# Pseudo-Anosov subgroups of fibered 3-manifold groups

### Spencer Dowdall

University of Illinois at Urbana-Champaign, USA### Richard P. Kent IV

University of Wisconsin, Madison, USA### Christopher J. Leininger

University of Illinois at Urbana-Champaign, USA

## Abstract

Let $S$ be a hyperbolic surface and let $S˚$ be the surface obtained from $S$ by removing a point. The mapping class groups $Mod(S)$ and $Mod(S˚)$ fit into a short exact sequence

If $M$ is a hyperbolic $3$-manifold that fibers over the circle with fiber $S$, then its fundamental group fits into a short exact sequence

that injects into the one above. We show that, when viewed as subgroups of $Mod(S˚)$, finitely generated purely pseudo-Anosov subgroups of $π_{1}(M)$ are convex cocompact in the sense of Farb and Mosher. More generally, if we have a $δ$-hyperbolic surface group extension

any quasiisometrically embedded purely pseudo-Anosov subgroup of $Γ_{Θ}$ is convex cocompact in $Mod(S˚)$. We also obtain a generalization of a theorem of Scott and Swarup by showing that finitely generated subgroups of $π_{1}(S)$ are quasiisometrically embedded in hyperbolic extensions $Γ_{Θ}$.

## Cite this article

Spencer Dowdall, Richard P. Kent IV, Christopher J. Leininger, Pseudo-Anosov subgroups of fibered 3-manifold groups. Groups Geom. Dyn. 8 (2014), no. 4, pp. 1247–1282

DOI 10.4171/GGD/302