A Hurewicz-type theorem for quasimorphisms of countable approximate groups

Abstract

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 $(\Xi ,{\Xi }^{\infty })$ 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: $$\mathrm{asdim}\Xi \le \mathrm{asdim}\Lambda +\mathrm{asdim}\left(f^{-1}\left(f(e_{\Xi })D(f{)}^{-1}D(f)\right)\right),$$ 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 $\mathrm{asdim}G\le \mathrm{asdim}H+\mathrm{asdim}\left(f^{-1}\left(f(e_{\Xi })D(f{)}^{-1}D(f)\right)\right)$.

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