A Hurewicz-type formula for asymptotic-dimension-lowering symmetric quasimorphisms of countable approximate groups
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| Format: | Preprint |
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2024
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| _version_ | 1866910506456973312 |
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| author | Tonić, Vera |
| author_facet | Tonić, Vera |
| contents | 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 $f$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hurewicz-type formula for asymptotic-dimension-lowering symmetric quasimorphisms of countable approximate groups Tonić, Vera Group Theory Metric Geometry 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 $f$. |
| title | A Hurewicz-type formula for asymptotic-dimension-lowering symmetric quasimorphisms of countable approximate groups |
| topic | Group Theory Metric Geometry |
| url | https://arxiv.org/abs/2406.19909 |