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 $\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}G\le \mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}H+\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}(\ker f)$. In this paper we establish a similar formula for certain quasimorphisms of countable approximate groups: if $(\mathrm{\Xi },{\mathrm{\Xi }}^{\mathrm{\infty }})$ and $(\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\infty }})$ are countable approximate groups and if $f:(\mathrm{\Xi },{\mathrm{\Xi }}^{\mathrm{\infty }})\to (\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\infty }})$ is a symmetric unital quasimorphism, we show that $\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{\Xi }\le \mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{\Lambda }+\mathrm{a}\mathrm{s}\mathrm{d}\mathrm{i}\mathrm{m}(f^{-1}\text{\hspace{0.17em}⁣}(D(f)))$, where $D(f)$ is the defect set of $f$.