Superexponential Dehn functions inside CAT(0) groups

TL;DR

This paper constructs specific high-dimensional CAT(0) groups with subgroups exhibiting superexponential Dehn functions, revealing new complex geometric behaviors within these groups.

Contribution

It introduces the first examples of CAT(0) groups with subgroups having superexponential Dehn functions, broadening understanding of their geometric properties.

Findings

Existence of 4D CAT(0) groups with Dehn functions $ ext{exp}^{(n)}(x^m)$.

Existence of 6D CAT(0) groups with Dehn functions $ ext{exp}^{(n)}(x^eta)$ for dense $eta$.

Expansion of known geometric behaviors of subgroups in CAT(0) groups.

Abstract

We construct 4-dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^m)$ for integers $n, m \geq 1$ and 6-dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^\alpha)$ for integers $n \geq 1$ and $\alpha$ dense in $[1,\infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.