Small cancellation and the Assouad-Nagata dimension of finitely generated groups

Abstract

We prove that the Assouad-Nagata dimension of any finitely generated (but not necessarily finitely presented) $C'(\sfrac{1}{6})$ group is at most 2. Using this result along with techniques of classical small cancellation theory, we construct, for every $k, m, n \in \mathbb N \cup {\infty}$ with $4 \leq k \leq m \leq n$, a finitely generated group with asymptotic dimension $k$ and Assouad-Nagata dimension $m$, which contains a finitely generated subgroup of Assouad-Nagata dimension $n$. This simultaneously answers two previously open questions in asymptotic dimension theory.