Cannon-Thurston fibers for iwip automorphisms of $F_N$

Abstract

For any atoroidal iwip $ϕ\in Out(F_N)$ the mapping torus group $G_ϕ=F_N\rtimes_ϕ e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_ϕ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_ϕ$. We prove that for any $ϕ$ as above, the map $\hat ι$ is finite-to-one and that the preimage of every point of $\partial G_ϕ$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_ϕ$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ 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 $\partial G_ϕ$ (for the translation action of $F_N$ on $\partial G_ϕ$). By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_ϕ$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$.