Today I went to U. of Chicago to see a talk by Lee
Mosher on train tracks. He is writing a monograph (Train track expansions
of measured foliations, available at his web site) on train tracks and
the solution to the conjugacy problem in the mapping class group, which
is already at 297 pages, even though he hasn't started describing his solution
to the conjugacy problem. His thesis and early papers were on the topic
of the conjugacy problem, and he has made some important contributions
to understanding the combinatorial structure of the mapping class groups,
namely that they have an automatic structure. In his talk, he explained
how to determine if a measured foliation is arational, which
is in a precise way analogous to an irrational number (this generalizes
the case of measured foliations on the torus, which are parametrized by
irrational numbers - see the introduction to his monograph for definitions).
He then gave a nice example of a particular set of train tracks, and how
to determine whether laminations carried by them are arational, by discussing
a sequence of "left" and "right" splittings. Then the lamination would
be described by a sequence LRLLRRRRLLLL... . The lamination is arational
iff this sequence forms a regular language, one accepted by a certain finite-state
automaton. Later, I'll try to draw a picture of his example.
(added 2/28/03: here's Mosher's
example added 3/13/03: Lee tells me that his example is incomplete,
so reader beware. The automaton for recognizing arationality is more complicated
than described.)
One interesting thing mentioned by Lee at lunchtime is that the number of conjugacy classes of a matrix element in SL(2,Z) is the class number of the quadratic field generated over Q by the eigenvalues of the matrix. Roger Alperin said that this follows by taking the eigenvectors of the matrix as a basis of an ideal, and somehow one can show that the shape of this ideal as a lattice in R^2 (its ideal class) is a complete invariant of the conjugacy class. This seems similar to the fact that the number of cusps of the Bianchi groups PSL(2,O_d), where O_d is the ring of integers in the field (-d)^(1/2), is the class number of the field. I'll try to understand this and explain it at a later date.
Lee told me that one day he came into Thurston's office, and Thurston told him that he could solve the word problem in groups by using grep, a Unix command that searches for repeated patterns in some text. Apparently, the computer creates a little automaton which searches for the pattern. Thurston and others developed a theory of automatic groups, groups for which any element could be written in a special normal form in which multiplications by generators preserving normal form could be recognized by a finite state automaton. This is typical of Thurston, where he understands a problem by computing or dissecting a special case, and then creates a general theory based on this detailed knowledge. Lee Mosher also seems to do math this way, and likes to give illuminating examples in his talks.