.

Theorem A. Let M be a closed, orientable hyperbolic 3-manifold. Suppose that M contains an incompressible surface S which is not a fibroid. Let λ be a positive number less than log 3. Then either (i) M contains a non-trivial closed geodesic of length < λ which is homotopic in M to a closed curve in M − S, or (ii) M contains a hyperbolic ball of radius λ/2.

The proof of Theorem A, which is given in Section 10, depends on a geometric observation made in [2]. We may regard M as the quotient of hyperbolic space H^{3} by a discrete torsion-free group of isometries. For each maximal cyclic subgroup X of Γ, the elements of X have a common axis A_{H^{3}}(X); and if X is “short” in the sense that it has a generator of length < λ, then there is a neighborhood Z_{λ}(X) of A_{H^{3}}(X) in H^{3} consisting of all those points that are displaced a distance less than λ by some element of X. According to [2, Proposition 3.2], if conclusion (ii) of Theorem A does not hold then H^{3} is the union of the sets Z_{λ}(X) where X ranges over all short maximal cyclic subgroups of Γ. In particular, the closures of the Z_{λ}(X) form a locally finite family of closed sets covering H^{3}. In this paper we call this family a “plating” of H^{3} rather than a “covering” in order to avoid confusion with the notion of a covering space.

By an elementary construction that is reviewed in Section 3, any incompressible surface S in M defines an action of Γ ≅ π_{1}(M) on a tree T. If conclusion (i) of Theorem A does not hold then every short maximal cyclic subgroup X of Γ has an axis A_{T} (X) in T, i.e. a line which is invariant under X and on which X acts by translations. A key ingredient in the proof of Theorem A is Proposition 10.3, which implies—under the hypothesis λ < log 3— that if X and Y are short maximal cyclic subgroups of Γ such that Z_{λ}(X) ∩ Z_{λ}(Y) ≠ ∅, then A_{T} (X) and A_{T} (Y) have a common edge in T. Proposition 10.3 is proved by combining the geometrical results of [1] with an elementary fact about group actions on trees ([4,Proposition 4.2], reproduced below as Proposition 2.6), and a theorem of Simon’s on compactification of covering spaces of 3-manifolds (a special case of [15, Theorem 3.1], re-interpreted in terms of trees in Proposition 3.6 below).

It follows from all this that a counterexample to Theorem A would give a plating Z of the universal covering space M̃ = H^{3} of M, indexed by certain maximal cyclic subgroups of Γ, such that if the sets in the plating corresponding to two maximal cyclic subgroups X and Y meet each other then the axes of X and Y in T overlap at least in an edge. Furthermore, the plating would be Γ-equivariant in a natural sense, and the set corresponding to each maximal cyclic subgroup X would be X-invariant and would have a compact quotient by X. One completes the proof of Theorem A by showing that the existence of a plating with these properties implies that the surface S from which the tree T is constructed is a fibroid. This step is embodied in Theorem 9.1. The latter theorem is purely topological in statement and proof. It depends on the hypothesis that M is simple (see 1.6), which is a topological consequence of hyperbolicity. (Thurston’s uniformization theorem implies that any simple 3-manifold which contains an incompressible surface has a hyperbolic structure, but the hyperbolic structure is not used in the proof of 9.1.)

Section 9 is devoted to the proof of Theorem 9.1. The proof involves an intriguing set-theoretical manipulation that constructs from the given plating Z a new plating W, indexed by edges of T. The plating W has order at most 2, in the sense that every point lies in at most 3 = 2 + 1 sets of the plating. The proof that W has order at most 2 ultimately depends on a simple combinatorial fact about trees which we prove as Proposition 2.4. The plating W has formal properties, such as equivariance, somewhat similar to those of Z; on the other hand, because it is indexed by edges, W is related to the tree T in a simpler way than Z is. This, together with the fact that W has order at most 2, allows one to use 3-manifold arguments to prove that S is a fibroid. These arguments are presented in Section 8, and are summed up in Proposition 8.2. It is here that one uses the characteristic submanifold theory and Scott’s theorem.

In Section 1 we establish some conventions that are used in the paper. In Sections 2 and 3 we prove the facts about trees and incompressible surfaces that will be needed. In Section 4 we review a somewhat weak and special form of the characteristic submanifold theory that is sufficient for the applications that we need and less technical than the general theory. In Sections 5 and 6 we develop a systematic theory of platings of spaces that behave equivariantly under group actions; this facilitates the manipulations in Sections 8 and 9. In Section 7 we prove some more specialized results about equivariant platings in the case where the group is locally extended residually finite, and then deduce from Scott’s theorem the result about platings of 2-manifolds that is needed in Section 8.

We are indebted to P. Papasoglou for pointing out that a finiteness hypothesis is needed for some of the results in Section 2. We would also like to thank Peter Scott for useful and encouraging discussions of the material.

Section 1. Conventions

1.1. Since most of the work in the paper is topological, we have generally followed the standard conventions of topology. In particular a manifold may have a boundary unless we specify otherwise. (In Sections 10 and 11, where we consider a hyperbolic manifold M, it is explicitly assumed that M is closed. We do not have occasion to consider hyperbolic manifolds with boundary.)

1.2. The outer category. The following conventions will be useful for dealing with fundamental groups. Let Γ_{1} and Γ_{2} be groups. By an outer homomorphism from Γ_{1} to Γ_{2} we mean an equivalence class of homomorphisms from Γ_{1} to Γ_{2}, where two homomorphisms h and h′ are said to be equivalent if there is an inner automorphism i of Γ_{1} such that h′ = h ∘ i. This equivalence relation is respected by composition of homomorphisms; hence if f_{1} : Γ_{1} → Γ_{2} and f_{2} : Γ_{2} → Γ_{3} are outer homomorphisms between groups, there is a well-defined outer homomorphism f_{2} ∘ f_{1} : Γ_{1} → Γ_{3}. This defines a category in which the objects are groups and the morphisms are outer homomorphisms. The automorphism group of an object in this category is the familiar outer automorphism group of a group.

An outer subgroup of a group Γ is a conjugacy class of subgroups of Γ. A group in the conjugacy class will be said to realize the given outer subgroup. If A and B are outer subgroups of Γ and if some representative of A is contained in a representative of B, we shall say that A is contained in B. The index of an outer subgroup A is the index of an arbitrary subgroup realizing A. If X is a genuine subgroup of Γ, the outer subgroup realized by X will often be referred to simply as “the outer subgroup X.”

The image of an outer homomorphism f : Γ_{1} → Γ_{2} is well-defined as an outer subgroup of Γ_{2}; this outer subgroup will be denoted by f(Γ_{1}). If f(Γ_{1}) is the outer subgroup Γ_{2} we shall say that f is surjective. The kernel of an outer homomorphism is a well-defined normal subgroup of Γ_{1}. An outer homomorphism is injective if its kernel is trivial.

1.3. The fundamental group. If Ω is a path-connected space, we shall often write π_{1}(Ω) for the fundamental group of Ω with an unspecified base point. We regard π_{1}(Ω) as an object in the category described above. Any continuous map F : Ω_{1} → Ω_{2} between path-connected spaces induces a well-defined outer homomorphism f_{♯} : π_{1}(Ω_{1}) → π_{1}(Ω_{2}). For example, the “homomorphisms” appearing in the definitions of incompressibility and boundary-irreducibility given in the introduction are in reality outer homomorphisms.

If α : S^{1} → Ω is a closed curve in a path connected space Ω, the outer homomorphism α_{♯} assigns to the clockwise generator of π_{1}(S^{1}) a conjugacy class in π_{1}(Ω). This gives a bijective correspondence between conjugacy classes in π_{1}(Ω) and free homotopy classes of closed curves in Ω.

1.4. Γ-spaces. Let Γ be a group. Recall that a Γ-set is a set equipped with an action of Γ. If S is a Γ-set then for each s ∈ S we shall denote the stabilizer of s in S by Γ_{s}.

A Γ-space is a topological space equipped with an action of Γ by homeomorphisms. A Γ-space C will be termed free if the action of Γ on C is free. We shall say that a Γ-space C is uniform if there is a compact set R ⊂ C such that

Γ · R = C. If C is uniform then any closed, invariant subset C′ of C is uniform. (Indeed, if R ⊂ C is compact and Γ · R = C, then R′ = R ∩ C′ is compact and Γ · R′ = C′.)

Note also that if C is a uniform Γ-space and Γ_{0} ≤ Γ is a subgroup of finite index, then C is a uniform Γ_{0}-space. Indeed, if Φ ⊂ Γ is a finite set such that Γ_{0}Φ = Γ, and if R ⊂ C is a compact set such that Γ · R = C, then R_{0} = Φ · R is compact and Γ_{0} · R_{0} = C.

1.5. Piecewise linear Γ-manifolds. Let Γ be a group and n a positive integer. By a piecewise linear Γ-manifold of dimension n we shall mean a PL n-manifold equipped with an action of Γ which is simplicial with respect to some triangulation that defines the given PL structure of M. In particular a PL Γ-manifold has the structure of a Γ-space.

If M̃ is a PL Γ-manifold which is free (as a Γ-space), then the quotient space M̃/Γ is a PL manifold of the same dimension as M̃, the manifold M̃ is a covering space of M via the quotient map p : M̃ → M, and the group of deck transformations is Γ. Conversely, the universal covering space of a PL manifold is a free PL Γ-manifold in a natural way, where Γ denotes the group of deck transformations. In this paper the language of free PL Γ-manifolds will sometimes be more convenient than the language of covering space theory.

If M̃ is a free PL Γ-manifold then there is a natural outer homomorphism ν : π_{1}(M̃/Γ) → Γ. The kernel of ν is isomorphic to π_{1}(M̃); in particular, ν is an outer isomorphism if M̃ is 1-connected.

A PL Γ-manifold M̃ will be termed Γ-orientable if the action of Γ on M̃ preserves orientation. Note that a free PL Γ-manifold M̃ is orientable if and only if M̃/Γ is orientable.

1.6. A topological 3-manifold M is termed irreducible if M is connected and every locally flat 2-sphere in M bounds a 3-ball. In this paper we shall say that a PL 3-manifold M is simple if it has the following properties:

(i) M is irreducible; (ii) M is boundary-irreducible (see introduction); and (iii) π_{1}(M) contains no free abelian subgroup of rank 2.

This definition is more restrictive than the one used, for example, in [8].

Note that if an orientable, closed PL 3-manifold M is simple then M contains no incompressible tori.

Section 2. Trees

2.1. Graphs. By a graph we shall mean a CW complex of dimension ≤ 1. The 0-cells and 1-cells of a graph Ψ are called vertices and edges. In particular, an edge is a subset of Ψ and is homeomorphic to an open interval. The boundary of an edge is a subset of the 0-skeleton of Ψ consisting of one or two points. An edge whose boundary is a single point is called a loop. A vertex v of Ψ is called an end point if v is in the boundary of a unique edge e, and e is not a loop.

2.2. Trees. A tree is a 1-connected graph. In particular a tree contains no loops, and may therefore be regarded as a simplicial complex. If e is an edge of a tree T then its closure ē is homeomorphic to a closed interval.

If T is any tree we shall denote by E(T) the set of all edges of T.

We may regard the real line R as a tree by triangulating it so that the vertices are precisely the integer points. A line is a tree L which is simplicially isomorphic to R. A translation of L is a simplicial automorphism which is simplicially conjugate to an integer translation of R.

A segment is a tree σ which is simplicially isomorphic to a (connected) subcomplex of R; we may take the subcomplex to be one of the intervals (−∞, ∞), [0, ∞) or [0, n], where n is a non-negative integer. In the latter case we have n = card(E(σ)) < ∞; we shall refer to n as the length of the finite segment σ. A segment of length 0 consists of a single vertex, and will be termed degenerate.

A triod is a tree which is simplicially isomorphic to a cone on three vertices. The cone point will be called the center of the triod. Note that a triod contains exactly three segments of length 2.

A subcomplex of a tree T which is a segment (or a triod) will be called simply a segment (or a triod) in T.

2.3. Proposition. Let V be a finite, non-empty collection of subtrees of a tree T. Suppose that for all V, V′ ∈ V we have V ∩ V′ ≠ ∅. Then ⋂ V is a subtree of T. In particular, ⋂ V ≠ ∅.

Proof: It suffices to show that ⋂ V has exactly one connected component. First we show that it has at most one. Let s and t be vertices of ⋂ V. Since T is a tree, there is a unique segment α ⊂ T with endpoints s and t. Since each V ∈ V is connected we must have α ⊂ V for every V ∈ V. Hence α ⊂ ⋂ V. This shows that ⋂ V has at most one connected component.

It remains to show that ⋂ V ≠ ∅. If card V ≤ 2 this is trivial. Suppose that card V = 3 and write V = {V, V′, V′′}. Let us choose vertices s ∈ V ∩ V′ and t ∈ V ∩ V′′. There is a unique segment σ ⊂ T with endpoints s and t. Since V is connected we have σ ⊂ V. Since V′ and V′′ are connected and V′ ∩ V′′ ≠ ∅, the set V′ ∪ V′′ is connected and hence σ ⊂ V′∪V′′. Thus we may write σ as the union of the two closed subsets σ∩V′ and σ∩V′′. Since σ is connected, these two subsets must intersect, i.e. ∅ ≠ σ ∩ V′ ∩ V′′ ⊂ V ∩ V′ ∩ V′′. This proves the assertion when card V = 3.

Finally, suppose that card V = n > 3 and that the assertion is true for sets of cardinality < n. Write V = {V_{1},…, V_{n}}, and set W_{i} = V_{i} ∩ V_{n} for i = 1,…, n − 1. Since we have proved the proposition for n = 2, each W_{i} is a subtree of T. Since we have also proved the proposition for n = 3 we have W_{i} ∩ W_{j} = V_{i} ∩ V_{j} ∩ V_{n} ≠ ∅ for all i, j ∈ {1,…, n − 1}. Hence by the induction hypothesis we have ∅ ≠ W_{1} ∩ ⋯ ∩ W_{n−1} = V_{1} ∩ ⋯ ∩ V_{n}.

2.3.1. Corollary. Let L be a finite, non-empty collection of lines in a tree T. Suppose that for all L, L′ ∈ L we have L ∩ L′ ≠ ∅. Then ⋂ L is a (possibly degenerate and possibly infinite) segment in T. In particular, ⋂ L ≠ ∅.

2.4. In Section 9 we shall need to apply Corollary 2.3.1 under the more restrictive condition that E(L) ∩ E(L′) ≠ ∅ for all L, L′ ∈ L. In this case we will need to know exactly how the segment ⋂ L can degenerate. This question is answered by the following result.

Proposition. Let L be a finite, non-empty collection of lines in a tree T. Suppose that for all L, L′ ∈ L we have E(L)∩E(L′) ≠ ∅. Let v be a vertex of T. Then we have ⋂ L = {v} if and only if there is a triod Y centered at the vertex v such that every segment of length 2 in Y is contained in some line in L. If such a triod Y does exist, it is uniquely determined by L, and it has the property that every line in L intersects Y in a segment of length 2.

Proof: First suppose that there is a triod Y centered at {v} such that every length-2 segment in Y is contained in a line in L. Let e_{0}, e_{1}, e_{2} be the edges of Y. Then there is a line L_{0} ∈ L containing e_{1} ∪ e_{2}. By Corollary 2.3.1, ⋂ L is a segment; clearly v ∈ ⋂ L ⊂ L_{0}. Hence if ⋂ L is non-degenerate it must contain e_{1} or e_{2}. But there is a line in L which contains e_{0} and e_{2}, and hence cannot contain e_{1}; thus e_{1} ⊄ ⋂ L. Similarly e_{2} ⊄ ⋂ L. Hence ⋂ L = {v}.

For the rest of the proof we assume that ⋂ L = {v}. We shall construct a triod with the asserted properties, and prove that it is unique. We begin by choosing an arbitrary line L_{0} ∈ L. Then L_{0} contains exactly two edges of T incident to v, say f_{1} and f_{2}. Now for i = 1, 2 there must exist a line L_{i} ∈ L which does not contain the edge f_{i}; otherwise we would have f_{i} ⊂ ⋂ L, contradicting the hypothesis.

Let us set σ_{0} = L_{1} ∩ L_{2}, σ_{1} = L_{0} ∩ L_{2} and σ_{2} = L_{0} ∩ L_{1}. By Corollary 2.3.1, each σ_{i} is a segment. Each segment σ_{i} contains the vertex v, and is non-degenerate by the hypothesis of the proposition. Hence each σ_{i} contains an edge e_{i} incident to v. The edge e_{1} is contained in L_{0}, is incident to v, and is distinct from f_{2} since f_{2} ⊄ L_{2}. Hence e_{1} = f_{1}. Similarly e_{2} = f_{2}.

For i = 1, 2 we have e_{0} ⊂ L_{i} and e_{i} = f_{i} ⊄ L_{i}. Hence e_{0} is distinct from e_{1} and e_{2}. Thus Y = ē_{0} ∪ ē_{1} ∪ ē_{2} is a triod. Note that ē_{0} ∪ ē_{1} ⊂ L_{2}, ē_{0} ∪ ē_{2} ⊂ L_{1} and ē_{1} ∪ ē_{2} ⊂ L_{0}. Thus every segment of length 2 in Y in contained in L_{i}, and is in fact equal to Y ∩ L_{i}, for some i ∈ {0, 1, 2}.

Now let L be any line in L. We shall show that L meets Y in a length-2 segment. Assume this is false. Then there are at least two edges of Y that are not contained in L. Hence there is a length-2 segment in Y that has no edges in common with L; we may write this length-2 segment in the form Y ∩ L_{i} for some i ∈ {0, 1, 2}. By Corollary 2.3.1, the set σ = L_{i} ∩ L is a segment in T. By the hypothesis of the Proposition, σ is non-degenerate and contains v. But the two edges of L_{i} that are incident to v are both contained in Y, and hence neither of them is an edge of L. This is a contradiction.

It remains to prove the uniqueness assertion. Let Y′ be any triod in T, centered at v, such that every length-2 segment in Y′ is contained in a line in L. Since every line in L meets the closed star of v in a length-2 segment contained in Y, it follows that every length-2 segment in the triod Y′ is contained in the triod Y. This implies that Y = Y′.

2.5. Γ-trees. Let Γ be a group. By a Γ-tree we mean a tree T equipped with an action of Γ by simplicial automorphisms. Recall from [13] that Γ is said to act without inversions on T if for every x ∈ Γ and every edge e of T which is x-invariant, x fixes the vertices of T. In this case we shall say that the Γ-tree T is non-inversive. Recall that if Γ acts without inversions on T, then for every x ∈ Γ exactly one of the following alternatives holds:

(i) Fix(x) ≠ ∅; or (ii) x is T -hyperbolic, i.e. there is a unique x-invariant line A ⊂ T, called the axis of x in

T, and x|A is a non-trivial translation. When x is T -hyperbolic, we shall denote the axis of x in T by A_{T} (x). A cyclic subgroup X of Γ will be termed T -hyperbolic if it has a T -hyperbolic generator.

Note that if X is a T -hyperbolic cyclic subgroup then X is infinite cyclic. Furthermore, it is clear that all non-trivial elements of X are T -hyperbolic and have the same axis in T. If a cyclic subgroup X is T -hyperbolic we shall write A_{T} (X) for the common axis of the non-trivial elements of X.

2.6. Proposition. Let Θ be a group generated by two elements ξ and η. Let T be a non-inversive Θ-tree, and suppose that ξ and η are both T -hyperbolic. Suppose also that E(A_{T} (ξ)) ∩ E(A_{T} (η)) = ∅. Then Θ is free on the generators ξ and η, and the action of Θ on T is free.

Proof: This is a special case of [4, Proposition 4.2].

Section 3. Surfaces in 3-manifolds

3.1. Bi-collared surfaces and dual graphs. Let M be an orientable 3-manifold without boundary. By a bi-collared surface in M we shall mean a map c : S × [−1, 1] → M, where S is a 2-manifold without boundary and c maps S × [−1, 1] homeomorphically onto a closed subset of M. In the case that M has a PL structure, we shall say that c is a PL bi-collared surface if S has a PL structure such that c is a PL map. The set c(S × {0}) is called the core of c. Any locally flat 2-manifold without boundary embedded as a closed subset of M is the core of a bi-collared surface c, and c is determined up to isotopy by the given 2-manifold. Studying bi-collared surfaces in M is therefore essentially equivalent to studying locally flat orientable 2-manifolds without boundary in M. However, for the constructions in this paper, including the construction of the dual tree to an incompressible surface, it is more convenient to take bi-collared surfaces as the basic objects.

A bi-collared surface in a closed, orientable 3-manifold M is said to be incompressible if its core S is incompressible in the sense defined in the Introduction. (In particular S is then connected and is not a 2-sphere.)

If M is an orientable 3-manifold without boundary and c : S × [−1, 1] → M is a bicollared surface in M, we shall write |c| = c(S × [−1, 1]) and Split(c) = M − c(S × (−1, 1)). The 3-manifold Split(c) may be thought of as being obtained by splitting M along the core of c.

We shall denote by Dual(c) the quotient space of M obtained by identifying each component of the closure of M − |c| to a point, and identifying c(S × {t}) to a point for each component S of S and each t ∈ [−1, 1]. Then Dual(c) has the structure of a graph, of which the vertices are in natural one-one correspondence with the components of M − |c|, and the edges are in natural one-one correspondence with the components of |c|. We shall denote by q_{c} the quotient map from M to Dual(c).

3.2. The tree associated to an incompressible surface. Now let c be an incompressible bi-collared surface in a closed, orientable 3-manifold M. Let M̃ denote the universal covering space of M, let p : M̃ → M denote the covering projection, and let Γ ≅ π_{1}(M) denote the group of deck transformations. Let c̃ denote the bi-collared surface in M̃ lying above c. Since M̃ is 1-connected, Dual(c̃) is also 1-connected; that is, it is a tree. We shall write T (c) = Dual(c̃). The action of Γ on M̃ induces a natural simplicial action without inversions on T (c). We shall always regard T (c) as a non-inversive Γ-tree by equipping it with this action.

For any vertex v of T = T (c) we shall denote by M̃_{v} the set q−1 ; c̃({v}). Note that M̃_{v} is a component of Split(c̃), and covers a component of Split(c). This component of Split(c) will be denoted by M_{v}. For each edge s of T we shall denote by M̃_{s} the set q−1 ; c̃(s̄); this set is a component of |c̃|, and covers a component of |c|, which will be denoted by M_{s}.

Let v be any vertex of T. The components of ∂M̃_{v} are the sets of the form M̃_{v} ∩ M̃_{s}, where s ranges over the edges incident to v. We shall denote the set M̃_{v} ∩ M̃_{s} by ∂_{s}M̃_{v}. Each ∂_{s}M̃_{v} covers a component of ∂ Split(c) = ∂|c|, which will be denoted by ∂_{s}M_{v}. Since c is incompressible, the components of ∂M̃_{v} are simply connected. As M̃ is also simply connected, it follows from van Kampen’s theorem that M̃_{v} is simply connected.

The group of deck transformations of the simply connected covering space M̃_{v} of M_{v} is Γ_{v}. Likewise, the group of deck transformations of the simply connected covering space ∂_{s}M̃_{v} of ∂_{s}M_{v} is Γ_{s}. Thus we have a natural outer isomorphism from π_{1}(M_{v}) to Γ_{v}, which will be denoted by ν_{v}, Similarly, there is a natural outer isomorphism ν_{s,v} : π_{1}(∂_{s}M_{v}) → Γ_{s} for every edge s incident to v. We have a commutative diagram of groups and outer homomorphisms

Γ_{s}  ⟶ Γ_{v} ⟶ Γ
↑^{ν_{s,v}}  ↑^{ν_{v}}  ↑^{ν}
π_{1}(∂_{s}M_{v}) ⟶ π_{1}(M_{v}) ⟶ π_{1}(M)

where ν is the natural outer isomorphism between π_{1}(M) and Γ, and where the horizontal arrows in the bottom row represent outer homomorphisms induced by inclusion and those in the top row represent inclusions of groups.

It follows from the commutativity of the right-hand half of the above diagram that the outer subgroup im(π_{1}(M_{v}) → π_{1}(M)) of π_{1}(M) is mapped to the outer subgroup Γ_{v} of Γ by the outer isomorphism ν. In particular an element x of Γ has a fixed point in T if and only if the conjugacy class in π_{1}(M) corresponding to the conjugacy class of x in Γ is represented by a closed curve in Split(c).

3.3. Let Γ be a group. A Γ-tree is said to be minimal if it has no proper Γ-invariant subtree.

Proposition. Let c be an incompressible bi-collared surface in a closed, orientable PL 3-manifold M. Let Γ denote the group of deck transformations of the universal covering space of M. Then the Γ-tree T (c) is minimal.

Proof: Let S denote the core of c. If S does not separate the given closed 3-manifold M then T = T (c) contains only one Γ-orbit of vertices. In this case T is obviously minimal. Now suppose that S separates M. Since S is connected, there is only one Γorbit of edges in T. Hence a proper Γ-invariant subtree would consist of a single vertex v, fixed by the entire group Γ. Choose an edge e incident to v, and let w be the second vertex of e. We have Γ_{w} = Γ_{e}. In view of the commutative diagram in 3.2, this implies that the natural outer homomorphism π_{1}(∂_{e}M_{w}) → π_{1}(M_{w}), induced by inclusion, is an isomorphism. In particular the inclusion homomorphism H_{1}(∂_{e}M_{w}) → H_{1}(M_{w}) is an isomorphism. According to Poincaré-Lefschetz duality, as M_{w} is a compact 3-manifold whose boundary is ∂_{e}M_{w}, the groups H_{1}(∂_{e}M_{w}) and H_{1}(M_{w}) can be isomorphic only if S has genus 0; but this contradicts the incompressibility of c.

3.4. It is a standard observation that if an irreducible 3-manifold M contains an incompressible surface S, there is a covering space of M which deform-retracts to some surface that maps homeomorphically to S under the covering projection. The following result is a somewhat more precise formulation of this observation from the arboreal point of view.

Proposition. Let c : S × [−1, 1] → M be an incompressible PL bi-collared surface in a closed, irreducible, orientable PL 3-manifold M. Let M̃ denote the universal covering space of M, and let Γ denote the group of deck transformations. Let s be any edge of T (c). Then M̃_{s}/Γ_{s} is a deformation retract of M̃/Γ_{s}, and M̃_{s}/Γ_{s} is mapped homeomorphically onto |c| under the induced covering map M̃/Γ_{s} → M. In particular, M̃/Γ_{s} is homotopy equivalent to S.

Proof: The group Γ_{s} is the stabilizer of M̃_{s} ⊂ M̃ in Γ. The manifold M̃ is connected, and since c is incompressible, M̃_{s} is also simply connected. By elementary covering space theory it follows that π_{1}(M̃_{s}/Γ_{s}) → π_{1}(M̃/Γ_{s}) is an isomorphism, and that M̃_{s}/Γ_{s} is mapped homeomorphically onto |c| by the induced covering map. But the incompressibility of c also implies that S has positive genus; hence S is aspherical and π_{1}(S) is infinite. Since c : S×[−1, 1] → M induces a monomorphism of fundamental groups, π_{1}(M) is also infinite. It follows from the sphere theorem [6] that an irreducible, orientable PL 3-manifold with infinite fundamental group is aspherical. The covering space M̃/Γ_{s} of the aspherical manifold M is automatically aspherical. Thus M̃_{s}/Γ_{s} ↪ M̃/Γ_{s} is a map between aspherical triangulable spaces which induces an isomorphism of fundamental groups, and is therefore a homotopy equivalence.

3.4.1. In view of Proposition 3.4, 3-manifolds of the homotopy type of closed surfaces arise naturally as covering spaces of manifolds containing incompressible surfaces. The following result gives a property of such manifolds that will be useful in Section 8.

Proposition. Let M be an orientable PL 3-manifold, let S be a closed, connected, orientable surface, and suppose that η : M → S is a homotopy equivalence. Let F ⊂ M be a closed, connected, orientable PL surface. Then either

(a) F bounds a compact PL 3-manifold in M, or

(b) η|F : F → S has degree one. If (b) holds then the inclusion F ↪ M induces an epimorphism of fundamental groups.

Proof: Suppose that (a) does not hold. Then the image u of the fundamental class of F in H_{2}(M; Z) is primitive, i.e. it is not of the form nu_{0} with u_{0} ∈ H_{2}(M; Z) and n > 1. Now since η is a homotopy equivalence we have H_{2}(M; Z) ≅ Z, and hence u is a generator of H_{2}(M; Z). This shows that (b) holds.

Now assume that (b) holds but that F ↪ M does not induce an epimorphism of fundamental groups. Then η|F : F → S does not induce an epimorphism of fundamental groups. Hence η|F lifts to some non-trivial covering space S̃ of S. If S̃ is compact then the covering projection S̃ → S has a finite degree d > 1; it follows that d divides the degree of η|F : F → S, which contradicts (b). If S̃ is non-compact then H_{2}(S̃) = 0 and hence η|F : F → S has degree 0, and again (b) is contradicted.

3.5. The following interesting property of the tree associated to an incompressible bicollared surface will also be needed in Section 8.

Proposition. Let c : S×[−1, 1] → M be an incompressible bi-collared surface in a closed, irreducible, orientable PL 3-manifold M. Let Γ denote the group of deck transformations of the universal covering space of M. Let s be any edge of T = T (c). Then the graph T/Γ_{s} is 1-connected and has no end points.

Proof: Let M̃ denote the universal covering space of M. Let c̃/Γ_{s} denote the bi-collared surface in M̃/Γ_{s} lying over c. We may identify T/Γ_{s} canonically with Ψ = Dual(c̃/Γ_{s}).

To show that Ψ is 1-connected it suffices to show that every edge separates. Let S denote the core of c and S̃ the core of c̃. The edges of Ψ are in canonical one-one correspondence with the components of S̃/Γ_{s}. To show that every edge e separates Ψ, it suffices to show that every component of S̃/Γ_{s} separates M̃/Γ_{s}.

Let F̃_{0} denote the component of S̃ contained in M̃_{s} and set F_{0} = F̃_{0}/Γ_{s}. By Proposition 3.4, F_{0} is a deformation retract of M̃/Γ_{s}. Now let F denote an arbitrary component of S̃/Γ_{s}. Then either F = F_{0} or F ∩ F_{0} = ∅. In either case, F is isotopic to a surface disjoint from F_{0}. Since F_{0} carries H_{1}(M̃/Γ_{s}; Z_{2}), the mod 2 intersection number of F with any class in H_{1}(M̃/Γ_{s}; Z_{2}) is zero. This implies that F separates M̃/Γ_{s}.

It remains to show that Ψ has no end points. Suppose that Ψ does have an endpoint and let v be a vertex of T which maps to it under the quotient map T → Ψ = T/Γ_{s}. Then M̃_{v}/Γ_{s} ⊂ M̃/Γ_{s} is a component of Split(c̃/Γ_{s}) whose boundary consists of a single component Φ of ∂|c̃/Γ_{s}|. We may write M̃/Γ_{s} = (M̃_{v}/Γ_{s})∪X, where X is a PL 3-manifold such that X ∩ (M̃_{v}/Γ_{s}) = ∂X = Φ. Since c is incompressible, π_{1}(Φ) → π_{1}(M̃/Γ_{s}) is injective. Thus if ∗ is a base point in Φ we may identify π_{1}(M̃/Γ_{s}, ∗) with an amalgamated free product G = A ∗_{C} B, where A = π_{1}(X, ∗), B = π_{1}(M̃_{v}/Γ_{s}, ∗) and C = π_{1}(Φ, ∗). In particular we have A ∩ B = C. But since M̃_{s}/Γ_{s} ⊂ |c̃/Γ_{s}| ⊂ X, and since M̃_{s}/Γ_{s} is a deformation retract of M̃/Γ_{s}, we have A = G. Hence B = C, i.e. the inclusion homomorphism π_{1}(Φ, ∗) → π_{1}(M̃_{v}/Γ_{s}, ∗) is an isomorphism. It follows that every path in M̃_{v}/Γ_{s} with end points in Φ is fixed-endpoint homotopic to a path in Φ.

Now M̃_{v}/Γ_{s} covers M_{v}. Since M̃_{v}/Γ_{s} has connected boundary, so does M_{v}. By the covering homotopy property, every path in M_{v} with endpoints in ∂M_{v} is fixed-endpoint homotopic to a path in ∂M_{v}. In particular the inclusion homomorphism π_{1}(∂M_{v}) → π_{1}(M_{v}) is surjective. It is injective since c is incompressible. Thus π_{1}(∂M_{v}) → π_{1}(M_{v}) is an isomorphism, and hence so is H_{1}(∂M_{v}) → H_{1}(M_{v}). According to Poincaré-Lefschetz duality, as M_{v} is compact, H_{1}(∂M_{v}) and H_{1}(M_{v}) can be isomorphic only if ∂M_{v} has genus 0; but this contradicts the incompressibility of c.

3.6. The tree T (c) associated with an incompressible bi-collared surface c in a closed PL 3-manifold M gives a useful way of formulating certain results about covering spaces of M. This is illustrated by the following proposition, which will be used in Section 10.

By a handlebody we mean a compact 3-manifold which is homeomorphic to the regular neighborhood of a graph in R^{3}.

Proposition. Let c be an incompressible bi-collared surface in a closed, orientable 3manifold M. Let Θ be a finitely generated subgroup of the group Γ of deck transformations of the universal cover M̃ of M, and suppose that the natural action of Γ on T = T (c) restricts to a free action of Θ on T. Then the covering space M̃/Θ of M is homeomorphic to the interior of a handlebody.

Proof: Let us fix a PL structure in which the bi-collared surface c is PL. We can write M as a union M_{1} ∪ M_{2}, where M_{1} = |c| and M_{2} = Split(c).

Theorem 3.1 of [15] deals with a compact PL 3-manifold written in the form M = M_{1} ∪ M_{2}, where the M_{i} are compact PL 3-manifolds. The theorem gives sufficient conditions for a covering space of such a manifold M, given in the form M̃/Θ where Θ is a finitely generated subgroup of the group Γ of the universal covering space M̃, to be homeomorphic to the interior of a compact 3-manifold. We shall show that the conditions of [15, Theorem 3.1] hold under the hypotheses of Proposition 3.6 if M_{1} and M_{2} are defined as above and if Θ acts freely on T. Condition (i) of [15, Theorem 3.1] asserts, in the orientable case, that M_{1} and M_{2} are irreducible. This holds in our situation because M is irreducible and c is incompressible.

Condition (ii), that the components of M_{1} ∩ M_{2} are incompressible, also follows from the incompressibility of c. Condition (iii) is equivalent to the condition that if S̃ is any component of the preimage of M_{1} ∩ M_{2} in M̃/Θ, the intersection of Θ with the stabilizer of S̃ in Γ is finitely generated. In the notation of 3.2 we have S̃ = ∂_{s}M_{v} for some edge s of T and some vertex v of s. The intersection of Θ with the stabilizer of S̃ is Θ_{s}. Since Θ is assumed to act freely on T, the group Θ_{s} is trivial. Finally, condition (iv) is equivalent to the condition that if A is a component of the preimage of M_{1} or M_{2} in M̃/Θ, such that the intersection Θ_{A} of Θ with the stabilizer of A is finitely generated, then A/Θ_{A} is PL homeomorphic to a manifold of the form Q − K, where Q is a compact manifold and K is a closed subset of ∂Q. We have either A = M_{s}, where s is an edge of T, or A = M_{v}, where v is a vertex of T. Hence Θ_{A} is the stabilizer of s or v respectively, and is therefore trivial since Γ acts freely on T. Thus A = A/Θ_{A} is the universal cover of a compact, irreducible manifold with non-empty boundary. By [16, Theorem 8.1], A is PL homeomorphic to Q − K, where Q is a closed 3-ball and K is a closed subset of ∂Q.

Thus M̃/Θ is PL homeomorphic to the interior of some compact PL 3-manifold J. But by [16, Theorem 8.1], M̃ is homeomorphic to R^{3}. It follows that M̃/Θ is irreducible, and hence that J is irreducible. Furthermore, π_{1}(J) is isomorphic to the group Θ, which acts freely on the tree T and is therefore a free group. It follows from [6, Theorem 5.2] that if a compact, orientable, irreducible 3-manifold J has a free fundamental group then J is a handlebody or a 3-sphere. Since M contains an incompressible surface, π_{1}(M) is infinite, and thus J cannot be a 3-sphere.

Section 4. The characteristic submanifold

In 3.2 we considered an incompressible bi-collared surface c in a closed, orientable 3manifold M. For each edge s incident to a vertex v of T = T (c), we defined a natural outer isomorphism ν_{s,v} : π_{1}(∂_{s}M_{v}) → Γ_{s}. If s and e are two distinct edges incident to a vertex v of T, we will need a topological interpretation of the outer subgroup ν_{s,v}^{−1}(Γ_{s} ∩ Γ_{e}) of π_{1}(∂_{s}M_{v}). This requires the use of the theory of the characteristic submanifold [9,8]. The relevant material is reviewed below.

4.1. The following relatively weak and special version of the characteristic submanifold theorem of [9] or [8] contains the information that we need.

Proposition. Let N be a simple, compact, orientable PL 3-manifold with non-empty boundary. Then there exists a (possibly empty) compact PL 3-manifold Σ ⊂ N having the following properties:

(i) Φ = Σ ∩ ∂N is a compact PL 2-manifold; (ii) each component of the frontier ∂Σ − int Φ of Σ is an annulus A such that the natural

outer homomorphism π_{1}(A) → π_{1}(N) is injective; (iii) each component of Σ is either (a) a solid torus Σ_{i} such that Φ_{i} = Σ_{i}∩Φ is a non-empty

disjoint union of annuli, all homotopically non-trivial in Σ_{i}, or (b) an I-bundle Σ_{i} over a non-simply connected, compact, possibly non-orientable PL 2-manifold, such that Φ_{i} = Σ_{i} ∩ Φ is the associated ∂I-bundle;

(iv) if C is any compact, connected, non-simply-connected, orientable 2-manifold, and F : (C ×I, C ×∂I) → (M, ∂M) is any map of pairs such that the outer homomorphism F_{♯} : π_{1}(C × I) → π_{1}(M) is non-trivial, and if F is not homotopic (as a map of pairs) to a map from (C × I, C × ∂I) to (∂M, ∂M), then F is homotopic (as a map of pairs) to a map from (C × I, C × ∂I) to (Σ, Φ).

Proof: This is included, for example, in the Characteristic Pair Theorem stated in the introduction to Chapter V of [8]. In this theorem one is given a pair (M, T), where M is a sufficiently large, compact, irreducible, orientable PL 3-manifold and T ⊂ ∂M is a 2-manifold with boundary such that π_{1}(T_{i}) → π_{1}(M) is injective for every component T_{i} of T. Here we take (M, T) = (N, ∂N). Since the compact, irreducible, orientable PL 3-manifold N has non-empty boundary, it is automatically sufficiently large. The injectivity condition is included in our definition 1.6 of a simple manifold. The conclusion of the Characteristic Pair Theorem of [8] gives a pair (Σ, Φ) such that every component of Σ is either an I-bundle or a Seifert fibered space. For every component Σ_{i} of Σ the natural outer homomorphism π_{1}(Σ_{i}) → π_{1}(M) is injective. In our case, since M = N is simple, π_{1}(Σ_{i}) has no free abelian subgroup of rank 2. But the only Seifert fiber space whose fundamental group has no free abelian subgroup of rank 2 is a solid torus. Now that it has been established that the components of (Σ, Φ) are I-bundles and solid tori, the information about how they intersect ∂N is included in the statement of the Characteristic Pair Theorem of [8]. Conclusion (iv) is also included in the latter statement.

If Σ is any submanifold of N having the properties stated in the Proposition above, we shall refer to the pair (Σ, Φ) as a characteristic pair of the simple manifold pair (N, ∂N). (This is a weaker definition than the one given in [8] or [9]: we do not require Φ to be “perfectly embedded” in the sense of [8]. In particular, with the present definition, (Σ, Φ) is unique up to ambient isotopy. However, this definition is well-adapted to the applications in the present paper.)

4.2. Proposition. Let c be an incompressible bi-collared surface in a closed, orientable, irreducible PL 3-manifold M. Let v be any vertex of T = T (c). Let (Σ_{v}, Φ_{v}) be a characteristic pair of (M_{v}, ∂M_{v}). Let s and e be any two distinct edges incident to v. Then Γ_{s} ∩ Γ_{e} is finitely generated. Furthermore, if Γ_{s} ∩ Γ_{e} ≠ {1} then there is a component Φ_{e} ⊂ ∂_{s}M_{v} of Φ_{v} such that the outer subgroup ν_{s,v}^{−1}(Γ_{s} ∩ Γ_{e}) of π_{1}(∂_{s}M_{v}) is contained in the outer subgroup im(π_{1}(Φ_{e}) → π_{1}(∂_{s}M_{v})).

Proof: According to the commutative diagram in 3.2, the outer isomorphism ν_{v}^{−1} : Γ_{v} → π_{1}(M_{v}) carries the outer subgroup Γ_{s} onto the outer subgroup im(π_{1}(∂_{s}M_{v}) → π_{1}(M_{v})). As ∂_{s}M_{v} is a boundary component of the compact PL 3-manifold M_{v} and the natural outer homomorphism π_{1}(∂_{s}M_{v}) → π_{1}(M_{v}) is injective, it follows from [7, Proposition 1.4] that a subgroup realizing the outer subgroup im(π_{1}(∂_{s}M_{v}) → π_{1}(M_{v})) has finitely generated intersection with every finitely generated subgroup of π_{1}(M_{v}). Thus Γ_{s} has finitely generated intersection with every finitely generated subgroup of Γ_{v}. But Γ_{e} ≤ Γ_{v} is isomorphic to π_{1}(∂_{e}M_{v}) and is therefore finitely generated. Thus Γ_{s} ∩ Γ_{e} is finitely generated.

Now suppose that Γ_{s} ∩ Γ_{e} is non-trivial. Consider the PL 3-manifold J = M̃_{v}/(Γ_{s} ∩ Γ_{e}), which is a covering space of M_{v}. The universal covering space of J is M̃_{v}, and the group of covering transformations obviously leaves both ∂_{s}M̃_{v} and ∂_{e}M̃_{v} invariant. It follows that ∂_{s}M̃_{v} and ∂_{e}M̃_{v} project to distinct components Q_{s} and Q_{e} of ∂J, and that the inclusioninduced outer homomorphisms π_{1}(Q_{s}) → π_{1}(J) and π_{1}(Q_{e}) → π_{1}(J) are both surjective. It follows from the incompressibility of c that these outer homomorphisms are in fact both outer isomorphisms.

Since π_{1}(Q_{s}) ≅ π_{1}(J) ≅ Γ_{s} ∩ Γ_{e} is finitely generated, there exists a compact connected PL 2-manifold C ⊂ Q_{s} such that π_{1}(C) → π_{1}(J) is surjective. We may take C to have nonempty boundary, so that C is homotopy-equivalent to a 1-complex. Since π_{1}(Q_{e}) → π_{1}(J) is surjective, the inclusion C ↪ Q_{s} is homotopic in J to a map of C into Q_{e}. Let F̃ : C × [0, 1] → J be a homotopy such that F̃_{0} is the inclusion and F̃_{1}(C) ⊂ Q_{e}. Then F̃ projects via the covering map p : J → M_{v} to a homotopy F : C × [0, 1] → M_{v}.

Since the lift F̃ of F maps C × {0} and C × {1} to distinct components Q_{s} and Q_{e} of ∂J, the map of pairs F : (C × [0, 1], C × {0, 1}) → (M_{v}, ∂M_{v}) is not homotopic to a map into (∂M_{v}, ∂M_{v}). Furthermore, since we have assumed that Γ_{s} ∩ Γ_{e} is non-trivial, the outer homomorphism F_{♯} : π_{1}(C × [0, 1]) → π_{1}(M) is non-trivial. By Property (iv) of the characteristic pair from 4.1, it follows that F = p ∘ F̃ is homotopic as a map of pairs to a map into (Σ_{v}, Φ_{v}). In particular, p|C is homotopic in ∂_{s}M_{v} to a map of C into some component Φ_{e} ⊂ ∂_{s}M_{v} of Φ_{v}. Hence the outer subgroup ν_{s,v}^{−1}(Γ_{s} ∩ Γ_{e}) = (p|C)_{♯}(π_{1}(C)) of π_{1}(∂_{s}M_{v}) is contained in the outer subgroup im(π_{1}(Φ_{e}) → π_{1}(∂_{s}M_{v})), as asserted.

4.3. The following characterization of books of I-bundles in terms of the characteristic submanifold theory will be used in Section 8.

Proposition. Let N be a compact, simple, orientable, PL 3-manifold with non-empty boundary. Let (Σ, Φ) be a characteristic pair of (N, ∂N), and suppose that every component of ∂N − int Φ is an annulus. Then N is a book of I-bundles.

Proof: Let W denote the union of all components of Σ which are solid tori, and let E denote the union of all other components of Σ. Then according to 4.1, each component of E is an I- bundle over a compact PL 2-manifold-with-boundary, and intersects ∂N in the associated ∂I-bundle. Hence we may regard E as an I-bundle over a possibly disconnected PL 2-manifold-with-boundary B, and the frontier of B is the preimage of ∂B under the bundle projection E → B.

Now let V denote the closure of N − E. The boundary of V is the union of the three sets X = W ∩ ∂N, Y = ∂N − int Φ, and the frontier Z of E in N. By 4.1, the components of X are annuli. The hypothesis of the proposition says that the components of Y are annuli. The components of Z are in particular components of the frontier of Σ, and are therefore annuli by 4.1. Thus ∂V is a union of annuli in a natural way, and it is clear that any component of intersection of two of these annuli is a common boundary component. Thus each component of ∂V has Euler characteristic 0. Since N is orientable it follows that ∂V is a disjoint union of tori.

Let V_{i} be any component of V. It is clear from 4.1 that ∂V_{i} ≠ ∅. Choose any component S of ∂V_{i}. Then S is a torus. On the other hand, by 4.1, each component of the frontier of V_{i} is an annulus A such that the natural outer homomorphism π_{1}(A) → π_{1}(N) is injective. This implies that the natural outer homomorphism π_{1}(V_{i}) → π_{1}(N) is injective. Since N is simple it follows that π_{1}(V_{i}) has no free abelian subgroup of rank 2. Hence the natural outer homomorphism π_{1}(S) → π_{1}(V_{i}) has a non-trivial kernel. It then follows from the loop theorem (see [6]) that there is a PL disk D ⊂ V_{i} such that ∂D = D ∩ ∂V_{i} is a homotopically non-trivial simple closed curve in S. If R is a regular neighborhood of D in V_{i}, the closure of V_{i} − R is a PL 3-manifold whose boundary contains a 2-sphere; using the irreducibility of N we conclude that \overline{V_{i} − R} is a ball, and hence that V_{i} is a solid torus. In view of the definition it is now clear that N is a book of I-bundles.

Section 5. Platings

5.1. Let Ω be a topological space. Recall that a family of closed subsets of Ω is said to be locally finite if every point of Ω has a neighborhood which meets only finitely many sets in the family. By a plating of Ω we shall mean a locally finite family C = (C_{i})_{i∈I} of non-empty closed subsets of Ω, indexed by some set I, such that ⋃_{i∈I} C_{i} = Ω. (We have avoided the word “covering” in this context because in the applications Ω will often arise as a covering space.)

A plating C = (C_{i})_{i∈I} of a space Ω will be said to have finite order if there is an integer m such that for any point P ∈ Ω there are at most m + 1 distinct indices i ∈ I for which P ∈ C_{i}. The smallest such integer m will then be called the order of C.

5.2. Γ-platings. Now suppose that Γ is a group and that Ω is a Γ-space. By a Γ-plating of Ω we shall mean a triple (I, G, C), where I is a Γ-set, G = (G_{i})_{i∈I} is a family of subgroups of Γ and C = (C_{i})_{i∈I} is a plating of Ω, and the following conditions are satisfied:

(i) G_{i} is contained in the stabilizer Γ_{i} of i for every i ∈ I; (ii) γG_{i}γ^{−1} = G_{γ·i} for every i ∈ I and every γ ∈ Γ; and (iii) γ · C_{i} = C_{γ·i} for every i ∈ I and every γ ∈ Γ.

Recall that for any i in the Γ-set I, Γ_{i} ≤ Γ denotes the stabilizer of i ∈ I. It is clear from the definition of a Γ-plating that C_{i} is invariant under G_{i}.

5.3. Proposition. Let Γ be a group and let Ω be a uniform Γ-space. Suppose that (I, G, C) is a Γ-plating of Ω. Then C has finite order.

Proof: Let R ⊂ Ω be a compact set such that Γ · R = Ω. Write C = (C_{i})_{i∈I}. Let I_{0} denote the set of all indices i ∈ I such that C_{i} ∩ R ≠ ∅. Since the plating C is by definition locally finite, the cardinality k of I_{0} is finite. In particular, for every P ∈ R there are at most k indices i such that P ∈ C_{i}. It now follows from the equivariance condition (iii) in the definition of a Γ-plating that for any P ∈ Ω there are at most k indices i such that P ∈ C_{i}.

The order of a Γ-plating (I, G, C) of a uniform Γ-space Ω is defined to be the order of C.

5.4. Proposition. Let Γ be a group and let (I, G, C) be a Γ-plating of a uniform Γ-space Ω, where C = (C_{i})_{i∈I}. Then the Γ-set I contains only finitely many Γ-orbits.

More generally, for any positive integer d, let J_{d} denote the subset of the Cartesian power I^{d} consisting of all d-tuples (i_{1},…, i_{d}) such that C_{i_{1}} ∩ ⋯ ∩ C_{i_{d}} ≠ ∅. Then J_{d} is a union of finitely many Γ-orbits under the diagonal action of Γ on I^{d}.

Proof: Let R ⊂ Ω be a compact set such that Γ · R = Ω. Let Φ_{d} denote the subset of J_{d} consisting of all d-tuples (i_{1},…, i_{d}) such that C_{i_{1}} ∩ ⋯ ∩ C_{i_{d}} ∩ R ≠ ∅. Since C is locally finite, Φ_{d} is finite. But for any (i_{1},…, i_{d}) ∈ J_{d} there is a point P ∈ C_{i_{1}} ∩ ⋯ ∩ C_{i_{d}}, and there is an element γ of Γ such that γ · P ∈ R. It follows that (γ · i_{1},…, γ · i_{d}) ∈ Φ_{d}. This proves the final assertion of the proposition. Since the C_{i} are non-empty according to the definition of a plating, we have J_{1} = I, and so the first assertion is a special case of the final assertion.

5.5. Uniform Γ-platings. We shall say that the Γ-plating (I, G, C) of Ω is uniform if for each i ∈ I the G_{i}-set C_{i} is uniform.

5.6. Induced platings of invariant subsets. Suppose that (I, G, C) is a Γ-plating of a uniform Γ-space Ω, where C = (C_{i})_{i∈I} and G = (G_{i})_{i∈I}. Suppose that Ω′ is a closed Γ-invariant subspace of Ω. Let I′ denote the set of all indices i ∈ I such that C_{i} ∩ Ω′ ≠ ∅, and set C′ = (C_{i} ∩ Ω′)_{i∈I_{Ω′}} and G′ = (G_{i})_{i∈Ω′}. Then it is clear that (I′, G′, C′) is a Γplating of Ω′ and that its order is at most the order of (I, G, C). If (I, G, C) is uniform then it follows from 1.4 that (I′, G′, C′) is uniform.

5.7. Boundary platings. Let Γ be a group, let Ω be a uniform Γ-space, and let (I, G, C) be a Γ-plating of Ω, where C = (C_{i})_{i∈I} and G = (G_{i})_{i∈I}. Let s be an index in I, and let Q_{s} denote the frontier of C_{s}. Since C_{s} is G_{s}-invariant by 5.2, Q_{s} is also G_{s}-invariant. It follows that the set I^{s} ⊂ I, consisting of all indices i ≠ s such that C_{i} ∩ Q_{s} ≠ ∅, is also G_{s}-invariant, and therefore has the structure of a G_{s}-set. Let us set G^{s} = (G_{i} ∩ G_{s})_{i∈I^{s}} and C^{s} = (C_{i} ∩ Q_{s})_{i∈I^{s}}.

Proposition. With the above notation, (I^{s}, G^{s}, C^{s}) is a G_{s}-plating of Q_{s}, and its order is strictly less than the order of (I, G, C). If (I, G, C) is uniform then Q_{s} is a uniform G_{s}-space.

Proof: Since the plating C is by definition a locally finite family of closed subsets of Ω, it is clear that C^{s} is a locally finite family of closed subsets of Q_{s}. To show that C^{s} is a plating of Q_{s} we must show that Q_{s} ⊂ ⋃_{i∈I_{s}} C_{i}. For this purpose, note that the set D_{s} = ⋃_{s≠i∈I} C_{s} is the union of a locally finite family of closed subsets of Ω and is therefore closed. Since Ω = C_{s} ∪ D_{s}, it follows that the frontier Q_{s} of C_{s} is contained in D_{s}. This immediately implies that Q_{s} ⊂ ⋃_{i∈I_{s}} C_{i}, so that C^{s} is a plating of Q_{s}. Since the triple (I, G, C) satisfies conditions (i)–(iii) of Definition 5.2, the triple (I^{s}, G^{s}, C^{s}) satisfies the same conditions with G_{s} in place of Γ. Thus (I^{s}, G^{s}, C^{s}) is a G_{s}-plating.

Now let m < ∞ denote the order of C, and let P be any point of Q_{s}. There are at most m + 1 indices i ∈ I such that P ∈ C_{i}. But s is one such index, since P ∈ Q_{s} ⊂ C_{s}. Hence there are at most m indices i ≠ s such that P ∈ Q_{i}. It follows at once that C^{s} has order at most m.

If (I, G, C) is uniform then by definition C_{s} is a uniform G_{s}-space. Since Q_{s} is a closed, G_{s}-invariant subset of C_{s}, it follows from 1.4 that Q_{s} is also a uniform G_{s}-space.

The G_{s}-plating (I^{s}, G^{s}, C^{s}) will be called the boundary plating of Q_{s} induced by the Γ-plating (I, G, C) of Ω.

5.7.1. Doubly uniform platings. It will be convenient to formulate a sufficient condition for a boundary plating to be uniform.

Let (I, G, C) be a Γ-plating of a Γ-space Ω, where G = (G_{i})_{i∈I} and C = (C_{i})_{i∈I}. Note that for any two indices i, i′ ∈ I the set C_{i} ∩ C_{i′} is G_{i} ∩ G_{i′}-invariant. We shall say that the Γ-plating (I, G, C) is doubly uniform if for all i, i′ ∈ I the set C_{i} ∩ C_{i′} is a uniform G_{i} ∩ G_{i′}-space.

Proposition. Let Γ be a group, and let (I, G, C) be a doubly uniform Γ-plating of a Γ-space Ω, where C = (C_{i})_{i∈I} and G = (G_{i})_{i∈I}. Then (I, G, C) is uniform. Furthermore, for any index s ∈ I, the boundary-plating of the frontier of C_{s} induced by (I, G, C) is a uniform G_{s}-plating.

Proof: By applying the above definition when i = i′, we see that C_{i} is a uniform G_{i}-space for every i ∈ I. This proves that (I, G, C) is uniform.

Now let s be any index in I, and let Q_{s} denote the frontier of C_{s} and let I^{s} be defined as in 5.7. Since (I, G, C) is doubly uniform, C_{i} ∩ C_{s} is a uniform (G_{i} ∩ G_{s})-space for every i ∈ I^{s}. Hence by 1.4, the G_{i} ∩ G_{s}-invariant closed subset C_{i} ∩ Q_{s} of C_{i} ∩ C_{s} is also a uniform (G_{i} ∩ G_{s})-space.

Section 6. Neighborhoods of platings

6.1. Let C = (C_{i})_{i∈I} be a plating of a topological space Ω. A neighborhood of C is a family C′ = (C′_{i})_{i∈I} of subsets of Ω, indexed by the same set I, such that C_{i} ⊂ int C′_{i} for every i ∈ I. A neighborhood C′ = (C′_{i})_{i∈I} of C will be termed closed if C′_{i} is a closed subset of Ω for every i ∈ I. Note that if a neighborhood C′ of a plating C is closed and is a locally finite family, then it is itself a plating of Ω.

6.2. Now let Γ be a group and let (I, G, C) be a Γ-plating of a Γ-space Ω. A neighborhood of (I, G, C) is a triple of the form (I, G, C′), where

(i) C′ = (C′_{i})_{i∈I} is a neighborhood of C, and (ii) γ · C_{i} = C_{γ·i} for every i ∈ I.

Note that by definition a neighborhood of a Γ-plating involves the same underlying Γ-set and the same indexed family of subgroups G as the given Γ-plating.

If (I, G, C′) is a neighborhood of the Γ-plating (I, G, C) such that C′ is a locally finite family, we shall say that (I, G, C′) is locally finite. Similarly, if C′ is closed we shall say that (I, G, C′) is closed. Note that any locally finite closed neighborhood of a Γ-plating of Ω is itelf a Γ-plating of Ω.

Any Γ-plating (I, G, C) of a Γ-space Ω has the neighborhood (I, G, U) given by U = (Ω)_{i∈I}. This neighborhood is never locally finite unless I is finite.

6.3. Proposition. Let Γ be a group, let M̃ be a uniform PL Γ-manifold of some dimension n > 0, and let (I, G, C) be a Γ-plating of M̃, where C = (C_{i})_{i∈I}. Let (I, G, C^{+}) be any neighborhood of (I, G, C). Then there is a closed, locally finite neighborhood (I, G, C′) of (I, G, C), where C′ = (C′_{i})_{i∈I}, such that

(i) (I, G, C^{+}) is a neighborhood of (I, G, C′); (ii) C′_{i} is a PL n-manifold for every i ∈ I; (iii) there is a Γ-invariant triangulation of M̃ in which all the C_{i} are subcomplexes; and (iv) if i_{1},…, i_{r} are indices in I such that C_{i_{1}} ∩ ⋯ ∩ C_{i_{r}} = ∅, then C′_{i_{1}} ∩ ⋯ ∩ C′_{i_{r}} = ∅. Furthermore, if (I, G, C) is uniform (or doubly uniform), we can take (I, G, C′) to be uniform (or, respectively, doubly uniform).

6.3.1. Remark. It follows from condition (iv) that if (I, G, C) has order m < ∞ then (I, G, C′) also has order m. This will be important when we apply Proposition 6.3 and its corollaries 6.4 and 6.5.

6.3.2. The next four lemmas will be used in the proof of Proposition 6.3. In these lemmas, Γ will denote a group, M̃ will denote a PL Γ-manifold of some dimension n > 0, and T will denote a Γ-invariant triangulation of M̃ which defines its given PL structure. For each integer d ≥ 0 we shall denote by T^{(d)} the d-th barycentric subdivision of T . For each closed set C ⊂ M̃ and each d ≥ 0 we shall let N_{d}(C) denote the union of all closed simplices in T^{(d)} that meet C_{i}, and we shall let N_{d}^{∗}(C) denote the union of all closed simplices of T^{(d+2)} that meet N_{d}(C). Then N_{d}^{∗}(C) is a regular neighborhood of N_{d}(C); in particular, it is a PL n-manifold and a neighborhood of C. It is clear that N_{d+1}^{∗}(C) ⊂ N_{d}(C) for every d ≥ 0, and that ⋂_{d≥0} N_{d}(C) = C. Hence ⋂_{d≥0} N_{d}^{∗}(C) = C. It is also clear that N_{d}(γ · C) = γ · N_{d}(C) and N_{d}^{∗}(γ · C) = γ · N_{d}^{∗}(C) for every γ ∈ Γ and every d ≥ 0. In particular, if the closed set C ⊂ M̃ is invariant under a subgroup G of Γ then N_{d}(C) and N_{d}^{∗}(C) are G-invariant.

6.3.3. Lemma. If C is a closed subset of M̃ invariant under a a subgroup G of Γ, and if C is a uniform Γ-space, then N_{d}(C) and N_{d}^{∗}(C) are uniform for every d ≥ 0.

Proof: Let R ⊂ C be a compact set such that G · R = C. Then N_{d}(R) and N_{d}^{∗}(R) are compact, and we have G · N_{d}(R) = N_{d}(C) and G · N_{d}^{∗}(R) = N_{d}^{∗}(C).

6.3.4. Lemma. Suppose that (C_{i})_{i∈I} is a locally finite indexed family of closed subsets of M̃. Then (N_{0}(C_{i}))_{i∈I} is also a locally finite family.

Proof: It is enough to show that for every open simplex σ of T there are only finitely many indices i ∈ I such that C_{i} ∩ σ ≠ ∅. But if C_{i} ∩ σ ≠ ∅ then C_{i} meets some closed simplex having σ as a face; that is, C_{i} ∩ K ≠ ∅, where K is the closure of the star of σ in T . Since K is compact, the assertion follows.

6.3.5. Lemma. Let (I, G, C) be a Γ-plating of M̃, and let t ≥ 0 be an integer. Then there is an integer d ≥ 0 such that for any set Φ ⊂ I we have ⋂_{i∈Φ} N_{d}(C_{i}) ⊂ N_{t}(⋂_{i∈Φ} C_{i}).

Proof: Suppose that t is an integer for which there exists no d with the stated property. Then for every d ≥ 0 we can find a set Φ_{d} ⊂ I and a point P_{d} ∈ ⋂_{i∈Φ_{d}} N_{d}(C_{i}) which does not lie in N_{t}(⋂_{i∈Φ_{d}} C_{i}). Now since M̃ is Γ-uniform, there is a compact set R ⊂ M̃ such that Γ · R = M̃. For each d ≥ 0 there is an element γ_{d} of Γ such that γ_{d} · P_{d} ∈ R. We have

γ_{d} · P_{d} ∈ ⋂_{i∈Φ_{d}} γ_{d} · N_{d}(C_{i}) = ⋂_{i∈Φ_{d}} N_{d}(C_{γ_{d}·i}) = ⋂_{i∈γ_{d}^{−1}·Φ_{d}} N_{d}(C_{i}).

Similarly

γ_{d} · P_{d} ∉ N_{t}( ⋂_{i∈γ_{d}^{−1}·Φ_{d}} C_{i}).

Thus after replacing the P_{d} by the γ_{d} · P_{d} and the Φ_{d} by the γ_{d}^{−1}(Φ_{d}) we may assume that all the P_{d} lie in R.

In particular, R meets N_{d}(C_{i}) ⊂ N_{0}(C_{i}) for every i ∈ Φ_{d}. Since the family (N_{0}(C_{i}))_{i∈I} is locally finite by Lemma 6.3.4, it follows that ⋃_{d≥0} Φ_{d} is a finite subset of I. Hence the sequence Φ_{0}, Φ_{1},… contains only finitely many distinct sets; in particular there is a sequence d_{1} < d_{2} < … such that all the Φ_{d_{i}} are equal to the same set Φ. The sequence (P_{d_{k}})_{k≥0} lies in the compact set R. Hence after replacing the sequence (d_{k})_{k≥0} by a subsequence we may assume that (P_{d_{k}})_{k≥0} converges to some point P ∈ M̃.

We have P_{d_{k}} ∈ N_{d_{k}}(C_{i}) for every i ∈ Φ and every k ≥ 1. Since d_{k} tends to infinity with k it follows that P ∈ C_{i} for every i ∈ Φ, i.e. P ∈ ⋂_{i∈Φ} C_{i}. But we have P_{d_{k}} ∉ N_{t}(⋂_{i∈Φ} C_{i}). This is a contradiction.

6.3.6. Lemma. Let (I, G, C) be a Γ-plating of M̃, where C = (C_{i})_{i∈I}. Let (I, G, C^{+}) be a neighborhood of (I, G, C), and write C^{+} = (C_{i}^{+})_{i∈I}. Then there is an integer d ≥ 0 such that N_{d}(C_{i}) ⊂ int C^{+} for every i ∈ I.

Proof: We proceed in the same spirit as in the proof of Lemma 6.3.5. Assume that there is no integer d with the stated property. Then for every d ≥ 0 there exist an index i_{d} ∈ I and a point P_{d} which lies in N_{d}(C_{i_{d}}) but not in C_{i_{d}}^{+}. Let R be a compact subset of M̃ such that Γ · R = M̃, and choose γ_{d} ∈ Γ such that γ_{d} · P_{d} ∈ R. Then γ_{d} · P_{d} lies in N_{d}(C_{γ_{d}·i_{d}}) but not in C_{γ_{d}·i_{d}}^{+}. Thus after replacing the P_{d} by γ_{d} · P_{d} and the i_{d} by γ_{d} · i_{d} we may assume that all the P_{i} lie in R. Now for each d we have P_{d} ∈ N_{d}(C_{i_{d}}) ⊂ N_{0}(C_{i_{d}}), and by Lemma 6.3.4 the family (N_{0}(C_{i}))_{i∈I} is locally finite. Since R is compact it follows that the sequence (i_{d})_{d≥0} contains only finitely many distinct terms. Hence there is a sequence d_{1} < d_{2} < … such that all the i_{d_{k}} are equal to a single index s. After replacing (d_{k})_{k≥0} by a subsequence we may assume that (P_{d_{k}}) converges to a point P. Since P_{d_{k}} ∈ N_{d_{k}}(C_{s}) and d_{k} tends to infinity with k, we have P ∈ C_{s}. Since the P_{d_{k}} lie outside the neighborhood C_{s}^{+} of C_{s}, we have a contradiction.

6.3.7. Proof of Proposition 6.3: We fix a Γ-invariant triangulation T of M̃ and use the notation of 6.3.2. For each integer d ≥ 0 we have a family C^{d} = (N_{d}^{∗}(C_{i}))_{i∈I} of closed subsets of M̃. For every i and every d the set N_{d}^{∗}(C_{i}) is a neighborhood of C_{i}. By the remarks in 6.3.2 we have γ · N_{d}^{∗}(C_{i}) = N_{d}^{∗}(γ · C_{i}) = N_{d}^{∗}(C_{γ·i}) for all i, d and γ. Thus C^{d} is a neighborhood of C for every d.

We obviously have N_{d}^{∗}(C_{i}) ⊂ N_{0}(N_{0}(C_{i})) for every i. By two successive applications of Lemma 6.3.4 the family (N_{0}(N_{0}(C_{i})))_{i∈I} is locally finite. Hence C^{d} is locally finite for every d. Furthermore, it is clear that for every d conditions (ii) and (iii) hold for C′ = C^{d}. We shall show that for sufficiently large d the other conclusions of the proposition hold as well.

Lemma 6.3.6 shows that if d is sufficiently large then for every index i ∈ I we have N_{d}(C_{i}) ⊂ int C_{i}^{+}. Since N_{d+2}^{∗}(C_{i}) ⊂ N_{d}(C_{i}) for all d and i, it follows that if d is sufficiently large then for every index i ∈ I we have N_{d}^{∗}(C_{i}) ⊂ int C_{i}^{+}. This shows that (i) holds for large d.

Lemma 6.3.5 implies that if d is sufficiently large, then for any indices i_{1},…, i_{r} in I we have N_{d}(C_{i_{1}}) ∩ ⋯ ∩ N_{d}(C_{i_{r}}) ⊂ N_{1}(C_{i_{1}} ∩ ⋯ ∩ C_{i_{r}}). In particular if C_{i_{1}} ∩ ⋯ ∩ C_{i_{r}} = ∅ then N_{d}(C_{i_{1}}) ∩ ⋯ ∩ N_{d}(C_{i_{r}}) = ∅. Again using that N_{d+2}^{∗}(C_{i}) ⊂ N_{d}(C_{i}) for all d and i, we conclude that conclusion (iv) holds for large d,

Now suppose that (I, G, C) is uniform. Then for each i ∈ I the G_{i}-space C_{i} is uniform. By Lemma 6.3.3, N_{d}^{∗}(C_{i}) is uniform for every d. This shows that C^{d} is uniform for every d.

Finally, suppose that (I, G, C) is doubly uniform. By Lemma 6.3.5, if d is sufficiently large then for all indices i, i′ ∈ I we have N_{d}(C_{i}) ∩ N_{d}(C_{i′}) ⊂ N_{0}(C_{i} ∩ C_{i′}). Since N_{d+2}^{∗}(C_{i}) ⊂ N_{d}(C_{i}) for all d and i, it follows that if d is sufficiently large then for all i, i′ ∈ I we have N_{d}^{∗}(C_{i}) ∩ N_{d}^{∗}(C_{i′}) ⊂ N_{0}(C_{i} ∩ C_{i′}). Since (I, G, C) is doubly uniform the G_{i} ∩ G_{i′}-space C_{i} ∩C_{i′} is uniform. Hence by Proposition 6.3.3, the G_{i} ∩G_{i′}-space N_{0}(C_{i} ∩C_{i′}) is uniform; by 1.4, its closed invariant subspace N_{d}^{∗}(C_{i}) ∩ N_{d}^{∗}(C_{i′}) is also uniform. This proves that C^{d} is doubly uniform for large d.

6.4. We can apply Proposition 6.3 to any Γ-plating (I, G, C) of M̃ by taking (I, G, C^{+}) to be the neighborhood (I, G, U) of (I, G, C), where U = (M̃)_{i∈I} (cf. 6.2). We record the special case in which (I, G, C) is uniform, which will be needed in Section 9.

Corollary. Let Γ be a group, let M̃ be a PL Γ-manifold of some dimension n > 0, and let (I, G, C) be a uniform Γ-plating of M̃, where C = (C_{i})_{i∈I}. Then there is a closed, locally finite neighborhood (I, G, C′) of (I, G, C), where C′ = (C′_{i})_{i∈I}, such that

(i) C′_{i} is a PL n-manifold for every i ∈ I; (ii) there is a Γ-invariant triangulation of M̃ in which all the C_{i} are subcomplexes; (iii) if i_{1},…, i_{r} are indices in I such that C_{i_{1}} ∩ ⋯ ∩ C_{i_{r}} = ∅, then C′_{i_{1}} ∩ ⋯ ∩ C′_{i_{r}} = ∅; and (iv) (I, G, C′) is uniform.

6.5. Another special case of Proposition 6.3 is the case in which Γ is the trivial group.

Corollary. Let M be a compact PL n-manifold, and let C = (C_{i})_{i∈I} be a finite plating of M. Let C^{+} be a neighborhood of C. Then there is a closed neighborhood C′ = (C_{i})_{i∈I} of C such that

(i) C^{+} is a neighborhood of C′; (ii) C′_{i} is a PL n-manifold for every i ∈ I; and (iii) if i_{1},…, i_{r} are indices in I such that C_{i_{1}} ∩ ⋯ ∩ C_{i_{r}} = ∅, then C′_{i_{1}} ∩ ⋯ ∩ C′_{i_{r}} = ∅.

Section 7. Precise platings, LERF groups and 2-manifolds

7.1. Precise Γ-platings. Let (I, G, C) be a Γ-plating of a Γ-space Ω, where G = (G_{i})_{i∈I} and C = (C_{i})_{i∈I}. We will say that (I, G, C) is precise if for every i ∈ I and every γ ∈ Γ−G_{i} the set C_{γ·i} = γ · C_{i} is disjoint from C_{i}.

7.1.1. Observe that if (I, G, C) is precise then for every i ∈ I and every component K of C_{i}, the stabilizer of K in Γ is a subgroup of G_{i}.

7.2. Proposition. Let Γ be a group and let M̃ be a free, uniform PL Γ-manifold of dimension n. Set M = M̃/Γ, and let p : M̃ → M denote the quotient map. Suppose that M̃ has a precise, uniform Γ-plating (I, G, C) where C is a family of PL subsets and G = (G_{i})_{i∈I}. Then M has a finite plating (B_{1},…, B_{r}), having the same order as (I, G, C), such that for each j ∈ {1,…, r} the set B_{j} is a compact, connected PL subset of M and each component of p^{−1}(B_{j}) is a component of C_{i} for some i ∈ I.

Proof: Let m denote the order of (I, G, C). First note that it suffices to construct a finite plating (A_{j})_{j∈J} of M, with order m, such that for each j ∈ J the set A_{j} is a compact (but possibly disconnected) PL subset of M, and each component of p^{−1}(A_{j}) is a component of C_{i} for some i ∈ I. Indeed, if (A_{j})_{j∈J} is such a plating, the connected components of the A_{j} form a plating with the required properties.

Since M̃ is a uniform Γ-space, it follows from Proposition 5.4 that I contains only finitely many Γ-orbits. Let J ⊂ I be a complete set of orbit representatives, in the sense that each Γ-orbit in I contains a unique element of J. Set A_{j} = p(C_{j}) for each j ∈ J. We have

p^{−1}(A_{j}) = ⋃_{x∈Γ}(x · C_{j}) = ⋃_{i∈Γ·j} C_{i}.

Since the plating C is by definition a locally finite family, p^{−1}(A_{j}) is the union of a locally finite family of closed PL subsets of M̃, and is therefore itself a closed PL subset of M̃.

Hence A_{j} is a closed PL subset of M. Note also that since I = ⋃_{j∈J} Γ · j, we have

⋃_{j∈J} p^{−1}(A_{j}) = ⋃_{i∈I} C_{i} = M̃,

and hence ⋃_{j∈J} A_{j} = M. Thus (A_{j})_{j∈J} is a plating of M.

Since (I, G, C) is precise we have C_{i} ∩ C_{i′} = ∅ for any two indices i ≠ i′ in the orbit Γ · j. Thus p^{−1}(A_{j}) is the union of a disjoint locally finite family of sets C_{i}, where i ranges over Γ · j. Hence every component of p^{−1}(A_{j}) is a component of some C_{i}.

Now let τ be any point of M. Then there is a point τ̃ ∈ M̃ such that p(τ̃) = τ. For any j ∈ J such that τ ∈ A_{j}, we have τ̃ ∈ p^{−1}(A_{j}), and hence τ̃ ∈ C_{i} for some i ∈ Γ · j. But since C has order m, there are at most m + 1 indices i ∈ I such that τ̃ ∈ C_{i}. Since the indices in J are in distinct orbits, it follows that there are at most m + 1 indices j ∈ J such that τ ∈ A_{j}. This shows that (A_{j})_{j∈J} has order at most m. (Notice that this step does not depend on the assumption that C is precise.)

It remains to show that the order of (A_{j})_{j∈J} is at least m. Since C has order m, there is a point τ̃ ∈ M̃ which lies in C_{i} for m + 1 distinct values of i, say for i = i_{0},…, i_{m}. For any two distinct integers s, t ∈ {0,…, m} we have i_{s} ≠ i_{t} and C_{i_{s}} ∩ C_{i_{t}} ≠ ∅. Since (I, G, C) is precise it follows that i_{0},…, i_{m} lie in distinct Γ-orbits. Hence we may write i_{t} = γ_{t} · j_{t} for t = 0, ⋯, m, where j_{0}, ⋯ j_{m} are distinct elements of J. We have p(τ̃) ∈ ⋂_{t=0}^{m} C_{j_{t}}; this shows that (A_{j})_{j∈J} has order at least m, and completes the proof of the proposition.

7.3. LERF groups. Recall that a group Γ is said to be locally extended residually finite if every finitely generated subgroup of Γ is an intersection of subgroups of finite index. According to [12], the fundamental group of any connected 2-manifold is locally extended residually finite.

Proposition. Let Γ be a locally extended residually finite group, and let M̃ be a free PL Γ-manifold. Suppose that (I, G, C) is a uniform Γ-plating of M̃, where G = (G_{i})_{i∈I}. Suppose that G_{i} is finitely generated for every i ∈ I. Then there is a subgroup Γ_{0} of finite index in Γ such that the triple (I, G^{0}, C), where G^{0} = (G_{i} ∩ Γ_{0})_{i∈I}, is a precise, uniform Γ_{0}-plating of M̃.

Remark. The hypothesis that the G_{i} are finitely generated is easily seen to hold automatically if each set in the family C has a finite number of connected components. In general, however, this may not be the case, and so the hypothesis of finite generation is needed.

Proof of Proposition 7.3: Let us write C = (C_{i})_{i∈I}. By 6.4, there is a closed, locally finite neighborhood (I, G, C′) of (I, G, C), where C′ = (C′_{i})_{i∈I}, such that (I, G, C′) is a uniform plating of M̃ and all the C′_{i} are subcomplexes in a fixed Γ-invariant triangulation of M̃. If Γ_{0} is a finite-index subgroup of Γ, and if the Γ_{0}-plating (I, G^{0}, C′), where G^{0} = (G_{i}∩Γ_{0})_{i∈I}, is precise and uniform, then it follows immediately that (I, G^{0}, C) is precise and uniform. Hence we may assume without loss of generality that the C_{i} are all subcomplexes in a fixed triangulation T of M̃.

By Proposition 5.4, the Γ-set I contains only finitely many Γ-orbits. Thus we may write I = Γ · S, for some finite subset S of I. Since (I, G, C) is uniform, C_{s} is a uniform G_{s}-space for every s ∈ S. For each s ∈ S we choose a finite subcomplex R_{s} of C_{s} (in the triangulation T ) such that G_{s} · R_{s} = C_{s}. For each s ∈ S we define Φ_{s} to be the set of all elements y ∈ Γ − G_{s} such that y · R_{s} ∩ R_{s} ≠ ∅. For any y ∈ Φ_{s} there is a vertex v of R_{s} such that y · v is also a vertex of R_{s}. Since R_{s} has finitely many vertices and Γ acts freely on M̃ it follows that Φ_{s} is a finite subset of Γ for every s ∈ S.

For each s ∈ S and each y ∈ Φ_{s} we have y ∈ Γ − G_{s}, where G_{s} is a finitely generated subgroup of the locally extended residually finite group Γ. Hence there is a subgroup Θ_{s,y} of finite index in Γ such that G_{s} ≤ Θ_{s,y} but y ∉ Θ_{s,y}. For each s ∈ S we set Θ_{s} = ⋂_{y∈Φ_{s}} Θ_{s,y}. Note that Θ_{s} has finite index in Γ, that G_{s} ≤ Θ_{s} and that Θ_{s} ∩ Φ_{s} = ∅.

Now set Γ_{0} = ⋂_{s∈S} Θ_{s}. Then Γ_{0} has finite index in Γ.

We set G^{0} = (G_{i} ∩ Γ_{0})_{i∈I}. It is clear that (I, G^{0}, C) is a Γ_{0}-plating of M̃. Since G_{i} ∩ Γ_{0} has finite index in G_{i} for every i ∈ I, it follows from 1.4 that (I, G^{0}, C) is uniform. We must show that (I, G^{0}, C) is precise. Thus for every i ∈ I and every x ∈ Γ_{0} such that C_{i} meets C_{x·i} = x · C_{i}, we must show that x ∈ G_{i}.

We first consider the case where i = s ∈ S. Assume that C_{s} ∩ x · C_{s} ≠ ∅, and let v be a vertex of C_{s} such that x · v ∈ C_{s}. We may write v = g_{1} · w_{1} and x · v = g_{2} · w_{2}, where w_{1}, w_{2} ∈ R_{s} and g_{1}, g_{2} ∈ G_{s}. Then g_{2}^{−1}xg_{1} · w_{1} = w_{2}, and so R_{s} ∩ g_{2}^{−1}xg_{1} · R_{s} ≠ ∅. Hence by the definition of Φ_{s} we have either g_{2}^{−1}xg_{1} ∈ G_{s} or g_{2}^{−1}xg_{1} ∈ Φ_{s}. But we have g_{1}, g_{2} ∈ G_{s} ≤ Θ_{s} and x ∈ Γ_{0} ≤ Θ_{s}. Hence g_{2}^{−1}xg_{1} ∈ Θ_{s}. Since Θ_{s} ∩ Φ_{s} = ∅ we cannot have g_{2}^{−1}xg_{1} ∈ Φ_{s}. Hence we must have g_{2}^{−1}xg_{1} ∈ G_{s}. Since g_{1}, g_{2} ∈ G_{s}, it follows that x ∈ G_{s}, as required.

Now consider an arbitrary index i ∈ I. Suppose that for a given x ∈ Γ_{0} we have C_{i} ∩ C_{x·i} ≠ ∅. We may write i = γ · s for some s ∈ S and some γ ∈ Γ. Then we have C_{s} ∩ C_{γ^{−1}xγ·s} = γ^{−1}(C_{i} ∩ C_{x·i}) ≠ ∅, and hence γ^{−1}xγ ∈ G_{s}, by the case of the assertion already proved. Now by condition (ii) in the definition of a Γ-plating (5.2) it follows that x ∈ G_{i}.

7.4. The following elementary lemma about 2-manifolds will be combined with Propositions 7.2 and 7.3 to prove Proposition 7.4.2 below, which will find a crucial application in Section 8.

7.4.1. Lemma. Let f : F → S be a map between closed, connected orientable, PL 2manifolds. Let Ξ ⊂ S be a compact, possibly disconnected PL 2-manifold; assume that no component of ∂Ξ is a homotopically trivial curve in S. Let B = (B_{j})_{j∈J} be a finite plating of F by PL sets, having order ≤ 1. Assume that for each j ∈ J the map f|B_{j} is homotopic in S to a map whose image is contained in Ξ. Then either f has degree 0, or every component of S − int Ξ is an annulus.

Proof: Let N_{j} be a regular neighborhood of B_{j} in F for each j ∈ J. Then (N_{j})_{j∈J} is a neighborhood of B. By Corollary 6.5 and Remark 6.3.1, there is a closed neighborhood B′ = (B′_{j})_{j∈J} of B such that (i) B′_{j} ⊂ N_{j} for every j ∈ J, (ii) B′_{j} is a PL 2-manifold for every j ∈ J and (iii) B′ has order ≤ 1. Since N_{j} is a regular neighborhood of B_{j}, the map f|B′_{j} is homotopic to a map with image contained in Ξ for every j ∈ J. Thus the hypotheses all continue to hold if B is replaced by B′, and we may therefore assume without loss of generality that each B_{j} is a PL 2-manifold and that F = ⋃_{j∈J} int B_{j}.

Since the plating B = (B_{j})_{j∈J} of F by PL 2-manifolds has order at most 1 and F = ⋃_{j∈J} int B_{j}, the closed PL 1-manifolds ∂B_{j} are pairwise disjoint. Hence L = ⋃ ∂B_{j} is a closed PL 1-manifold. Thus if A is a regular neighborhood of L in F, the components of A are annuli. Set R = F − int A. Then each component of R is contained in some B_{j}. Hence after modifying f within its homotopy class we may assume that f(R) ⊂ int Ξ.

Suppose that there is a component Q of S−Ξ which is not an annulus. After modifying f by a small homotopy we may assume that it is transverse to ∂Q. Set P = f^{−1}(Q) ⊂ int A, let P_{1},…, P_{k} denote the components of P, and set f_{i} = f|P_{i} : P_{i} → Q. The f_{i} are boundary-preserving maps. If we assign to each P_{i} the orientation induced from F and to Q the orientation induced from S, we have deg f = ∑_{i=1}^{k} deg f_{i}. Now each P_{i} is contained in an annulus component A_{i} of A. If we choose consistent base points in P_{i} ⊂ A_{i} and Q ⊂ S, we have a commutative diagram of groups and outer homomorphisms

π_{1}(P_{i}) ⟶^{(f_{i})_{#}} π_{1}(Q)
↓  ↓
π_{1}(A_{i}) ⟶_{(f|A_{i})_{#}} π_{1}(S)

in which the vertical arrows represent outer homomorphisms induced by inclusion. The outer homomorphism π_{1}(Q) → π_{1}(S) is injective since no component of ∂Q ⊂ ∂Ξ is homotopically trivial in S. Since π_{1}(A_{i}) is cyclic it follows that the outer group Z = (f_{i})_{#}(π_{1}(P_{i})) ≤ π_{1}(Q) is cyclic. Since Q is not an annulus, Z has infinite index in π_{1}(Q). Hence f_{i} lifts to an infinite-sheeted covering space Q̃ of Q. We have H_{2}(Q̃, ∂Q̃) = 0, and thus deg f_{i} = 0. Summing over i we get deg f = 0.

7.4.2. Proposition. Let Γ be a group, let F̃ and S̃ be free, uniform, Γ-orientable, connected 2-dimensional PL Γ-manifolds without boundary, and let f̃ : F̃ → S̃ be a Γequivariant continuous map. Set S = S̃/Γ and F = F̃/Γ. Assume that genus S > 0. Suppose that S̃ is simply connected, and let ν_{S} : π_{1}(S) → Γ denote the natural outer isomorphism. Suppose that the map f : F → S induced by f̃ has non-zero degree. Let Ξ ⊂ S be a compact PL 2-manifold. Suppose that no component of ∂Ξ is homotopically trivial in S. Suppose that there is a uniform Γ-plating (I, G, C) of F̃, where G = (G_{i})_{i∈I}, such that

(i) C has order at most 1, (ii) G_{i} is finitely generated for every i ∈ I, and (iii) for each i ∈ I the outer subgroup ν_{S}^{−1}(G_{i}) of π_{1}(S) is contained in the outer subgroup

im(π_{1}(Ξ_{i}) → π_{1}(S)) for some component Ξ_{i} of Ξ. Then every component of S − int Ξ is an annulus.

Proof: We have Γ ≅ π_{1}(S). Hence by [12], Γ is locally extended residually finite. By 7.3, there is a subgroup Γ_{0} of finite index in Γ such that the triple (I, G_{0}, C), where G^{0} = (G_{i} ∩ Γ_{0})_{i∈I}, is a precise, uniform Γ-plating of F̃.

Set F_{0} = F̃/Γ_{0}. Then F_{0} is a finite-sheeted regular covering space of F. In particular F_{0} is compact and the covering projection r : F_{0} → F has non-zero degree. Hence f_{0} = f ∘ r also has non-zero degree.

Let ν_{F} : π_{1}(F) → Γ and ν_{F_{0}} : π_{1}(F_{0}) → Γ_{0} denote the natural outer homomorphisms. Letting ι : Γ_{0} → Γ denote the inclusion, we then have a diagram of groups and outer homomorphisms

which we may think of as a triangle subdivided into left-hand, right-hand and bottom subtriangles. This is a commutative diagram of outer homomorphisms. Indeed, the full triangle is commutative by virtue of the Γ-equivariance of f̃; the left-hand triangle is commutative by elementary covering space theory, and the bottom triangle is commutative by the definition of f_{0}. It follows that the right-hand triangle of outer homomorphisms also commutes.

Let p_{0} : F̃ → F_{0} denote the covering projection. By Proposition 7.2, the closed 2manifold F_{0} has a finite plating (B_{1},…, B_{r}) of order 1, where for each j ∈ {1,…, r} the set B_{j} is a compact, connected, PL subset of F_{0}, and each component of p_{0}^{−1}(B_{j}) is a component of C_{i} for some i ∈ I. For each j let Θ_{j} denote the outer subgroup im(π_{1}(B_{j}) → π_{1}(F)), a subgroup of π_{1}(F_{0}). For each j the outer subgroup ν_{F_{0}}(Θ_{j}) of Γ_{0} is realized by the stabilizer of a component R_{j} of p_{0}^{−1}(B_{j}). Since R_{j} is a component of C_{i} for some i, the observation 7.1.1, applied to the precise Γ_{0}-plating (I, G_{0}, C), gives that ν_{F_{0}}(Θ_{j}) is contained in the outer subgroup G_{i} of Γ_{0} for some i ∈ I. In particular the outer subgroup ιν_{F_{0}}(Θ_{j}) of Γ is contained in the outer subgroup G_{i} of Γ for some i.

By the commutativity of the right-hand subtriangle in the diagram above, we have (f_{0})_{♯}(Θ_{j}) = ν_{S}^{−1}ιν_{F_{0}}(Θ_{j}). Thus the outer subgroup (f_{0})_{♯}(Θ_{j}) of π_{1}(S) is contained in the outer subgroup ν_{S}^{−1}(G_{i}) for some i. By condition (ii) in the hypothesis of the proposition it follows that the outer subgroup (f_{0})_{♯}(Θ_{j}) of π_{1}(S) is contained in the outer subgroup im(π_{1}(Ξ_{j}) → π_{1}(S)) for some component Ξ_{j} of Ξ. Now S is aspherical since genus S > 0, and π_{1}(Ξ_{i}) → π_{1}(S) is injective since no component of ∂Ξ is homotopically trivial. We can therefore conclude that for each j the map f_{0}|B_{j} is homotopic in S to a map whose image is contained in Ξ. Since f_{0} has non-zero degree, it now follows from Lemma 7.4.1 that every component of S − Ξ is an annulus.

Section 8. Fibroids and platings, I.

8.1. If Γ is a group and T is a Γ-tree, the action of Γ on T induces an action on the set E(T) of edges of T. We shall regard E(T) as a Γ-set by equipping it with this action.

8.2. This section is devoted to the proof of the following topological result, which gives a sufficient condition for an incompressible surface to be a fibroid.

Proposition. Let M be a simple, closed, orientable 3-manifold containing an incompressible bi-collared surface c. Let Γ ≅ π_{1}(M) denote the group of deck transformations of the universal covering space M̃ of M, and let T denote the Γ-tree T (c). Suppose that there is a doubly uniform Γ-plating (E(T), G, W) of the Γ-space M̃, where G = (Γ_{e})_{e∈E(T)} and W = (W_{e})_{e∈E(T)}, which satisfies the following conditions.

(i) The order of W is at most 2. (ii) There is a Γ-equivariant map f : M̃ → T such that for every e ∈ E(T) we have

f(W_{e}) ⊂ ē. Then the core of c is a fibroid.

8.3. Throughout the section, M will denote a manifold, and c a bi-collared surface, satisfying the hypotheses of Proposition 8.2. We shall fix a PL structure on M in which c is a PL bi-collared surface. We shall denote the core of c by S. As in the statement of the proposition, M̃ will denote the universal covering space of M and Γ its group of deck transformations, and T will denote the Γ-tree T_{(}c). We shall set q = q_{c} : M̃ → T. We shall write p : M̃ → M for the covering projection. The bi-collared surface in M̃ lying over c (3.1) will be denoted by c̃. We shall write E = E(T), and we shall fix a doubly uniform Γ-plating W = (W_{e})_{e∈E} which is indexed by the Γ-set E and satisfies conditions (i) and (ii) of proposition 8.2.

8.4. Lemma. We have f(M̃) = T.

Proof: Since f is equivariant, f(M̃) is a Γ-invariant subtree of T. But T is a minimal Γ-tree by 3.3.

8.5. Lemma. For any e ∈ E we have f^{−1}(e) ⊂ W_{e} ⊂ f^{−1}(ē).

Proof: By the hypothesis of Proposition 8.2 we have f(W_{e}) ⊂ ē and hence W_{e} ⊂ f^{−1}(ē). Now let P be any point of f^{−1}(e). Since W is a plating we have P ∈ W_{e′} for some e′ ∈ E. It follows that f(P) ∈ ē′ and hence that e′ = e. This proves that f^{−1}(e) ⊂ W_{e}.

8.6. For any subcomplex Λ of T we shall denote by nbhd_{1} Λ the union of Λ with all (open) edges of the first barycentric subdivision of T that share a vertex with Λ.

Lemma. There is a locally finite closed neighborhood (E, G, W′) of (E, G, W), where W′ = (W′_{e})_{e∈E}, having the following properties:

(i) f(W′_{e}) ⊂ nbhd_{1} ē for every e ∈ E; (ii) (E, G, W′) is doubly uniform; (iii) W′_{e} is a PL 3-manifold for every e ∈ E; and (iv) W′ has order at most 2.

Proof: For every e ∈ E(T), the set nbhd_{1}(ē) is a neighborhood of e in T, and for any γ ∈ Γ we have γ · nbhd_{1} ē = nbhd_{1}(γ · ē). This naturality property, and the properties of the map f given in hypothesis (ii) of Proposition 8.2, imply that (E, G, (f^{−1}(nbhd_{1} ē))_{e∈E}) is a neighborhood of (E, G, W′). We can now apply Proposition 6.3, with (E, G, (f^{−1}(nbhd_{1} ē))_{e∈E}) playing the rôle of (I, G, C^{+}). The resulting locally finite closed neighborhood (E, G, W′) of (E, G, W) has the property (i) stated in the conclusion of the proposition because it has (E, G, (f^{−1}(nbhd_{1} ē))_{e∈E}) as a neighborhood. Property (iv) follows from Remark 6.3.1, and the others are immediate from Proposition 6.3.

For the rest of the section we fix a neighborhood (E, G, W′) of (E, G, W) having the properties stated in the above lemma.

8.7. If s is any edge of T, we shall denote by p/Γ_{s} : M̃/Γ_{s} → M the covering map induced by p. We shall let c̃/Γ_{s} denote the bi-collared surface in M̃/Γ_{s} lying over c. According to Proposition 3.4, the component M̃_{s}/Γ_{s} of |c̃/Γ_{s}| is a deformation retract of M̃/Γ_{s}, and p/Γ_{s} maps M̃_{s}/Γ_{s} homeomorphically onto |c|. In particular, M̃/Γ_{s} is homotopy equivalent to S.

If v is any vertex of T incident to s, it follows that ∂_{s}M̃_{v}/Γ_{s} is a deformation retract of M̃/Γ_{s}, and that p/Γ_{s} maps ∂_{s}M̃_{v}/Γ_{s} homeomorphically onto ∂_{s}M_{v}. We shall let r_{s,v} denote a deformation retraction (chosen arbitrarily) of M̃/Γ_{s} to ∂_{s}M̃_{v}/Γ_{s}.

8.8. Lemma. Let s be any edge of T, and let v be any vertex of T incident to s. Then there is a component F̃ of the boundary of W′_{s}/Γ_{s} such that

(i) F̃ is invariant under Γ_{s}; (ii) F̃ is a uniform, Γ_{s}-orientable PL Γ_{s}-manifold; (iii) the map r_{s,v}|(F̃/Γ_{s}) : F̃/Γ_{s} → ∂_{s}M_{v}/Γ_{s} has degree one; and (iv) for every edge e of T not incident to v, we have F̃ ∩ W′_{e} = ∅.

Proof: Since W′ is doubly uniform according to Lemma 8.6, it is uniform by Proposition 5.7.1. Thus W′_{s} is a uniform Γ_{s}-space, i.e. W′_{s} ⊂ M̃/Γ_{s} is compact. Hence the PL 2-manifold ∂(W′_{s}/Γ_{s}) is closed. Note also that since M̃/Γ_{s} is orientable, ∂(W′_{s}/Γ_{s}) is orientable.

The map f induces a map f_{s} : M̃/Γ_{s} → T/Γ_{s}. We have a commutative diagram

M̃ ⟶^{f} T
↓  ↓
M̃/Γ_{s} ⟶^{f_{s}} T/Γ_{s}

in which the vertical arrows represent quotient maps. Since Γ_{s} fixes s, there is an edge s_{0} of the graph T/Γ_{s} whose preimage in T is precisely s. One vertex of s_{0} is the image of v in T/Γ_{s}, which we shall denote by v_{0}; let us denote the other vertex of s_{0} by w_{0}. By Lemma 8.5, we have f^{−1}(s) ⊂ W_{s} ⊂ W′_{s}. From the above commutative diagram it follows that f_{s}^{−1}(s_{0}) ⊂ W′_{s}/Γ_{s}.

By Proposition 3.5, T/Γ_{s} is a tree without end points. In particular, (T/Γ_{s}) − s_{0} has two connected components, each of which is a tree with infinitely many vertices. Let Λ_{v} and Λ_{w} denote the components of (T/Γ_{s}) − s_{0} containing v_{0} and w_{0} respectively. Since f_{s}^{−1}(s_{0}) ⊂ W′_{s}/Γ_{s}, each component of M̃/Γ_{s} − W′_{s}/Γ_{s} is mapped by f_{s} into either Λ_{v} or Λ_{w}.

By Lemma 8.4, f maps M̃ onto T. Hence f_{s} maps M̃/Γ_{s} onto T/Γ_{s}. Since each of the trees Λ_{v} and Λ_{w} has infinitely many vertices, f^{−1}(Λ_{v}) and f^{−1}(Λ_{w}) are non-compact. But since W′_{s}/Γ_{s} is compact and M̃/Γ_{s} is connected, M̃/Γ_{s} − int W′_{s}/Γ_{s} has only finitely many components. Since f^{−1}(Λ_{v}) is non-compact, f must map some non-compact component N of M̃/Γ_{s} − int W′_{s}/Γ_{s} into Λ_{v}.

We claim that there is a component F of ∂N which does not bound a compact 3-manifold in M̃/Γ_{s}. Assume to the contrary that every component C of ∂N bounds a compact 3manifold J_{C}. No J_{C} can contain N since N is non-compact. Since M̃_{e} is connected it follows that M̃_{e} = N ∪ ⋃_{C} J_{C}, where C ranges over the components of ∂N. This means that M̃ −int N is compact, which is impossible since f^{−1}(Λ_{w}) is non-compact. This proves the claim.

Since ∂_{s}M_{v}/Γ_{s} is a deformation retract of M̃/Γ_{s} via r = r_{s,v}, we may apply Proposition 3.4.1 to conclude that π_{1}(F) → π_{1}(M̃/Γ_{s}) is surjective and that r|F : F → ∂_{s}M_{v}/Γ_{s} has degree 1. Since π_{1}(F) → π_{1}(M̃/Γ_{s}) is surjective, the preimage F̃ of F in M̃ is connected and is therefore a component of ∂W′_{s}. We have F = F̃/Γ_{s}. Since W′_{s} is a uniform Γ_{s}-space, its Γ_{s}-invariant closed subset F̃ is also a uniform Γ_{s}-space by 1.4. Furthermore, since M̃ is Γ-orientable, W′_{s} is Γ_{s}-orientable, and hence its Γ_{s}-invariant boundary component F̃_{s} is Γ_{s}-orientable as well. It remains to check conclusion (iv) of the lemma.

By Lemma 8.6, we have f(W′_{e}) ⊂ nbhd_{1} ē for every edge e of T. By our choice of the surface F ⊂ ∂W′_{s} we have f_{s}(F) ⊂ Λ_{v}; hence f(F̃) ⊂ Λ̃_{v}, where Λ̃_{v} is the component of T − s containing v. It follows that f(F̃) ⊂ Λ̃_{v} ∩ nbhd_{1} s̄ ⊂ nbhd_{1}{v}. Hence if e is any edge of T not incident to v, we have f(F̃ ∩ W′_{e}) ⊂ nbhd_{1}{v} ∩ nbhd_{1} ē = ∅, and hence F̃ ∩ W′_{e} = ∅.

8.9. Proof of Proposition 8.2: We are required to prove that the core of c is a fibroid. According to the definitions, this is equivalent to showing that every component of Split(c) is a book of I-bundles.

Consider an arbitrary component of Split(c), which we may write in the form M_{v} for some vertex v of T = T (c). Since M is simple and c is incompressible, it follows from the definitions that M_{v} is simple. Let (Σ, Φ) denote the characteristic pair of (M_{v}, ∂M_{v}). According to Proposition 4.3, we need only show that every component of ∂M_{v} − int Φ is an annulus.

Consider an arbitrary component of ∂M_{v}, which we may write in the form ∂_{s}M_{v} for some edge s incident to v. Set Ξ_{0} = ∂_{s}M_{v} ∩Φ. We are required to show that every component of ∂_{s}M_{v} − int Ξ_{0} is an annulus. For technical reasons that will become apparent we consider a PL 2-manifold Ξ ⊂ ∂M_{v} constructed as follows. If Ξ_{0} ≠ ∅ we set Ξ = Ξ_{0}. If Ξ = ∅ we take Ξ to be an arbitrarily chosen homotopically non-trivial annulus in ∂_{s}M_{v}; such an annulus exists because the incompressibility of c guarantees that ∂_{s}M_{v} is not a 2-sphere. Note that since M is simple, ∂_{s}M_{v} cannot be a torus. Hence if we can show that every component of ∂_{s}M_{v} − int Ξ is an annulus, it will follow that Ξ = Ξ_{0} and hence that every component of ∂_{s}M_{v} − int Ξ_{0} is an annulus, as required. The rest of the argument will be devoted to proving that every component of ∂_{s}M_{v} − int Ξ is an annulus.

Let us choose a component F̃_{s,v} of the boundary of W′_{s} which satisfies conditions (i)–(iv) of Lemma 8.8. Set r = r_{s,v}, and let r̃ : M̃ → F̃ be a retraction covering r. We have a Γ_{s}-equivariant map f̃ = r̃|F̃ : F̃ → ∂_{s}M̃_{v}. We shall show that ∂_{s}M̃_{v}, F̃, f̃ and Ξ ⊂ ∂_{s}M_{v} satisfy the hypotheses of Proposition 7.4.2, with the group Γ_{s} playing the rôle of Γ. It will then follow from Proposition 7.4.2 that every component of ∂_{s}M_{v} − int Ξ is an annulus, as required.

By Lemma 8.8, F̃ is a uniform, Γ_{s}-orientable Γ_{s}-manifold. Since ∂_{s}M_{v} is a closed, orientable 2-manifold, the PL Γ-manifold ∂_{s}M̃_{v} is also uniform and Γ_{s}-orientable. Furthermore, since ∂_{s}M_{v} is incompressible, ∂_{s}M̃_{v} is simply connected and ∂_{s}M_{v} is not a 2-sphere. We have observed that f̃ : F̃ → ∂_{s}M̃_{v} is Γ_{s}-equivariant. The induced map f : F → ∂_{s}M_{v} is the map r_{s,v}|(F̃/Γ_{s}), which by Lemma 8.8 has degree 1.

It remains to construct a uniform Γ-plating of F̃ satisying conditions (i)—(iii) of Proposition 7.4.2.

Set Q_{s} = ∂W′_{s}. According to Proposition 5.7, the Γ-plating (E, G, W′) of M̃ induces a boundary plating of Q_{s}. This boundary plating is a Γ_{s}-plating and will be denoted (E^{s}, G^{s}, (W′)^{s}). Here E^{s} is the set of all edges e ≠ s of T such that W′_{e} ∩ Q_{s} ≠ ∅; and we have G^{s} = (Γ_{e} ∩ Γ_{s})_{e∈E^{s}} and (W′)^{s} = (W′_{e} ∩ Q_{s})_{e∈E^{s}}. Since (E, G, W) has order at most 2 by Lemma 8.6, it follows from Proposition 5.7 that (E^{s}, G^{s}, (W′)^{s}) has order at most 1. Moreover, since (E, G, W) is doubly uniform by Lemma 8.6, it follows from Proposition 5.7.1 that the plating (E^{s}, G^{s}, (W′)^{s}) is uniform.

It now follows from 5.6 that the Γ_{s}-space F̃ inherits a Γ_{s}-plating (EF̃, GF̃, W′ ; F̃) from the Γ_{s}-plating (E^{s}, G^{s}, (W′)^{s}) of Q_{s}. Here EF̃ consists of all edges e ≠ s of T such that W′_{e}∩F ≠ ∅; and we have GF̃ = (Γ_{e} ∩ Γ_{s})_{e∈EF̃} and (W′)F̃ = (W′_{e} ∩ F)_{e∈EF̃}. Since (E^{s}, G^{s}, (W′)^{s}) is uniform and has order at most 1, (EF̃, GF̃, W′ ; F̃) has the same properties. That (EF̃, GF̃, W′ ; F̃) has order at most 1 is condition (i) of Proposition 7.4.2. We shall complete the proof by showing that (EF̃, GF̃, W′ ; F̃) satisfies conditions (ii) and (iii) of Proposition 7.4.2.

Consider any element e of the index set EF̃. The group in the family GF̃ indexed by e is Γ_{s} ∩ Γ_{e}.

The definition of EF̃ implies that e is an edge distinct from s. Furthermore, we have W′_{e} ∩ F ≠ ∅; this implies that e is incident to v, since by condition (iv) of Lemma 8.8 we have F̃ ∩ W′_{s} = ∅ for every edge e of T not incident to v. It now follows from Proposition 4.2 that Γ_{s} ∩ Γ_{e} is finitely generated. This is condition (ii) of 7.4.2.

Finally we must verify condition (iii), that there is a component Ξ_{e} of Ξ such that the outer subgroup ν_{s,v}^{−1}(Γ_{s} ∩ Γ_{e}) of π_{1}(∂_{s}M_{v}) is contained in the outer subgroup im(π_{1}(Ξ_{e}) → π_{1}(∂_{s}M_{v})). If Γ_{s} ∩ Γ_{e} ≠ {1}, it follows from Proposition 4.2 that the outer subgroup ν_{s,v}^{−1}(Γ_{s} ∩ Γ_{e}) of π_{1}(∂_{s}M_{v}) is contained in the outer subgroup im(π_{1}(Φ_{e}) → π_{1}(∂_{s}M_{v})) for some component Φ_{e} ⊂ ∂_{s}M_{v} of Φ_{v}. In this case it follows from the definitions of Ξ and Ξ_{0} that Φ_{e} is a component of Ξ_{0} and hence of Ξ. If Γ_{s} ∩ Γ_{e} = {1}, then since Ξ ≠ ∅, it is trivially true that the (trivial) outer subgroup ν_{s,v}^{−1}(Γ_{s} ∩ Γ_{e}) is contained in the outer subgroup im(π_{1}(Ξ_{e}) → π_{1}(∂_{s}M_{v})) for some component Ξ_{e} of Ξ. Thus condition (iii) of 7.4.2 is verified in all cases. This completes the proof of Proposition 8.2.

Section 9. Fibroids and platings, II.

9.1. This section is devoted to the proof of the following result, which like Proposition 8.2 is purely topological and gives a sufficient condition for an incompressible surface to be a fibroid.

Theorem. Let M be a simple, closed, orientable 3-manifold containing a connected incompressible bi-collared surface c. Let Γ ≅ π_{1}(M) denote the group of deck transformations of the universal covering space M̃ of M, and set T = T (c). Suppose that there is a uniform Γ-plating (I, X , Z) of the Γ-space M̃, where Z = (Z_{i})_{i∈I} and X = (X_{i})_{i∈I}, satisfying the following conditions.

(i) For each i ∈ I the group X_{i} is a T -hyperbolic cyclic subgroup of Γ. (ii) For any i, i′ ∈ I such that Z_{i} ∩ Z_{i′} ≠ ∅, we have E(A_{T} (X_{i})) ∩ E(A_{T} (X_{i′})) ≠ ∅. Then the core of c is a fibroid.

9.2. Throughout the section, M will denote a manifold, and c a bi-collared surface, satisfying the hypotheses of Theorem 9.1. We shall fix a PL structure on M in which c is a PL bi-collared surface. As in the statement of the theorem we shall denote the universal covering space of M by M̃ and its group of deck transformations by Γ, and we shall set T = T (c). We shall fix a uniform Γ-plating Z = (Z_{i})_{i∈I} of M̃ satisfying conditions (i) and (ii) of the theorem.

9.3. Corollary 6.4 gives a closed, locally finite neighborhood (I, X , Z′) of (I, X , Z) which is itself a uniform plating of M̃. Furthermore, if we write Z′ = (Z′_{i})_{i∈I}, then the Z′_{i} are all subcomplexes of some Γ-invariant triangulation of M̃; and for any two indices i, j ∈ I such that Z_{i} ∩ Z_{j} = ∅ we have Z′_{i} ∩ Z′_{j} = ∅. It follows from this last property of Z′ that the hypotheses of Theorem 9.1 continue to hold if (I, X , Z) is replaced by (I, X , Z′). But since (I, X , Z′) is a neighborhood of (I, X , Z) we have M̃ = ⋃_{i∈I} int Z′_{i}. This means that in proving Theorem 9.1 we may assume, without loss of generality, that M̃ = ⋃_{i∈I} int Z_{i}, and that there is a Γ-invariant triangulation of M̃ in which the Z_{i} are all subcomplexes. We shall make these assumptions for the remainder of this section.

The proof of Theorem 9.1 proceeds by constructing from the Γ-plating (I, X , Z) a new Γplating (E(T), G, W) of M̃, indexed by the Γ-set E(T) (8.1), and satisfying the hypotheses of Proposition 8.2. The construction of the map f appearing in the hypothesis of Proposition 8.2 will precede the construction of the Γ-plating (E(T), G, W).

9.4. Lemma. There exists a continuous, Γ-equivariant map f : M̃ → T such that for every i ∈ I we have f(Z_{i}) ⊂ A_{T} (X_{i}).

Proof: For each P ∈ M̃ let us denote by I_{P} ⊂ I the set of all indices i such that P ∈ Z_{i}. Since the plating Z is by definition locally finite, I_{P} is a finite subset of I for each P ∈ M̃. By condition (ii) in the hypothesis of Theorem 9.1, for any two indices i, i′ ∈ I we have E(A_{T} (X_{i}))∩E(A_{T} (X_{i′})) ≠ ∅, and in particular A_{T} (X_{i})∩A_{T} (X_{i′}) ≠ ∅. Hence by Corollary 2.3.1, the set Ψ_{P} = ⋂_{i∈I_{P}} A_{T} (X_{i}) is a segment in T, and is in particular non-empty. Since Z_{γ·i} = γ · Z_{i} for all i ∈ I, γ ∈ Γ, we have I_{γ·P} = γ · I_{P} and hence Ψ_{γ·P} = γ · Ψ_{P} for all P ∈ M̃, γ ∈ Γ.

Let us equip M̃ with the Γ-invariant triangulation given by 9.3, in which the Z_{i} are all subcomplexes. Let M̃^{(d)} denote the d-skeleton of M̃ for d = 0,…, 3.

We claim that there is a Γ-equivariant map f^{0} : M̃^{(0)} → T such that f^{0}(P) ∈ Ψ_{P} for every P ∈ M̃^{(0)}. To prove this we use a set S^{0} ⊂ M^{(0)} which is a complete set of orbit representatives for the action of Γ on M̃^{(0)}, in the sense that M^{(0)} = ∐_{P ∈S^{0}} Γ · P. For each P ∈ S^{0} we choose a vertex v_{P} in the segment Ψ_{P}. Since Γ acts freely on M̃, and in particular on T^{0}, we may define f^{0} unambiguously by setting f^{0}(γ · P_{0}) = γ · v_{P_{0}} for all γ ∈ Γ, P ∈ S^{0}. The Γ-equivariance of f^{0} is clear. Furthermore, for any P ∈ M̃^{(0)}, writing P = γ · P_{0} with P_{0} ∈ S^{0}, we have f^{0}(P) = γ · v_{P_{0}} ∈ γ · Ψ_{P_{0}} = Ψ_{γ·P_{0}} = Ψ_{P}. This proves the claim.

For any finite set Φ of vertices of T let us denote by hull(Φ) the union of all segments in T having their endpoints in Φ. If Φ is non-empty then hull(Φ) is a connected subcomplex of T and hence a subtree. If the non-empty finite set Φ is contained in a line L ⊂ T, then hull(Φ) is a segment in L.

By induction on d ∈ {0,…, 3}, we shall show that f^{0} can be extended to a map f^{d} : M̃^{(d)} → T such that for every closed simplex τ of M̃ we have f(τ) ⊂ hull(f^{0}(τ ∩ M̃^{(0)})). For d = 0 the assertion is trivial. Suppose that f^{d} has been constructed for a given d ∈ {0, 1, 2}. Let S^{d+1} be a complete set of orbit representatives for the action of Γ on the set of all closed (d + 1)-simplices of M̃. For each τ ∈ S^{d+1}, and each proper face τ′ of τ, we have f^{d}(τ′) ⊂ hull(f^{0}(τ′ ∩ M̃^{(0)})) ⊂ hull(f^{0}(τ ∩ M̃^{(0)})). Thus f^{d} maps ∂τ into the tree hull(f^{0}(τ ∩ T^{0})). Since a tree is contractible, f^{d}|∂τ can be extended to a continuous map f_{τ} : τ → hull(f^{0}(τ ∩ M̃^{(0)})). Now Γ acts freely on the set of (d + 1)-simplices of T , so any (d + 1)-simplex τ can be written in a unique way as γ · τ_{0} with γ ∈ Γ and τ_{0} ∈ S^{d+1}. We define a continuous map f_{τ} : τ → T by setting f(γ · P) = γ · f(P) for every P ∈ τ. The Γ-equivariance of f^{d} implies that f_{τ} |∂τ = f^{d}|∂τ. Hence we may extend f^{d} to a continuous map f^{d+1} : M̃^{(d+1)} → T by setting f^{d+1}|τ = f_{τ} for every closed (d + 1)-simplex τ. It is now clear that f is Γ-equivariant and that f(τ) ⊂ hull(f^{0}(τ ∩ M̃^{(0)})) for every closed simplex τ of M̃^{(d+1)}. This completes the induction.

Set f = f^{3}. We shall complete the proof of the lemma by showing that f(Z_{i}) ⊂ A_{T} (X_{i}) for every i ∈ I. Since Z_{i} is a subcomplex of M̃ we need only show that if τ ⊂ Z_{i} is a closed simplex then f(τ) ⊂ A_{T} (X_{i}). For any vertex P of τ we have P ∈ Z_{i} and hence i ∈ I_{P}. Therefore f^{0}(P) ∈ Ψ_{P} ⊂ A_{T} (X_{i}). Thus f^{0}(τ ∩ T^{0}) ⊂ A_{T} (X_{i}). It follows that hull(f^{0}(τ ∩ T^{0})) is a segment in the line A_{T} (X_{i}). Hence f(τ) ⊂ hull(f^{0}(τ ∩ M̃^{(0)})) ⊂ A_{T} (X_{i}).

9.5. In this subsection we construct a closed set W_{e} ⊂ M̃ for each e ∈ E(T). Most of the section will be devoted to proving that the sets W_{e} constitute a plating, and that they define a Γ-plating satisfying the hypotheses of Proposition 8.2.

For each edge e of T, we define a set U_{e} ⊂ M̃ as follows. A point P ∈ M̃ lies in U_{e} if and only if for every i ∈ I such that P ∈ int Z_{i}, we have e ∈ E(A_{T} (X_{i})). Thus U_{e} is the complement of a union of sets of the form int Z_{i}, where i ranges over a certain subset of I. Since the sets int Z_{i} are open, it follows that U_{e} is closed.

For each triod Y in T we define a set V_{Y} ⊂ M̃ as follows. A point P ∈ M̃ belongs to V_{Y} if and only if for each length-2 segment σ ⊂ Y there is an index i ∈ I such that σ ⊂ A_{T} (X_{i}) and P ∈ Z_{i}. It is clear from this definition that V_{Y} is a union of sets of the form Z_{i} ∩ Z_{j} ∩ Z_{k}, where (i, j, k) ranges over some subset of I × I × I. But the plating Z is by definition a locally finite family of closed sets. It follows that V_{Y} is a locally finite union of closed sets, and hence a closed subset of M̃.

Now for each edge e of T, we set

W_{e} = (U_{e} ∩ f^{−1}(ē)) ∪ ⋃_{Y ∈Y_{e}} V_{Y} ,

where Y_{e} denotes the set of all triods containing e. Since the U_{e} and the V_{Y} are closed, and since f is continuous, it is clear that W_{e} is closed in M̃.

9.6. Lemma. For every edge e of T we have f(W_{e}) ⊂ ē.

Proof: By definition W_{e} is the union of U_{e} ∩ f^{−1}(ē) with the sets V_{Y}, where Y ranges over all triods having e as an edge. We therefore need only prove that if e is an edge of a triod Y then f(W_{e}) ⊂ ē. Actually we will prove slightly more, namely that f(W_{e}) = {v} where v is the center of the triod Y. Let σ, σ′, σ′′ be the length-2 segments contained in Y. Let P be any point of V_{Y}. By the definition of V_{Y} there are indices i, i′, i′′ ∈ I such that P ∈ Z_{i} ∩ Z_{i′} ∩ Z_{i′′} and such that σ ⊂ A_{T} (X_{i}), σ′ ⊂ A_{T} (X_{i′}) and σ′′ ⊂ A_{T} (X_{i′′}). By Lemma 9.4 we have f(P) ∈ f(Z_{i}) ∩ f(Z_{i′}) ∩ f(Z_{i′′}) ⊂ A_{T} (X_{i}) ∩ A_{T} (X_{i′}) ∩ A_{T} (X_{i′′}). But Proposition 2.4, applied to the set L = {A_{T} (X_{i}), A_{T} (X_{i′}), A_{T} (X_{i′′})}, shows that A_{T} (X_{i}) ∩ A_{T} (X_{i′}) ∩ A_{T} (X_{i′′}) = {v}. Hence f(P) = v as required.

9.7. Lemma. We have M̃ = ⋃_{e∈E(T)} W_{e}.

Proof: Given any point P ∈ M̃, we must show that P ∈ W_{e} for some edge e of T. We let I_{P} denote the set of all indices i ∈ I such that P ∈ Z_{i}. We have I_{P} ≠ ∅ since M̃ = ⋃_{i∈I} int Z_{i} by 9.3. But the plating Z is by definition a locally finite family; hence I_{P} is finite. Now let L denote the set of all lines of the form A_{T} (X_{i}) for i ∈ I_{P}. Then L is a finite, non-empty collection of lines in T. Since P ∈ Z_{i} for every i ∈ I_{P}, hypothesis (ii) of the theorem implies that E(L) ∩ E(L′) ≠ ∅ for all L, L′ ∈ L. By Corollary 2.3.1, ⋂ L is a segment in T, possibly degenerate.

Consider first the case where ⋂ L is non-degenerate. Note that for every i ∈ I_{P} we have P ∈ Z_{i} and hence f(P) ∈ f(Z_{i}) ⊂ A_{T} (X_{i}), in view of the defining property of f. By the definition of L this means that P ∈ ⋂ L. As ⋂ L is a non-degenerate segment, we have f(P) ∈ ē for some edge e ∈ E(⋂ L). But to say that e ∈ E(⋂ L) means that e ∈ E(A_{T} (X_{i})) for every i ∈ I_{P}; by the definition of U_{e} this says that P ∈ U_{e}. Hence in this case we have P ∈ U_{e} ∩ f^{−1}(ē) ⊂ W_{e}.

Now consider the case where the segment ⋂ L is degenerate. In this case, according to Proposition 2.4, there is a triod Y ⊂ T such that every length-2 segment contained in Y is contained in a line in L. This means that for every length-2 segment σ in Y there is an index i such that P ∈ int Z_{i} and σ ⊂ A_{T} (X_{i}). Since P ∈ int Z_{i} implies that in particular P ∈ Z_{i}, it follows from the definition of V_{Y} that in this case P ∈ V_{Y}, and so P ∈ W_{e} for any edge e of Y.

9.8. Lemma. Every point P ∈ M̃ has a neighborhood N in M̃ such that there are at most two edges e ∈ E(T) for which N ∩ U_{e} ∩ f^{−1}(ē) ≠ ∅.

Proof: By 9.3 there is an index i ∈ I such that P ∈ int Z_{i}. Let v be a vertex of T such that f(P) ∈ Star(v), where Star(v) denotes the open star of v in T. Then N = int Z_{i} ∩ f^{−1}(Star(v)) is a neighborhood of P in M̃. Let e be an edge such that N ∩U_{e} ∩f^{−1}(ē) ≠ ∅. Then in particular we have int Z_{i} ∩ U_{e} ≠ ∅, and it follows from the definition of U_{e} that e ∈ E(A_{T} (X_{i})). But we also have f^{−1}(Star(v)) ∩ f^{−1}(ē) ≠ ∅ and hence (Star(v)) ∩ ē ≠ ∅; thus the edge e must be incident to the vertex v. But there can be at most two closed edges that lie in the line A_{T} (X_{i}) and are incident to the vertex v.

9.9. Lemma. For any P ∈ M̃ there is at most one triod Y ⊂ T such that P ∈ V_{Y}. Furthermore, if P ∈ V_{Y} for a given triod Y and P ∈ U_{e} ∩ f^{−1}(ē) for a given edge e, then e ∈ E(Y).

Proof: As in the proof of Lemma 9.7, we let I_{P} denote the set of all indices i ∈ I such that P ∈ Z_{i}, and we let L denote the set of all lines of the form A_{T} (X_{i}) for i ∈ I_{P}. As we observed in the proof of Lemma 9.7, we have E(L) ∩ E(L′) ≠ ∅ for all L, L′ ∈ L. If Y is a triod such that P ∈ V_{Y} then every length-2 segment in Y is contained in a line in L. It follows from Proposition 2.4 that there is at most one triod with this property. This proves the first assertion.

Now suppose that P ∈ V_{Y} for a triod Y and that P ∈ U_{e} ∩ f^{−1}(ē) for an edge e. Let v denote the center of the triod Y. It follows from Proposition 2.4 that ⋂ L = {v}. By the defining property of the map f we have

f(P) ∈ ⋂_{i∈I_{P}} f(Z_{i}) ⊂ ⋂_{i∈I_{P}} A_{T} (X_{i}) = ⋂ L = {v},

i.e. f(P) = v. Hence the edge e is incident to v.

By 9.3 there is an index k ∈ I such that P ∈ int Z_{k}. Since P ∈ U_{e} it follows that e ∈ E(A_{T} (X_{k})). On the other hand, since P ∈ int Z_{k} we have in particular P ∈ Z_{k} and hence k ∈ I_{P}. Hence the line A_{T} (X_{k}) belongs to L. By Proposition 2.4, A_{T} (X_{k}) meets the triod Y in a length-2 segment. Since e is an edge of A_{T} (X_{k}) and is incident to v, it follows that e is an edge of Y.

9.10. Lemma. The indexed family (V_{Y})_{Y} of subsets of M̃, where Y ranges over all triods in T, is locally finite.

Proof: Since the plating (Z_{i})_{i∈I} is by definition a locally finite family, the family (Z_{i} ∩ Z_{j} ∩ Z_{k})_{(i,j,k)∈I} is also locally finite. Each set V_{Y}, where Y is a triod in T, is by definition a union of sets of the form Z_{i} ∩ Z_{j} ∩ Z_{k} for (i, j, k) ∈ I. Furthermore, for Y ≠ Y′ we have V_{Y} ∩ V_{Y′} = ∅ by Lemma 9.9. Hence if P is a point of M̃ and N is a neighborhood of P that meets Z_{i} ∩ Z_{j} ∩ Z_{k} for at most n < ∞ triples (i, j, k), then N meets V_{Y} for at most n triods Y.

9.11. Lemma. The family W = (W_{e})_{e∈E(T)} is a plating of M̃.

Proof: We observed in 9.5 that each W_{e} is closed. By Lemma 9.7 we have M̃ = ⋃_{e∈E(T)} W_{e}. It remains to verify that W is a locally finite family. Let P be any point of M̃. By Lemma 9.8, P has a neighborhood N in M̃ such that there are at most two edges e ∈ E(T) for which N ∩ U_{e} ∩ f^{−1}(ē) ≠ ∅. By Lemma 9.10 there is a neighborhood N′ of P such that N ∩N′ ≠ ∅ for only a finite number k of triods Y. Since each triod contains only three edges of T, it follows from the definition of the W_{e} that the neighborhood N ∩ N′ of P meets W_{e} for at most 3k + 2 edges e ∈ E(T). This proves local finiteness. (A slightly more careful application of the lemmas above would give a bound of 3 in place of 3k + 2: compare Lemma 9.12 below.)

9.12. Lemma. The plating W = (W_{e})_{e∈E(T)} has order at most 2.

Proof: Let P be any point of M̃. If there is no triod Y ⊂ T such that P ∈ V_{Y}, it follows from Lemma 9.8 that there are at most two edges e ∈ E(T) such that P ∈ W_{e}. On the other hand, if Y ⊂ T is a triod such that P ∈ V_{Y}, then it follows from Lemma 9.9 that the only edges of T for which P can lie in W_{e} are the three edges of Y. Thus in either case there are at most three edges e for which P ∈ W_{e}.

9.13. Now let us set G = (Γ_{e})_{e∈E(T)} for every e ∈ E(T).

Lemma. The triple (E(T), G, W) is a Γ-plating of the Γ-space M̃.

Proof: According to Lemma 9.11, W is a plating of M̃. Thus we need only establish conditions (i)–(iii) of Definition 5.2. Conditions (i) and (ii) are obvious since we have defined G = (Γ_{e})_{e∈E(T)}. Condition (iii) aserts that γ · W_{e} = W_{γ·e} for every e ∈ E(T) and every γ ∈ Γ. To prove this, we first show that γ · U_{e} = U_{γ·e}. From the definition of U_{e} we have

M̃ − γ · U_{e} = γ · ⋃_{\begin{matrix}i∈I \\ e∉A_{T} (X_{i}) \end{matrix}} Z_{i} = ⋃_{\begin{matrix}i∈I \\ e∉A_{T} (X_{i}) \end{matrix}} Z_{γ·i} = ⋃_{\begin{matrix}i∈I \\ e∉A_{T} (X_{γ^{−1}·i}) \end{matrix}} Z_{i} .

But for all i and γ we have A_{T} (X_{γ^{−1}·i}) = A_{T} (γ^{−1}X_{i}γ) = γ^{−1} · A_{T} (X_{i}). Hence

M̃ − γ · U_{e} = ⋃_{\begin{matrix}i∈I \\ e∉γ^{−1}·A_{T} (X_{i}) \end{matrix}} Z_{i} = ⋃_{\begin{matrix}i∈I \\ γ·e∉A_{T} (X_{i}) \end{matrix}} Z_{i} = M̃ − U_{γ·e},

so that γ·U_{e} = U_{γ·e}. A similar naturality argument shows that γ·V_{Y} = V_{γ·Y} for every triod Y ⊂ T and every γ ∈ Γ. The Γ-equivariance of f implies that γ · f^{−1}(e) = f^{−1}(γ · e) for all e and γ. Using the definition of the W_{e} one immediately concludes that γ · W_{e} = W_{γ·e}.

9.14. The next three lemmas will allow us to prove that the Γ-plating (E(T), G, W) is doubly uniform.

9.14.1. Lemma. Let σ be any segment in M̃, and let P be a point in ⋂_{e∈E(σ)} W_{e}. Then there is an index i ∈ I such that P ∈ Z_{i} and σ ⊂ A_{T} (X_{i}).

Proof: If e is any edge of σ, we have P ∈ W_{e}; thus according to the definition of W_{e}, either P ∈ U_{e} ∩ f^{−1}(e), or P ∈ V_{Y} for some triod Y containing e. We first consider the case in which P ∈ U_{e} ∩ f^{−1}(e) for every e ∈ E(σ), so that in particular P ∈ U_{e} ∩ f^{−1}(e) for every e ∈ E(σ). In this case, using 9.3, we choose an index i such that P ∈ int Z_{i}. By the definition of the U_{e} we have e ∈ E(A_{T} (X_{i})) for every edge e of σ; this means that σ ⊂ A_{T} (X_{i}), and the proof is complete in this case. Now consider the case in which there is an edge s of σ such that P ∉ U_{s}, and therefore P ∈ V_{Y} for some triod Y containing s. Let e be any edge of σ. If P ∈ U_{e} ∩ f^{−1}(e), then it follows from the second assertion of Lemma 9.9 that e is an edge of Y. On the other hand, if P ∈ V_{Y′} for some triod Y′ containing e, then the first assertion of Lemma 9.9 implies that Y′ = Y, so that again e is an edge of Y. Thus every edge of σ is an edge of Y. This means that σ ⊂ Y, and hence that σ ⊂ σ′ for some length-2 segment σ′ ⊂ Y. Since P ∈ V_{Y}, the definition of V_{Y} says that for some index i we have A_{T} (X_{i}) ⊃ σ′ ⊃ σ and P ∈ Z_{i}. Thus the proof is complete in this case as well.

9.14.2. For any segment σ ⊂ T we let I_{σ} denote the set of all indices i ∈ I such that σ ⊂ A_{T} (X_{i}). It is clear that I_{σ} is invariant under Γ_{σ}.

Lemma. For any segment σ ⊂ T, the Γ_{σ}-set I_{σ} contains only finitely many Γ_{σ}-orbits.

Proof: By Proposition 5.4, there are only finitely many Γ-orbits in I. Hence there is a finite set S ⊂ I_{σ} such that I_{σ} ⊂ Γ · S. Now let n denote the length of σ, and for any s ∈ S let Σ_{s} denote the set of all segments of length n in A_{T} (X_{s}). Then Σ_{s} is an X_{s}-set, and since the cyclic group X_{s} is T -hyperbolic, Σ_{s} is a finite union of X_{s}-orbits.

For each s ∈ S set Σ′_{s} = Σ_{s} ∩ Γ · {σ}. Since Σ_{s} contains only finitely many X_{s}-orbits, there is a finite set Q_{s} ⊂ Σ′_{s} such that Σ′_{s} ⊂ X_{s} · Q_{s}. Now we have Q_{s} ⊂ Σ′_{s} ⊂ Γ · σ. Hence for each s ∈ S and each ρ ∈ Q_{s}, we may choose an element γ_{ρ} ∈ Γ such that γ_{ρ} · ρ = σ. We have ρ ∈ Σ′_{s} ⊂ Σ_{s} and hence ρ ⊂ A_{T} (X_{s}). Thus σ = γ_{ρ} · ρ ⊂ γ_{ρ} · A_{T} (X_{s}) = A_{T} (γ_{ρ}X_{s}γ_{ρ}^{−1}) = A_{T} (X_{γ_{ρ}·s}). This shows that γ_{ρ} · s ∈ I_{σ}. Thus the finite set

J = ⋃_{s∈S}{γ_{ρ} · s : ρ ∈ Q_{s}}

is a subset of I_{σ}. We shall complete the proof by showing that Γ_{σ} · J = I_{σ}.

Let i ∈ I_{σ} be given. We may write i = γ · s for some s ∈ S and some γ ∈ Γ. We have σ ⊂ A_{T} (X_{i}) and hence γ^{−1} · σ ⊂ A_{T} (X_{s}). Thus γ^{−1} · σ ∈ Σ′_{s}. Thus we may write γ^{−1} · σ = x · ρ for some x ∈ X_{s} and some ρ ∈ Q_{s}. By definition we have γ_{ρ} · ρ = σ. Hence γxγ_{ρ}^{−1} ∈ Γ_{σ}. But since x ∈ X_{s} we have x · s = s. Therefore i = γ · s = (γxγ_{ρ}^{−1})(γ_{ρ} · s).

This shows that i is in the Γ_{σ}-orbit of γ_{ρ} · s, which by definition is an element of J. This completes the proof.

9.14.3. Lemma. For every non-degenerate segment σ ⊂ T, the Γ_{σ}-set W_{σ} = ⋂_{e∈E(σ)} W_{e} is uniform.

Proof: By Lemma 9.14.1, we have W_{σ} ⊂ ⋃_{i∈I_{σ}} Z_{i}. By Lemma 9.14.2, there is a finite set S ⊂ I_{σ} such that Γ_{σ} · S = I_{σ}. Thus W_{σ} ⊂ Γ_{σ} · ⋃_{i∈S} Z_{i}. As ⋂_{e∈E(σ)} W_{e} is Γ_{σ}-invariant, it follows that W_{σ} ⊂ Γ_{σ} ·⋃_{i∈S}(Z_{i} ∩W_{σ}). Hence it suffices to show that Z_{i} ∩W_{σ} is compact for every non-degenerate segment σ ⊂ T and every i ∈ I.

By the hypothesis of Theorem 9.1, Z is uniform. Thus there is a compact set R_{i} ⊂ Z_{i} such that X_{i} · R_{i} = Z_{i}. Let e_{0} be any edge of σ. By Lemma 9.6 we have W_{σ} ⊂ W_{e_{0}} ⊂ f^{−1}(\overline{e_{0}}). Now let x_{i} be a generator of X_{i}. Since x_{i} is T - hyperbolic and f(R) ⊂ T is compact, there is an integer N > 0 such that for every integer n with |n| > N we have x_{i}^{n} · R ∩ \overline{e_{0}} = ∅. It follows that

Z_{i} ∩ W_{σ} ⊂ Z_{i} ∩ f^{−1}(\overline{e}_{0}) ⊂ ⋃_{n=−N}^{N} x_{i}^{n} · R.

Thus Z_{i} ∩ W_{σ} is indeed compact.

9.14.4. Lemma. The Γ-plating (E(T), G, W) is doubly uniform.

Proof: We must show that if e and e′ are edges of T then the (Γ_{e} ∩ Γ_{e′})-set W_{e} ∩ W_{e′} is uniform. If e and e′ have no common vertex, then by Lemma 9.6 we have f(W_{e} ∩ W′_{e}) ⊂ ē ∩ ē′ = ∅; thus in this case, W_{e} ∩ W′_{e} is empty and the assertion follows trivially. If e and e′ have a common vertex, then σ = ē ∪ ē′ is a segment of length 1 or 2. According to Lemma 9.14.3 the Γ_{σ}-set W_{e} ∩W′_{e} is uniform. But Γ_{e} ∩Γ_{e′} is the subgroup of Γ_{σ} consisting of elements that preserve the orientation of σ. Hence Γ_{e} ∩ Γ_{e′} has index at most 2 in Γ_{σ}. It follows by 1.4 that the (Γ_{e} ∩ Γ_{e′})-set W_{e} ∩ W′_{e} is uniform.

9.15. Proof of Theorem 9.1: By Lemmas 9.13 and 9.14.4, (E(T), G, W) is a doubly uniform Γ-plating. By Lemma 9.12, W has order at most 2. By Lemma 9.4, f : M̃ → T is Γ-equivariant. By Lemma 9.6, we have f(W_{e}) ⊂ ē for every edge e of T. By Proposition 8.2 it follows that the core of c is a fibroid.

Section 10. Proof of Theorem A

10.1. Hyperbolic space. The proof of Theorem A will use some elementary properties of hyperbolic space H^{n}.

Recall that an isometry x of H^{n} is loxodromic if x leaves some line A ⊂ H^{n} invariant and x|A is a translation through some strictly positive distance l. The line A and the number l, which are uniquely determined by x, are called the axis and translation length of x; we shall write A=A_{H^{n}}(x) and l = l_{H^{n}}(x).

As in [2] and [3], we shall use the following notation. If x is an isometry of H^{n} and λ is a positive number, we shall denote by Z_{λ}(x) the open subset of H^{n} consisting of all points P such that dist(P, x^{m} · P) < λ for some m ≥ 1. The set Z_{λ}(x) is empty if λ ≤ l_{H^{n}}(x). If λ > l_{H^{n}}(x) then Z_{λ}(x) is a neighborhood of A_{H^{n}}(x).

We denote the closure of Z_{λ}(x) by Z̄_{λ}(x). It is clear that for 0 < λ < λ′ we have Z̄_{λ}(x) < Z̄′_{λ}(x).

10.2. Hyperbolic manifolds. If M is a closed, hyperbolic 3-manifold, we shall identify the universal covering space of M with H^{3}. The group Γ ≅ π_{1}(M) of deck transformations of the universal covering space is a discrete, torsion-free subgroup of the group PSL_{2}(C) of orientation-preserving isometries of H^{3}. Recall that since M is closed, each non-trivial element x of Γ is loxodromic. We may identify M with H^{3}/Γ.

Recall that for each non-trivial element x ∈ Γ, the centralizer of x is the unique maximal cyclic subgroup X of Γ containing x, and that X − {1} consists of all non-trivial elements of Γ having the same axis in H^{3} as x.

Note that since every element of Γ has a cyclic centralizer, π_{1}(M) ≅ Γ has no free abelian subgroup of rank 2. Furthermore, the universal cover H^{3} of M is homeomorphic to R^{3} and hence M is irreducible. Thus M is a simple 3-manifold. If X is any maximal cyclic subgroup of Γ and λ is any positive number, then as in [2] we shall write Z_{λ}(X) = Z_{λ}(x), where x is any generator of X; this is unambiguous because Z_{λ}(x^{−1}) = Z_{λ}(x).

The discreteness of Γ implies that for any P ∈ H^{3} and any λ > 0 there are only finitely many elements x ∈ Γ such that dist(P, x · P) < λ. Since each non-trivial element of Γ lies in a unique maximal cyclic subgroup, it follows that for any P ∈ H^{3} there are only finitely many maximal cyclic subgroups X of Γ such that P ∈ Z_{λ}(X). Thus for any λ the sets Z_{λ}(X), where X ranges over the maximal cyclic subgroups of Γ, form a locally finite family. Since for any λ and any X we have Z̄_{λ}(X) ⊂ Z_{λ+1}(X), it follows that for any λ the sets Z̄_{λ}(X), where X ranges over the maximal cyclic subgroups of Γ, again form a locally finite family.

10.3. Now suppose that we are given an incompressible surface S ⊂ M. Let c be a bi-collared surface with core S (3.1).

The following result, the proof of which will be based on the main theorem of [1], allows one to compare the action of Γ on the tree T (c) with its action on H^{3}.

Proposition. Let M = H^{3}/Γ be a closed, orientable hyperbolic 3-manifold, and let x and y be elements of Γ such that Z_{log 3}(x) ∩ Z_{log 3}(y) ≠ ∅. Suppose that M contains an incompressible bi-collared surface c; set T = T (c), and suppose that x and y are T - hyperbolic. Then E(A_{T} (x)) ∩ E(A_{T} (y)) ≠ ∅.

Proof: Let us fix a point P ∈ Z_{log 3}(x) ∩ Z_{log 3}(y). By the definition of Z_{log 3}(x) and Z_{log 3}(y), there are positive integers r and s such that dist(P, x^{r} ·P) < log 3 and dist(P, y^{s} · P) < log 3.

We set ξ = x^{r}, η = y^{s}, A = A_{T} (x) = A_{T} (ξ) and B = A_{T} (y) = A_{T} (η). We are required to show that E(A) ∩ E(B) ≠ ∅. If ξ and η commute, their axes A and B coincide. Thus we may assume that ξ and η do not commute.

Suppose that E(A) ∩ E(B) = ∅. Then by Proposition 2.6, the group Θ = ⟨ξ, η⟩ is free on ξ and η, and the action of Θ on T is free. Hence by Proposition 3.6, the hyperbolic 3manifold H^{3}/Θ is homeomorphic to the interior of a handlebody. In particular this means that the discrete group Θ ≤ PSL_{2}(C) is non-co-compact, and it is topologically tame in the sense that H^{3}/Θ is homeomorphic to the interior of a compact 3-manifold. Note also that since M is closed and Θ ≤ Γ, every non-trivial element of Θ is loxodromic.

Now we recall the main theorem of [1]. Suppose that ξ and η are non-commuting elements of PSL_{2}(C), and that the group generated by ξ and η is discrete, torsion-free, non-co-compact and topologically tame, and contains no parabolic elements. Then for any point P ∈ H^{3} we have

max(dist(P, ξ · P), dist(P, η · P)) ≥ log 3.

Applied to the elements η and ξ that we have defined, this gives a contradiction.

10.4. Proof of Theorem A: We write M = H^{3}/Γ, where Γ ≤ PSL_{2}(C) is discrete and torsion-free. Since M is closed, all the non-trivial elements of Γ are loxodromic. Let us say that an element x ∈ Γ − {1} is short if its translation length is less than λ. We choose a bi-collared surface c with core S, and we set T = T (c).

Consider the case in which some short element x ∈ Γ has a fixed point in T. Since x is short, its conjugacy class in Γ corresponds to a conjugacy class π_{1}(M) which is represented by a closed geodesic of length < λ. On the other hand, since x has a fixed point in T it follows from 3.1 that the conjugacy class in π_{1}(M) defined by x is also represented by a closed curve in M − |c|. Thus conclusion (i) of Theorem A holds in this case.

Now suppose that no short element of Γ has a fixed point in T. By 2.5 this means that every short element is T -hyperbolic. In this case we shall show that conclusion (ii) of Theorem A holds.

Let us say that a maximal cyclic subgroup X of Γ is short if it has a short generator, and let us denote by X the set of all short maximal cyclic subgroups of Γ. We set Z = (Z̄_{λ}(X))_{X∈X}.

Assume that conclusion (ii) of Theorem A does not hold, i.e. that M contains no hyperbolic ball of radius λ/2. Then by [2, Proposition 3.2] we have

H^{3} = ⋃_{X} Z_{λ}(X),

where X ranges over all maximal cyclic subgroups of Γ. Since Z_{λ}(X) = ∅ when X is not short, this means that

H^{3} = ⋃_{X∈X} Z_{λ}(X).

In particular we have H^{3} = ⋃_{X∈X} Z̄_{λ}(X). As we observed in 10.1, the family Z is a locally finite family, and by definition it consists of closed sets. Thus Z is a plating of H^{3}.

Now as Γ acts on X by conjugation, we may regard X as a Γ-set. If we set G = (X)_{X∈X}, it is clear from Definition 5.2 that (X , G, Z) is a Γ-plating of H^{3}. Since M is compact, it contains only finitely many conjugacy classes of short elements. Hence X is finite modulo Γ. Furthermore, each set Z̄_{λ}(X) is compact modulo X; this follows from [2, Proposition 1.6], which asserts that all the points of Z_{λ}(X) lie within a bounded radius of the axis of X in H^{3}. This shows that the Γ-plating (X , G, Z) is uniform.

We now apply Theorem 9.1 to the uniform Γ-plating (X , G, Z). Hypothesis (i) of 9.1 is immediate from the construction since we are in the case where every short element of Γ is T - hyperbolic. To see that hypothesis (ii) holds, note that by the hypothesis of Theorem A we have λ < log 3, and hence Z_{λ}(X) ⊂ Z_{log 3}(X) for every X ∈ X. Thus if X, X′ ∈ X satisfy Z_{λ}(X)∩Z_{X′} ≠ ∅, then Z_{log 3}(X)∩Z_{log 3}(X′) ≠ ∅, so that E(A_{T} (X))∩E(A_{T} (X′)) ≠ ∅ by Proposition 10.3. It therefore follows from Theorem 9.1 that S is a fibroid. But this contradicts the hypothesis of Theorem A.

Section 11. Proof of the main theorem

11.1. The Main Theorem of the introduction will be proved by combining Theorem A with the results of [3]. We begin with the following result, which is of independent interest and from which the Main Theorem will be deduced by specializing to a suitable value of λ.

Proposition. Let M be a closed, orientable hyperbolic 3-manifold. Let us write M = H^{3}/Γ, where Γ is a discrete, torsion-free group of isometries of H^{3}. Suppose that M contains a non-separating incompressible surface Σ which is not a fibroid. Let λ be a number such that 0 < λ < log 3. Then either

(i) there is an element x ∈ Γ of length < λ such that M contains an open set isometric to Z_{log 3}(x)/⟨x⟩ where x is some loxodromic isometry of H^{3}, or

(ii) M contains a hyperbolic ball of radius λ/2.

Proof: It is enough to show that conclusion (i) of Theorem A implies conclusion (i) of the present proposition. Suppose that some closed curve α : S^{1} → M − Σ is homotopic in M to a non-trivial closed geodesic of length < λ. Let a be an element of the conjugacy class in Γ corresponding to the free homotopy class of α in M. Since α is homotopic to a closed geodesic of length < λ we have length a < λ. On the other hand, since α is carried by the complement of the non-separating surface Σ, the element a lies in the kernel of some homomorphism of Γ onto Z. The centralizer X of a in Γ is infinite cyclic. Let x be a generator of X. Since a is a power of x, the translation length of x is less than λ, and x lies in the kernel of a homomorphism of Γ onto Z. By [3, Proposition 2.2], any two conjugates of x generate a group which is topologically tame and has infinite index in Γ. By [3, Proposition 2.1], it follows that M contains an open set isometric to Z_{log 3}(x)/⟨x⟩.

11.2. Proof of the Main Theorem: We apply Proposition 11.1, taking λ = 0.8. If conclusion (i) holds then it follows from [3, Proposition 4.1] that vol M > 0.34. If conclusion (ii) holds, i.e. if M contains a hyperbolic ball of radius 0.4, then using density estimates for sphere-packings as in [10], one computes that the volume of M is at least 0.35.

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