Publication Date:
2015-01-20
Description:
For any atoroidal iwip $\varphi \in {{\rm Out}}({F_N}),$ the mapping torus group $G_ \varphi =F_N\rtimes _ \varphi \langle t\rangle$ is hyperbolic, and, by a result of Mitra, the embedding $\iota : {F_N}{\mathop {\longrightarrow }\limits ^{\lhd }} G_ \varphi$ induces a continuous, ${F_N}$ -equivariant and surjective Cannon–Thurston map $\widehat \iota : \partial {F_N}\to \partial G_ \varphi$ . We prove that for any $\varphi$ as above, the map $\widehat \iota$ is finite-to-one and that the preimage of every point of $\partial G_ \varphi$ has cardinality at most $2N$ . We also prove that every point $S\in \partial G_ \varphi$ with at least three 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 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 $\partial F_N$ .
Print ISSN:
0024-6107
Electronic ISSN:
1469-7750
Topics:
Mathematics
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