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:(Ξ, Ξ^∞) → (Λ, Λ^∞), stating that asdim Ξ ≤ asdim Λ +asdim ([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, asdim G is equal to the supremum of asymptotic dimensions of finitely generated subgroups of G. Our version states that, if (Λ, Λ^∞) is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of Λ^∞, with these approximate subgroups contained in Λ².