Coherent groups and the perimeter of 2-complexes
by
Daniel Wise
Cornell University
Coauthors: Jon McCammond

A group is called coherent provided that every finitely generated subgroup is finitely presented. A word-hyperbolic group is called locally-quasiconvex if every finitely generated subgroup is quasiconvex.

We have recently developed a method to prove that various groups are coherent. An analagous but more stringent version of our theory shows that many of the same examples are actually locally quasiconvex. Note that every locally quasiconvex group is coherent.

The key proof uses a notion of the perimeter of a map between arbitrary 2-complexes. We have introduced this notion as a generalization of the usual notion of the length of the boundary of a 2-dimensional manifold. One way to describe our main result is as a generalization of the age-old fact that surface groups are coherent.

To date we have applied our method to many of the groups in the following classes:

1. One Relator Groups.
2. Coxeter Groups.
3. 3-Manifold Groups.
4. Small-Cancellation Groups.

It has been a long standing open question whether every one-relator group is coherent. Using our techniques, we are able to show that every group of the form <a₁ ... a_r | Wⁿ > is coherent so long as n >= |W|.

We show that a Coxeter group is coherent provided that for each two generators, the exponent of their product is at least one and a half times the number of generators. Note that there do exist incoherent Coxeter groups; the incoherent group F₂ ×F₂ embeds in many Coxeter groups with low exponents.

Using the same methods we are able to reprove the coherence and local quasiconvexity of many 3-manifold groups. The original proof that all 3-manifold groups are coherent is due to P. Scott. Finally, we can show that many small cancellation groups are coherent. This is the area which provides the richest application of our theory. Roughly speaking, we show that if a presentation has long relators which are sufficiently spread out among the generators then the group is coherent. We note that there do exist incoherent small-cancellation groups (Rips).

As a corollary of the proof, we find that if a complex X satisfies our perimeter condition, then for every finitely generated subgroup of \pi₁X , its corresponding based cover [^X] has a compact core. Another by-product of our main theorem is an algorithm for producing the finite presentation associated with a particular finite generating set. Specifically, the presentation for the subgroup generated by closed paths g₁, g₂, ... , g_n can be found in quadratic time in the total length of the generators g_i.

Date received: April 16, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabb-25.