Cannon-Thurston fibers for iwip automorphisms of FN

Abstract

For any atoroidal iwip φ ∈ Out ( F N ) , the mapping torus group G φ = F N ⋊ φ 〈 t 〉 is hyperbolic, and, by a result of Mitra, the embedding ι : F N ⟶ ⊲ G φ induces a continuous, F N -equivariant and surjective Cannon–Thurston map ι ^ : ∂ F N → ∂ G φ . We prove that for any φ as above, the map ι ^ is finite-to-one and that the preimage of every point of ∂ G φ has cardinality at most 2 N . We also prove that every point S ∈ ∂ G φ with at least three preimages in ∂ F N has the form ( w t m ) ∞ where w ∈ F N , m ≠ 0 , and that there are at most 4 N - 5 distinct F N -orbits of such singular points in ∂ G φ (for the translation action of F N on ∂ G φ ). By contrast, we show that for k = 1 , 2 , there are uncountably many points S ∈ ∂ G φ (and thus uncountably many F N -orbits of such S) with exactly k preimages in ∂ F N .

Locations

• Journal of the London Mathematical Society - View - PDF
• arXiv (Cornell University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View