Relative divergence of finitely generated groups

TL;DR

This paper introduces the concepts of upper and lower relative divergence of finitely generated groups with respect to subgroups, extending previous notions and exploring their behavior in various classes of groups.

Contribution

It generalizes existing divergence concepts by defining relative divergence, allowing for polynomial and exponential functions, and analyzes these in specific group classes.

Findings

Relative divergence can be polynomial or exponential.

Lower divergence can be any polynomial or exponential function.

Analysis of divergence in CAT(0) and relatively hyperbolic groups.

Abstract

We generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten's notion of divergence, and lower relative divergence generalizes a definition of Cooper-Mihalik. While the lower divergence of Cooper-Mihalik can only be linear or exponential, relative lower divergence can be any polynomial or exponential function. In this paper, we examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $\CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups.