A Hurewicz-type formula for asymptotic-dimension-lowering symmetric quasimorphisms of countable approximate groups

A well-known Hurewicz-type formula for asymptotic-dimension-lowering group homomorphisms, due to A. Dranishnikov and J. Smith, states that if $f:G\to H$ is a group homomorphism, then $\mathrm{asdim} G \leq \mathrm{asdim} H + \mathrm{asdim} (\ker f)$. In this paper we establish a similar formula for certain quasimorphisms of countable approximate groups: if $(\Xi, \Xi^\infty)$ and $(\Lambda, \Lambda^\infty)$ are countable approximate groups and if $f:(\Xi, \Xi^\infty)\to (\Lambda,\Lambda^\infty)$ is a symmetric unital quasimorphism, we show that $\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda + \mathrm{asdim} (f^{-1}\!(D(f)))$, where $D(f)$ is the defect set of $f$.