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

Abstract

We prove that for all $k,m,n\in \mathbb{N}\cup \{\infty \}$ with $4\le k\le m\le n$, 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.

33 pages, 7 figures This paper is a sequel to "Assouad-Nagata dimension of C'(1/6) groups" (1909.06646). Some of the results from the previous paper have been shifted to this one, since their proofs use similar techniques. The author believes this reorganization will make both papers easier to read and greatly reduce their combined length