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

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 $f:(\mathrm{\Xi },{\mathrm{\Xi }}^{\mathrm{\infty }})\to (\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\infty }})$, stating that $\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{\Xi }\le \mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{\Lambda }+\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}([\mathrm{k}\mathrm{e}\mathrm{r}f{]}_c)$. 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 $G$, $\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}G$ is equal to the supremum of asymptotic dimensions of finitely generated subgroups of $G$. Our version states that, if $(\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\infty }})$ is a countable approximate group, then $\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{\Lambda }$ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of ${\mathrm{\Lambda }}^{\mathrm{\infty }}$, with these approximate subgroups contained in ${\mathrm{\Lambda }}^2$.