Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups

Abstract

We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups $f:(\Xi, \Xi^\infty) \to (\Lambda, \Lambda^\infty)$, stating that $\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda +\mathrm{asdim} ([\mathrm{ker} f]_c)$. This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group $G$, $\mathrm{asdim} G$ is equal to the supremum of asymptotic dimensions of finitely generated subgroups of $G$. Our version states that, if $(\Lambda, \Lambda^\infty)$ is a countable approximate group, then $\mathrm{asdim} \Lambda$ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of $\Lambda^\infty$, with these approximate subgroups contained in $\Lambda^2$.Comment: The results in this paper were previously contained in the monograph titled "Foundations of geometric approximate group theory", by M. Cordes and the two authors listed here, but they had to be taken out of that monograph in order to shorten it. arXiv admin note: substantial text overlap with arXiv:2012.1530