On distortion of normal subgroups

TL;DR

This paper investigates how the distortion of finitely generated normal subgroups relates to group divergence, providing methods to compute this distortion in complex group structures like graph of groups and right-angled Artin groups.

Contribution

It introduces a method to compute normal subgroup distortion via group decomposition and applies it to various classes of groups, linking subgroup distortion with group divergence.

Findings

Established a connection between subgroup distortion and group divergence.

Developed a decomposition method for computing subgroup distortion.

Applied the method to specific groups like graph of groups and right-angled Artin groups.

Abstract

We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups of $\CAT(0)$ groups introduced by Macura and introduce a collection of normal subgroups of right-angled Artin groups. These groups provide a rich source to study the connection between subgroup distortion and group divergence on $\CAT(0)$ groups.