Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively Hyperbolic Groups

TL;DR

This paper proves that certain geometric structures of relatively hyperbolic groups are preserved under quasi-isometries, leading to invariant decompositions and insights into their automorphism groups.

Contribution

It establishes the quasi-isometry invariance of the coned and cusped spaces, enabling the construction of invariant JSJ-decompositions for relatively hyperbolic groups.

Findings

Quasi-isometry invariance of coned and cusped spaces.

Invariant JSJ-decomposition for relatively hyperbolic groups.

Splitting of the quasi-isometry groups related to these structures.

Abstract

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.