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- W180731593 abstract "Let $Phiin Aut(F_N)$ be an atoroidal iwip automorphism of a free group $F_N$, $Nge 3$ and let $M_Phi=F_Nrtimes_Phi $ be the mapping torus group of $Phi$. The group $M_Phi$ is Gromov-hyperbolic and it follows from the work of Mitra that the inclusion $iota:F_Nto M_Phi$ extends to a continuous surjective map $hat iota: partial F_Ntopartial M_Phi$, called the emph{Cannon-Thurston map}. We study the fibers of the map $hat iota$. We prove that for any $Phi$ as above, the map $hat iota$ is finite-to-one and that the preimage of every point of $partial M_Phi$ has cardinality $le 2N$. We also prove that every point $Sin partial M_Phi$ with $ge 3$ preimages in $partial F_N$ is rational and has the form $(xt^m)^infty$ where $xin F_N, mne 0$, and that there are at most $4N-5$ $F_N$-orbits of such points in $partial M_Phi$ (for the translation action of $F_N$ on $partial M_Phi$). We show that, by contrast, for $k=1,2$ there are uncountably many points $Sin partial M_Phi$ with exactly $k$ preimages in $partial F_N$." @default.
- W180731593 created "2016-06-24" @default.
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- W180731593 date "2012-07-15" @default.
- W180731593 modified "2023-09-27" @default.
- W180731593 title "On the fibers of the Cannon-Thurston map for free-by-cyclic groups" @default.
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