Martin Lustig – Cannon-Thurston pour Out(F_N)

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Date(s) - 15 février 2013

Catégories Pas de Catégories

(joint with I. Kapovich). For any hyperbolic fully irreducible (= iwip) automorphism $\phi \in \text{Out}(F_N)$ the mapping torus group $G_\phi$ of $F_N$ by $\phi$ is hyperbolic, and the embedding $\iota: F_N ;\triangleleft; G_\phi$ induces a continuous, $F_N$-equivariant and surjective Cannon-Thurston map $\hat \iota: \partial F_N \to \partial G_\phi$. We prove that for any $\phi$ as above, the map $\hat \iota$ is finite-to-one and that the preimage of every point of $\partial G_\phi$ has cardinality $\leq 2N$. We also prove that every point $S\in \partial G_\phi$ with $\geq 3$ preimages in $\partial F_N$ is rational and has the form $(wt^m)^\infty$ where $w\in F_N, m\neq 0$, $t$ is the stable letter representing $\phi$, and that there are at most $N-5$ $F_N$-orbits of such points in $\partial G_\phi$ (for the translation action of $F_N$ on $\partial G_\phi$). We show that, by contrast, for $k=1,2$ there are uncountably many points $S\in \partial G_\phi$ with exactly $k$ preimages in $\partial F_N$.