Subgroup structure of word-hyperbolic groups
by
Ilya Kapovich
University of Illinois at Urbana-Champaign
Coauthors: Richard Weidmann (Bochum University, Germany)

We generalize a claim from M.Gromov's book "Hyperbolic Groups" that in a torsion-free word-hyperbolic group G any k-generated non-free subgroup contains an element of ßhort" conjugacy length, where the ßhortness" constant depends only on k and G.

More precisely, we prove the following:

Theorem Let G be a word-hyperbolic group (not necessarily torsion-free) with a finite generating set X such that the Cayley graph \Gamma(G, X) is \delta-hyperbolic. Then for any integer k >= 1 there exists a constant C=C(k, \delta) > 0 with the following property.

Let M=(g₁, ..., g_k) be a k-tuple of elements of G. Let H be the subgroup of G generated by M. Then either H is free on M and quasiconvex in G or M is Nielsen-equivalent to a k-tuple M'=(f₁, ..., f_k) where the element f₁ is conjugate in G to an element of length at most C(k, \delta).

This shows that in a non-free k-generated subgroup the conjugacy length of the fist generator in a generating k-tuple can be made small. Further generalizations of this statement will also be discussed.

Date received: September 28, 2000

Copyright © 2000 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 Conferences Inc. Document # cafm-11.