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Mathematics > Geometric Topology
Title: On the geometry of the free factor graph for ${\rm{Aut}}(F_N)$
(Submitted on 6 Dec 2023 (v1), last revised 12 Mar 2024 (this version, v2))
Abstract: Let $\Phi$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $\Sigma$ with one boundary component. We show that if $b \in \pi_1(\Sigma)$ is the boundary word, $\phi \in {\rm{Aut}}(\pi_1(\Sigma))$ is a representative of $\Phi$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle \phi, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $\pi_1(\Sigma)$ 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.
Submission history
From: Richard D. Wade [view email][v1] Wed, 6 Dec 2023 14:59:28 GMT (17kb,D)
[v2] Tue, 12 Mar 2024 11:28:46 GMT (17kb,D)
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