A Hurewicz-type theorem for quasimorphisms of countable approximate groups
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914135634083840 |
|---|---|
| author | Tonić, Vera |
| author_facet | Tonić, Vera |
| contents | 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 $\operatorname{asdim} G\leq \operatorname{asdim} H + \operatorname{asdim} \left(f^{-1}\left(f(e_Ξ)D(f)^{-1}D(f)\right)\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Hurewicz-type theorem for quasimorphisms of countable approximate groups Tonić, Vera Group Theory Metric Geometry 51F30, 20F69 (Primary), 20N99 (Secondary) 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 $\operatorname{asdim} G\leq \operatorname{asdim} H + \operatorname{asdim} \left(f^{-1}\left(f(e_Ξ)D(f)^{-1}D(f)\right)\right)$. |
| title | A Hurewicz-type theorem for quasimorphisms of countable approximate groups |
| topic | Group Theory Metric Geometry 51F30, 20F69 (Primary), 20N99 (Secondary) |
| url | https://arxiv.org/abs/2511.02522 |