Cannon-Thurston fibers for iwip automorphisms of $F_N$

Abstract

For any atoroidal iwip $\varphi \in Out(F_N)$ the mapping torus group $G_{\varphi }=F_N{⋊}_{\varphi }<t>e$ is hyperbolic, and the embedding $\iota :F_N\stackrel{⊲}{⟶}{G}_{\varphi }$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat{\iota }:\partial F_N\to \partial G_{\varphi }$. We prove that for any $\varphi$ as above, the map $\hat{\iota }$ is finite-to-one and that the preimage of every point of $\partial G_{\varphi }$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_{\varphi }$ with $\ge 3$ preimages in $\partial F_N$ has the form $(w{t}^m{)}^{\infty }$ where $w\in F_N,m\ne 0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\partial G_{\varphi }$ (for the translation action of $F_N$ on $\partial G_{\varphi }$). By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_{\varphi }$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$.

final version; to appear in J. London Math. Soc.; 22 pages

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• Cubulating hyperbolic free-by-cyclic groups: the irreducible case
• Dynamics of Irreducible Endomorphisms of $F_n$
• Floyd maps to the boundaries of relatively hyperbolic groups
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Cited by in corpus (2)

• Invariant laminations for irreducible automorphisms of free groups
• On a Theorem of Scott and Swarup