Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity

TL;DR

This paper investigates the structure of relatively hyperbolic groups, showing their decomposition into star graphs with finitely generated central parts, and explores Floyd quasiconvexity, establishing properties of Floyd quasigeodesics and parabolic subgroups.

Contribution

It proves that relatively hyperbolic groups split as star graphs with finitely generated central groups and analyzes Floyd quasiconvexity using topological entourages and Floyd metrics.

Findings

Relatively hyperbolic groups split as star graphs with finitely generated central vertex groups.

Floyd quasigeodesics are tight and parabolic subgroups are Floyd quasiconvex.

Preimages of parabolic points under Floyd maps correspond to Floyd boundaries of stabilizers.

Abstract

The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-cocompact action by homeomorphisms on a compactum is finitely generated with respect to a system of parabolic subgroups. The second part essentially uses the methods of topological entourages developed in the first part. Using also Floyd metrics we obtain finer properties of finitely generated relatively hyperbolic groups. We show that there is a system of "tight" curves satisfying the property of horospherical quasiconvexity. We then prove that the Floyd quasigeodesics are tight and so the parabolic subgroups of $G$ are quasiconvex with respect to the Floyd metrics. As a corollary we obtain that the preimage of a parabolic point by the Floyd map is the Floyd boundary of its stabilizer.