Relative Dehn fuctions, hyperbolically embedded subgroups and combination theorems

TL;DR

This paper investigates how hyperbolically embedded subgroups and relative Dehn functions behave under group splittings, providing new combination theorems with bounds that generalize previous results.

Contribution

It establishes new combination theorems for hyperbolically embedded subgroups and relative Dehn functions in groups split as graphs of groups, relaxing previous assumptions.

Findings

Provides bounds for relative Dehn functions based on vertex groups.

Extends combination theorems to non-finitely generated edge groups.

Generalizes and improves existing results in the literature.

Abstract

Consider the following classes of pairs consisting of a group and a finite collection of subgroups: $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right\}$ and $ \mathcal{D}= \left\{ (G,\mathcal H) \mid \text{the relative Dehn function of $(G,\mathcal H)$ is well-defined} \right\}.$ Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a finite collection of subgroups $\mathcal{H}$ of $G$ such that: $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$, then $(G,\mathcal{H})$ is in $\mathcal C$. $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$, then $(G,\mathcal{H})$ is in $\mathcal D$. $\bullet$ For any vertex $v$ and for any $g\in G_v$, the element $g$ is conjugate to an element in some $Q\in\mathcal{H}_v$ if and only if $g$ is conjugate to an element in some $H\in\mathcal{H}$. That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.