Divergence spectra and Morse boundaries of relatively hyperbolic groups

TL;DR

This paper introduces the divergence spectrum as a new invariant for finitely generated groups, compares it with classical divergence notions, and explores its application to relatively hyperbolic groups and their Morse boundaries.

Contribution

It presents the divergence spectrum as a novel quasi-isometry invariant and analyzes its properties and distinctions among relatively hyperbolic groups and Coxeter groups.

Findings

Existence of infinitely many right-angled Coxeter groups with exponential divergence but distinct divergence spectra

Divergence spectrum differs among groups with similar divergence rates

Connections established between Morse boundaries and Bowditch boundaries in relatively hyperbolic groups

Abstract

We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence spectra of relatively hyperbolic groups. We show the existence of an infinite collection of right-angled Coxeter groups which all have exponential divergence but they all have different divergence spectra. We also study Morse boundaries of relatively hyperbolic groups and examine their connection with Bowditch boundaries.