Hierarchies of finitely presented groups and applications to geometry
by
Leonid Potyagailo
University of Lille 1
Coauthors: T. Delzant (University of Strasbourg)

In this joint work with Thomas Delzant we define a class C of subgroups of a finitely presented group G which we call elementary. In, particular this can be elementary subgroups (virtually abelian or finite) of a discrete group of isometries of the hyperbolic space or elementary (virtually cyclic or finite) subgroups of a word (Gromov) hyperbolic group etc.

To the group G we associate an invariant c(G) called complexity and we prove that it is strongly decreasing with respect to splittings of G as an amalgamated free product or HNN-extension over elementary subgroups. We deduce from it that the procedure of decompositions of the finitely presented group G over elementary subgroups stops after a finite number of steps. This is called finite hierarchy (or strong accessibility ) of G with respect to elementary subgroups. In a sense it is similar to well-known hierarchies of 3-dimensional Haken manifolds along a system of incompressible surfaces.

We then use hierarchies to describe the class of discrete subgroups of the isometry group of the real hyperbolic n-space (Kleinian groups) which do not contain proper subgroups isomorphic to themselves (so called co-Hopf property).

Date received: June 20, 1999

Copyright © 1999 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 # cacy-44.