Peripheral structures of relatively hyperbolic groups

TL;DR

This paper introduces parabolically extended structures for relatively hyperbolic groups, characterizes relative quasiconvexity using dynamical methods, and explores applications including Floyd boundary actions.

Contribution

It presents a new class of structures for relatively hyperbolic groups and provides a dynamical characterization of relative quasiconvexity, advancing understanding of boundary actions.

Findings

Groups acting geometrically finitely on Floyd boundaries are well-understood.

Dunwoody's inaccessible group does not act geometrically finitely on its Floyd boundary.

Abstract

In this paper, we introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained using dynamical methods. Some applications are discussed. The class of groups acting geometrically finitely on Floyd boundaries turns out to be easily understood. However, we also show that Dunwoody's inaccessible group does not act geometrically finitely on its Floyd boundary.