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

Abstract

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 $\mathrm{asdim}G\le \mathrm{asdim}H+\mathrm{asdim}(\ker f)$. 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 $\mathrm{asdim}\Xi \le \mathrm{asdim}\Lambda +\mathrm{asdim}(f^{-1}(D(f)))$, where $D(f)$ is the defect set of $f$.

arXiv comment11 pages. arXiv admin note: text overlap with arXiv:2404.01806