I study the algebra, geometry, and topology of hyperbolic 3-manifolds, and related subjects like character and representation varieties, hyperbolic groups and geometric flows.
We construct a family of hyperbolic link complements by gluing tangle complements along totally geodesic four-punctured spheres, and investigate the commensurability relation among its members. Each has trace field Q(i,\sqrt{2}). Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of manifolds with the same volume and scissors congruence class. Certain mutants are incommensurable, distinguished by cusp parameters. Others are non-isometric but commensurable. This is related to the presence of hidden symmetries. There are examples with integral and nonintegral traces.
If M is an orientable hyperbolic 3-manifold of finite volume, the family of finite covers of M associated to a map from \pi_1 M to Z has rank gradient 0 if and only if the Poincare-Lefschetz dual of the class in H^1(M) representing this map is represented by a fiber. This generalizes a theorem of M. Lackenby. If M is closed, we give an explicit lower bound for the rank gradient. The proof uses an acylindrical accessibility theorem due to R. Weidmann and the following result: if M is a closed, orientable hyperbolic 3-manifold and S is a connected incompressible surface that is not a fiber or semi-fiber of M, then the \pi_1 M action on the tree determined by S is (14g-12)-acylindrical, where g is the genus of S.
The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-metric" and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds, and integral convergence to hyperbolic for the metrics under consideration.
Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise. A variety of attractive properties follow: such manifolds are virtually fibered; their fundamental groups are LERF; and their geometrically finite subgroups are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We also provide examples that are not commensurable to any reflection group in hyperbolic 3-space.
If a compact, orientable hyperbolic manifold N with connected, totally geodesic boundary of genus 2 has Heegaard genus at least 5, its volume is greater than 6.89. This extends work of Kojima-Miyamoto. The following dichotomy is fundamental to the proof: either the shortest "return path" of N (defined by K-M) is long, or N has an embedded, codimension-0 submanifold X, which is not a book of I-bundles, with two incompressible boundary components, one the boundary of N. We apply this result to prove a theorem about closed manifolds extending the results of "Incompressible surfaces, hyperbolic volume, Heegaard genus, and homology."
An old theme in the study of hyperbolic 3-manifolds asserts that their volume tracks their topological complexity. We give evidence supporting this theme along the following lines: if M is a closed, orientable hyperbolic 3-manifold such that H_1(M:Z_2) has dimension at least 4, and if the image in H^2(M;Z_2) of the cup product map has dimension at most 1, then the volume of M is greater than 3.08.
The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with a fiber surface of genus 19. We characterize arithmeticity among the doubles and certain twisted doubles of the tripus and tetrus, and point out some consequences regarding towers of covers.
I give examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
We show that faithful representations exist in all topological components of the PSL_2(K) character variety of a surface group, where K=C or R, and in certain components of the PU(2,1) character variety. This answers a question of W. Goldman.
Chapter 1: background and a version of the Poincare polyhedron theorem for manifolds with totally geodesic boundary.
Chapter 2: computations of algebraic invariants of Kleinian groups associated to certain manifolds with totally geodesic boundary, including Thurston's Tripus and a family constructed by R. Frigerio.
Chapter 3: essentially the contents of "Totally geodesic surfaces and homology" above.
Chapter 4: questions, conjectures, and directions for future research.