Aperiodicity properties of automorphism groups of free products
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911358732206080 |
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| author | Guerch, Yassine |
| author_facet | Guerch, Yassine |
| contents | Let $G=G_1 \ast \ldots \ast G_k \ast F_N$ be a free product of finitely presented groups, where $F_N$ is a free group of rank $N \in \mathbb{N}$. Let $\mathrm{Out}(G,\mathcal{G})$ be the subgroup of $\mathrm{Out}(G)$ preserving the set of conjugacy classes $\mathcal{G}=\{[G_1],\ldots,[G_k]\}$. Under natural conditions on the groups $G_i$ with $i \in \{1,\ldots,k\}$, we prove that the group $\mathrm{Out}(G,\mathcal{G})$ has a finite index subgroup $\mathrm{IA}(G,\mathcal{G},3)$ with notable aperiodicity properties. We show that the group $\mathrm{IA}(G,\mathcal{G},3)$ is torsion free and, if $ϕ\in \mathrm{IA}(G,\mathcal{G},3)$, every $ϕ$-periodic conjugacy class of elements of $G$ is in fact fixed by $ϕ$ and every $ϕ$-periodic conjugacy class of free factors of $G$ is fixed by $ϕ$.
As an application, we prove that, for every toral relatively hyperbolic group $G$, the group $\mathrm{Out}(G)$ has a finite index subgroup $\mathrm{IA}(G,3)$ with the same above mentioned aperiodicity properties. We in particular give another proof of the theorem, due to Handel-Mosher, that the kernel of the action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$ satisfies natural aperiodicity properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03947 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Aperiodicity properties of automorphism groups of free products Guerch, Yassine Group Theory Geometric Topology 20E36, 20F65, 20F28, 20E08 Let $G=G_1 \ast \ldots \ast G_k \ast F_N$ be a free product of finitely presented groups, where $F_N$ is a free group of rank $N \in \mathbb{N}$. Let $\mathrm{Out}(G,\mathcal{G})$ be the subgroup of $\mathrm{Out}(G)$ preserving the set of conjugacy classes $\mathcal{G}=\{[G_1],\ldots,[G_k]\}$. Under natural conditions on the groups $G_i$ with $i \in \{1,\ldots,k\}$, we prove that the group $\mathrm{Out}(G,\mathcal{G})$ has a finite index subgroup $\mathrm{IA}(G,\mathcal{G},3)$ with notable aperiodicity properties. We show that the group $\mathrm{IA}(G,\mathcal{G},3)$ is torsion free and, if $ϕ\in \mathrm{IA}(G,\mathcal{G},3)$, every $ϕ$-periodic conjugacy class of elements of $G$ is in fact fixed by $ϕ$ and every $ϕ$-periodic conjugacy class of free factors of $G$ is fixed by $ϕ$. As an application, we prove that, for every toral relatively hyperbolic group $G$, the group $\mathrm{Out}(G)$ has a finite index subgroup $\mathrm{IA}(G,3)$ with the same above mentioned aperiodicity properties. We in particular give another proof of the theorem, due to Handel-Mosher, that the kernel of the action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$ satisfies natural aperiodicity properties. |
| title | Aperiodicity properties of automorphism groups of free products |
| topic | Group Theory Geometric Topology 20E36, 20F65, 20F28, 20E08 |
| url | https://arxiv.org/abs/2601.03947 |