Relatively hyperbolic groups with fixed peripherals

TL;DR

This paper introduces quasi-isometry invariants for relatively hyperbolic groups to identify hyperbolic components, and constructs infinitely many such groups with fixed peripheral structures using small cancellation theory.

Contribution

It develops new invariants for relatively hyperbolic groups and demonstrates the existence of infinitely many quasi-isometry types with fixed peripherals.

Findings

Quasi-isometry invariants detect hyperbolic parts of groups.

Existence of infinitely many quasi-isometry types with fixed peripherals.

Construction of groups using small cancellation over free products.

Abstract

We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any finite collection of finitely generated groups $\mathcal{H}$ each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi--isometry types of one--ended groups which are hyperbolic relative to $\mathcal{H}$. The groups are constructed using small cancellation theory over free products.