Prosolvable rigidity of surface groups
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917601347633152 |
|---|---|
| author | Jaikin-Zapirain, Andrei Morales, Ismael |
| author_facet | Jaikin-Zapirain, Andrei Morales, Ismael |
| contents | Surface groups are known to be the Poincaré Duality groups of dimension two since the work of Eckmann, Linnell and Müller. We prove a prosolvable analogue of this result that allows us to show that surface groups are profinitely (and prosolvably) rigid among finitely generated groups that satisfy $\mathrm{cd}(G)=2$ and $b_2^{(2)}(G)=0$. We explore two other consequences.
On the one hand, we derive that if $u$ is a surface word of a finitely generated free group $F$ and $v\in F$ is measure equivalent to $u$ in all finite solvable quotients of $F$ then $u$ and $v$ belong to the same $\mathrm{Aut}(F)$-orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let $F$ be a free group of rank $n\geq 3$ and let $w\in F$. Suppose that $G=F/\langle\!\langle w\rangle\!\rangle$ is a residually finite group all of whose finite-index subgroups are one-relator groups. Then $G$ is 2-free. Moreover, we show that if $H^2(G; \mathbb{Z})\neq 0$ then $G$ must be a surface group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12293 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prosolvable rigidity of surface groups Jaikin-Zapirain, Andrei Morales, Ismael Group Theory 20J06, 20E18 (Primary), 16K40, 20C07 (Secondary) Surface groups are known to be the Poincaré Duality groups of dimension two since the work of Eckmann, Linnell and Müller. We prove a prosolvable analogue of this result that allows us to show that surface groups are profinitely (and prosolvably) rigid among finitely generated groups that satisfy $\mathrm{cd}(G)=2$ and $b_2^{(2)}(G)=0$. We explore two other consequences. On the one hand, we derive that if $u$ is a surface word of a finitely generated free group $F$ and $v\in F$ is measure equivalent to $u$ in all finite solvable quotients of $F$ then $u$ and $v$ belong to the same $\mathrm{Aut}(F)$-orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let $F$ be a free group of rank $n\geq 3$ and let $w\in F$. Suppose that $G=F/\langle\!\langle w\rangle\!\rangle$ is a residually finite group all of whose finite-index subgroups are one-relator groups. Then $G$ is 2-free. Moreover, we show that if $H^2(G; \mathbb{Z})\neq 0$ then $G$ must be a surface group. |
| title | Prosolvable rigidity of surface groups |
| topic | Group Theory 20J06, 20E18 (Primary), 16K40, 20C07 (Secondary) |
| url | https://arxiv.org/abs/2312.12293 |