A Hurewicz-type theorem for quasimorphisms of countable approximate groups
Vera Tonić
TL;DR
This work generalizes the Hurewicz-type inequality for asymptotic dimension from group homomorphisms to quasimorphisms between countable approximate groups. It builds a coarse-geometric framework showing that quasimorphisms induce coarsely Lipschitz maps on base sets and that the defect set D(f) governs fiberwise dimension control. The main result states $asdim(Xi) ≤ asdim(Lambda) + asdim( f^{-1}( f(e_Xi) D(f)^{-1} D(f) ) )$, yielding a corollary for ordinary groups. Overall, the paper extends dimension-theoretic bounds to non-homomorphic coarse morphisms in approximate algebraic structures, with potential implications in geometric group theory.
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 $(Ξ, Ξ^\infty)$ and $(Λ,Λ^\infty)$ are countable approximate groups and $f:(Ξ, Ξ^\infty) \to (Λ,Λ^\infty)$ is a (general) quasimorphism, i.e., a quasimorphism which need not be symmetric nor unital, then the following formula is true: $$\operatorname{asdim} Ξ\leq \operatorname{asdim} Λ+ \operatorname{asdim} \left(f^{-1}\left(f(e_Ξ)D(f)^{-1}D(f)\right)\right),$$ 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 $\operatorname{asdim} G\leq \operatorname{asdim} H + \operatorname{asdim} \left(f^{-1}\left(f(e_Ξ)D(f)^{-1}D(f)\right)\right)$.