A Hurewicz-type theorem for quasimorphisms of countable approximate groups

TL;DR

This paper extends the classical Hurewicz theorem to quasimorphisms between countable approximate groups, providing a new inequality relating their asymptotic dimensions and the defect set of the quasimorphism.

Contribution

It establishes a Hurewicz-type formula for asymptotic dimension in the context of quasimorphisms between countable approximate groups, generalizing previous results for group homomorphisms.

Findings

Derived a formula relating asymptotic dimensions of approximate groups via quasimorphisms.

Showed that the inequality holds for countable groups as a special case.

Connected the asymptotic dimension of the domain to that of the codomain and the defect set.

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: $\operatorname{asdim} G \leq \operatorname{asdim} H + \operatorname{asdim} (\ker f)$. 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: $$\operatorname{asdim} \Xi \leq \operatorname{asdim} \Lambda + \operatorname{asdim} \left(f^{-1}\left(f(e_\Xi)D(f)^{-1}D(f)\right)\right),$$ 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 $\operatorname{asdim} G\leq \operatorname{asdim} H + \operatorname{asdim} \left(f^{-1}\left(f(e_\Xi)D(f)^{-1}D(f)\right)\right)$.