Properties of hyperbolic groups: free normal subgroups, quasiconvex subgroups and actions of maximal growth

Abstract

Hyperbolic groups are defined using the analogy between algebraic objects – groups – and hyperbolic metric spaces and manifolds. Our research involves the study and use of two very different, yet very natural, classes of subgroups in a hyperbolic group G: normal subgroups and quasiconvex subgroups. Normal subgroups are embedded “nicely” in G in the classical group theoretic sense, while the quasiconvex subgroups are embedded “hyperbolically” in G as geometric objects.

We prove that if R is a (not necessarily finite) set of words satisfying certain small cancellation condition in a hyperbolic group G then the normal closure of R is free. This result was first presented (for finite set R) by T. Delzant but the proof seems to require some additional argument. New applications are provided, the connection between different small cancellation techniques is studied.

One of our main results concerns the existence of highly transitive actions of non-elementary hyperbolic groups (i.e. the actions which are k–transitive for every natural k) on infinite countable sets. The construction of such examples involves limits of quasiconvex subgroups and some quantitative estimates on maximal growth of action. The main corollary is that almost every group admits a highly transitive action with finite kernel on a countable set. As a side-product of our approach we prove that for a non-elementary hyperbolic group G and a quasiconvex subgroup of infinite index H in G there exists g in G such that <H,g> is quasiconvex of infinite index and is isomorphic to the free product of H and <g> if and only if H and E(G) intersect trivially, where E(G) is the maximal finite normal subgroup of G.