On the action of relatively irreducible automorphisms on their train tracks
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916205939392512 |
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| author | Francaviglia, Stefano Martino, Armando Syrigos, Dionysios |
| author_facet | Francaviglia, Stefano Martino, Armando Syrigos, Dionysios |
| contents | Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $ϕ$ with exponential growth (relatively to ${\mathcal G}$). Such set is known to coincide with the minimally displaced set $\operatorname{Min}(ϕ)$ of $ϕ$.
Our main result is that $\operatorname{Min}(ϕ)$ is co-compact, under the action of the cyclic subgroup generated by $ϕ$.
Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of $\operatorname{Min}(ϕ)$ is in uniform distance from $\operatorname{Min}(ϕ^{-1})$. We also prove that the action of $G$ on the product of the attracting and the repelling trees for $ϕ$, is discrete. Finally, we get some fine insight about the local topology of relative outer space.
As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that $\operatorname{Min}(ϕ)$ is quasi-isometric to a line. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_01680 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the action of relatively irreducible automorphisms on their train tracks Francaviglia, Stefano Martino, Armando Syrigos, Dionysios Group Theory Geometric Topology Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $ϕ$ with exponential growth (relatively to ${\mathcal G}$). Such set is known to coincide with the minimally displaced set $\operatorname{Min}(ϕ)$ of $ϕ$. Our main result is that $\operatorname{Min}(ϕ)$ is co-compact, under the action of the cyclic subgroup generated by $ϕ$. Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of $\operatorname{Min}(ϕ)$ is in uniform distance from $\operatorname{Min}(ϕ^{-1})$. We also prove that the action of $G$ on the product of the attracting and the repelling trees for $ϕ$, is discrete. Finally, we get some fine insight about the local topology of relative outer space. As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that $\operatorname{Min}(ϕ)$ is quasi-isometric to a line. |
| title | On the action of relatively irreducible automorphisms on their train tracks |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2108.01680 |