Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912830802886656 |
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| author | Gaifullin, Alexander A. |
| author_facet | Gaifullin, Alexander A. |
| contents | Let $Σ_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$ Gaifullin, Alexander A. Group Theory Geometric Topology 20F28, 20F34 (Primary), 11H56, 20E05, 20F18, 57M07 (Secondary) Let $Σ_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group. |
| title | Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$ |
| topic | Group Theory Geometric Topology 20F28, 20F34 (Primary), 11H56, 20E05, 20F18, 57M07 (Secondary) |
| url | https://arxiv.org/abs/2507.20710 |