Divergence of finitely presented subgroups of CAT(0) groups

TL;DR

This paper constructs specific CAT(0) groups with finitely presented subgroups exhibiting a wide range of divergence behaviors, including polynomial and logarithmic divergence, and explores properties of contracting elements within these groups.

Contribution

It introduces new constructions of CAT(0) groups with diverse divergence functions and analyzes contracting elements in finitely presented subgroups.

Findings

Existence of CAT(0) groups with divergence functions of the form r^α for dense α in [2,∞)

Construction of CAT(0) groups with divergence r^q log(r) for integers q ≥ 2

Identification of contracting elements not contracting in certain finitely presented subgroups

Abstract

We construct families of $CAT(0)$ groups containing finitely presented groups whose divergence functions are of the form $r^\alpha$ for a dense set of exponents $\alpha \in [2,\infty)$ and $r^q\log(r)$ for integers $q \geq 2$. The same construction also yields examples of $CAT(0)$ groups containing contracting elements which are not contracting in certain finitely presented subgroups.