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arXiv · 2406.19909

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

Vera Tonić
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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 $\mathrm{asdim} G \leq \mathrm{asdim} H + \mathrm{asdim} (\ker f)$. In this paper we establish a similar formula for certain quasimorphisms of countable approximate groups: if $(Ξ, Ξ^\infty)$ and $(Λ, Λ^\infty)$ are countable approximate groups and if $f:(Ξ, Ξ^\infty)\to (Λ,Λ^\infty)$ is a symmetric unital quasimorphism, we show that $\mathrm{asdim} Ξ\leq \mathrm{asdim} Λ+ \mathrm{asdim} (f^{-1}\!(D(f)))$, where $D(f)$ is the defect set of $f$.

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Vera Tonić. 2024-06-28. A Hurewicz-type formula for asymptotic-dimension-lowering symmetric quasimorphisms of countable approximate groups. https://arxiv.org/abs/2406.19909

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