On the geometry of the free factor graph for ${\rm{Aut}}(F_N)$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911295557599232 |
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| author | Bestvina, Mladen Bridson, Martin R. Wade, Richard D. |
| author_facet | Bestvina, Mladen Bridson, Martin R. Wade, Richard D. |
| contents | Let $Φ$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $Σ$ with one boundary component. We show that if $b \in π_1(Σ)$ is the boundary word, $ϕ\in {\rm{Aut}}(π_1(Σ))$ is a representative of $Φ$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle ϕ, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $π_1(Σ)$ are quasi-isometrically embedded. It follows that for $N \geq 2$ the free factor graph for ${\rm{Aut}}(F_N)$ is not hyperbolic, in contrast to the ${\rm{Out}}(F_N)$ case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_03535 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the geometry of the free factor graph for ${\rm{Aut}}(F_N)$ Bestvina, Mladen Bridson, Martin R. Wade, Richard D. Geometric Topology Group Theory 20F65, 20E05 (Primary) 20E36, 51F30 (Secondary) Let $Φ$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $Σ$ with one boundary component. We show that if $b \in π_1(Σ)$ is the boundary word, $ϕ\in {\rm{Aut}}(π_1(Σ))$ is a representative of $Φ$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle ϕ, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $π_1(Σ)$ are quasi-isometrically embedded. It follows that for $N \geq 2$ the free factor graph for ${\rm{Aut}}(F_N)$ is not hyperbolic, in contrast to the ${\rm{Out}}(F_N)$ case. |
| title | On the geometry of the free factor graph for ${\rm{Aut}}(F_N)$ |
| topic | Geometric Topology Group Theory 20F65, 20E05 (Primary) 20E36, 51F30 (Secondary) |
| url | https://arxiv.org/abs/2312.03535 |