## Abstract

For any atoroidal iwip ℓ Out(F_{N}), the mapping torus group G_{ℓ} =F_{N}→ _{ℓ} t is hyperbolic, and, by a result of Mitra, the embedding iota: F_{N} hd G_{ℓ} induces a continuous, F_{N}-equivariant and surjective Cannon-Thurston map F_{N} to G_{ℓ}. We prove that for any as above, the map is finite-to-one and that the preimage of every point of G_{ℓ} has cardinality at most 2N. We also prove that every point S in G_{ℓ} with at least three preimages in F_{N} has the form (wt_{m}) where w F_{N}, m ne 0, and that there are at most 4N-5 distinct F_{N}-orbits of such singular points in G_{ℓ} (for the translation action of F_{N} on G_{ℓ}). By contrast, we show that for k=1,2, there are uncountably many points S G_{ℓ} (and thus uncountably many F_{N}-orbits of such S) with exactly k preimages in F_{N}.

Original language | English (US) |
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Pages (from-to) | 203-224 |

Number of pages | 22 |

Journal | Journal of the London Mathematical Society |

Volume | 91 |

Issue number | 1 |

DOIs | |

State | Published - Apr 17 2015 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)

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