Asymptotic and Assouad-Nagata dimension of finitely generated groups and their subgroups

TL;DR

This paper constructs specific finitely generated groups and subgroups to demonstrate the possible combinations of asymptotic and Assouad-Nagata dimensions, resolving two open questions in the field.

Contribution

It provides explicit examples of groups with prescribed asymptotic and Assouad-Nagata dimensions, answering previously open questions.

Findings

Existence of finitely generated groups with prescribed dimensions

Construction of subgroups with higher Assouad-Nagata dimension

Resolution of open questions in asymptotic dimension theory

Abstract

We prove that for all $k,m,n \in \mathbb N \cup \{\infty\}$ with $4 \leq k \leq m \leq n$, there exists a finitely generated group $G$ with a finitely generated subgroup $H$ such that the asymptotic dimension of $G$ is $k$, the Assouad-Nagata dimension of $G$ is $m$, and the Assouad-Nagata dimension of $H$ is $n$. This simultaneously answers two open questions in asymptotic dimension theory.