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: $\mathrm{asdim}G\le \mathrm{asdim}H+\mathrm{asdim}(\ker f)$. 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 $(\mathrm{\Xi },{\mathrm{\Xi }}^{\mathrm{\infty }})$ and $(\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\infty }})$ are countable approximate groups and $f:(\mathrm{\Xi },{\mathrm{\Xi }}^{\mathrm{\infty }})\to (\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\infty }})$ is a (general) quasimorphism, i.e., a quasimorphism which need not be symmetric nor unital, then the following formula is true: $$\mathrm{asdim}\mathrm{\Xi }\le \mathrm{asdim}\mathrm{\Lambda }+\mathrm{asdim}\left(f^{-1}(f(e_{\mathrm{\Xi }})D(f{)}^{-1}D(f))\right),$$ 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 $\mathrm{asdim}G\le \mathrm{asdim}H+\mathrm{asdim}\left(f^{-1}(f(e_{\mathrm{\Xi }})D(f{)}^{-1}D(f))\right)$.