On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913905755815936 |
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| author | Kapovich, Ilya |
| author_facet | Kapovich, Ilya |
| contents | Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in $\partial F_r$ of some abstract algebraic laminations associated with free group automorphisms.
For an exponentially growing outer automorphism $ϕ\in Out(F_r)$ we show that the set of endpoints $\mathcal E_{L}\subseteq \partial F_r$ of any of the \emph{attracting laminations} $L$ of $ϕ$ has Hausdorff dimension $0$ for any tree $T\in cv_r$ and any visual metric on the boundary $\partial T=\partial F_r$. If $ϕ\in Out(F_r)$ is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination $Λ_ϕ$ of $ϕ$ that gets collapsed by the Cannon-Thurston map $\partial F_r\to \partial G_ϕ$ for the associated free-by-cyclic group $G_ϕ=F_r\rtimes_ϕ\mathbb Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups Kapovich, Ilya Group Theory Dynamical Systems Geometric Topology Primary 20F65, Secondary 20F10, 20F67, 37B10, 37D99, 57M99 Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in $\partial F_r$ of some abstract algebraic laminations associated with free group automorphisms. For an exponentially growing outer automorphism $ϕ\in Out(F_r)$ we show that the set of endpoints $\mathcal E_{L}\subseteq \partial F_r$ of any of the \emph{attracting laminations} $L$ of $ϕ$ has Hausdorff dimension $0$ for any tree $T\in cv_r$ and any visual metric on the boundary $\partial T=\partial F_r$. If $ϕ\in Out(F_r)$ is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination $Λ_ϕ$ of $ϕ$ that gets collapsed by the Cannon-Thurston map $\partial F_r\to \partial G_ϕ$ for the associated free-by-cyclic group $G_ϕ=F_r\rtimes_ϕ\mathbb Z$. |
| title | On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups |
| topic | Group Theory Dynamical Systems Geometric Topology Primary 20F65, Secondary 20F10, 20F67, 37B10, 37D99, 57M99 |
| url | https://arxiv.org/abs/2410.02058 |