Theti Pseudo-Anosov subgroups of fibered 3-manifold groups

Spencer Dowdall, Richard P. Kent, Christopher J. Leininger

Research output: Contribution to journalArticlepeer-review

Abstract

Let S be a hyperbolic surface and let S be the surface obtained from S by o removing a point. The mapping class groups Mod(S) and Mod(S) fit into a short exact sequence 1→π1 (S) → Mod(S) → Mod(S) → 1. If M is a hyperbolic 3-manifold that fibers over the circle with fiber S, then its fundamental group its into a short exact sequence 1→π 1 (S)→π 1(M) → Z 1 that injects into the one above. We show that, when viewed as subgroups of Mod(S), finitely generated purely pseudo-Anosov subgroups of π1(M) are convex cocompact in the sense of Farb and Mosher. More generally, if we have a 1-hyperbolic surface group extension 1 → π1 (S) → ⌈φ → φ 1, any quasiisometrically embedded purely pseudo-Anosov subgroup of ⌈φ is convex coo compact in Mod(S). We also obtain a generalization of a theorem of Scott and Swarup by showing that finitely generated subgroups of π1(S) are quasiisometrically embedded in hyperbolic extensions ⌈φ.

Original languageEnglish (US)
Pages (from-to)1247-1282
Number of pages36
JournalGroups, Geometry, and Dynamics
Volume8
Issue number4
DOIs
StatePublished - 2014
Externally publishedYes

Keywords

  • Convex cocompact
  • Fibered 3-manifold
  • Gromov hyperbolic
  • Hyperbolic
  • Mapping class group
  • Pseudo-Anosov

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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