Commensurability of lattices in right-angled buildings
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914664795865088 |
|---|---|
| author | Shepherd, Sam |
| author_facet | Shepherd, Sam |
| contents | Let $Γ$ be a graph product of finite groups, with finite underlying graph, and let $Δ$ be the associated right-angled building. We prove that a uniform lattice $Λ$ in the cubical automorphism group Aut$(Δ)$ is weakly commensurable to $Γ$ if and only if all convex subgroups of $Λ$ are separable. As a corollary, any two finite special cube complexes with universal cover $Δ$ have a common finite cover. An important special case of our theorem is where $Γ$ is a right-angled Coxeter group and $Δ$ is the associated Davis complex. We also obtain an analogous result for right-angled Artin groups. In addition, we deduce quasi-isometric rigidity for the group $Γ$ when $Δ$ has the structure of a Fuchsian building. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01210 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Commensurability of lattices in right-angled buildings Shepherd, Sam Group Theory 20F65 (Primary) 20F55, 57M10 (Secondary) Let $Γ$ be a graph product of finite groups, with finite underlying graph, and let $Δ$ be the associated right-angled building. We prove that a uniform lattice $Λ$ in the cubical automorphism group Aut$(Δ)$ is weakly commensurable to $Γ$ if and only if all convex subgroups of $Λ$ are separable. As a corollary, any two finite special cube complexes with universal cover $Δ$ have a common finite cover. An important special case of our theorem is where $Γ$ is a right-angled Coxeter group and $Δ$ is the associated Davis complex. We also obtain an analogous result for right-angled Artin groups. In addition, we deduce quasi-isometric rigidity for the group $Γ$ when $Δ$ has the structure of a Fuchsian building. |
| title | Commensurability of lattices in right-angled buildings |
| topic | Group Theory 20F65 (Primary) 20F55, 57M10 (Secondary) |
| url | https://arxiv.org/abs/2203.01210 |