A Hurewicz-type theorem for quasimorphisms of countable approximate groups

In their theorem from 2006, A. Dranishnikov and J. Smith prove that if $f:G\to H$ is a group homomorphism, then the following formula for asymptotic dimension is true: $\operatorname{asdim} G \leq \operatorname{asdim} H + \operatorname{asdim} (\ker f)$. This result is known as the Hurewicz-type formula, after a 1927 theorem from classical dimension theory by W. Hurewicz, which inspired it. In this paper we establish a similar formula to the one by Dranishnikov and Smith, for the following setup: whenever $(\Xi, \Xi^\infty)$ and $(\Lambda,\Lambda^\infty)$ are countable approximate groups and $f:(\Xi, \Xi^\infty) \to (\Lambda,\Lambda^\infty)$ is a (general) quasimorphism, i.e., a quasimorphism which need not be symmetric nor unital, then the following formula is true: $$\operatorname{asdim} \Xi \leq \operatorname{asdim} \Lambda + \operatorname{asdim} \left(f^{-1}\left(f(e_\Xi)D(f)^{-1}D(f)\right)\right),$$ where $D(f)$ is the defect set of $f$. It follows as a corollary that if $f:G\to H$ is a quasimorphism of countable groups, then $\operatorname{asdim} G\leq \operatorname{asdim} H + \operatorname{asdim} \left(f^{-1}\left(f(e_\Xi)D(f)^{-1}D(f)\right)\right)$.