# Groups, Geometry, and Dynamics

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**Volume 8, Issue 4, 2014, pp. 1247–1282**

**DOI: 10.4171/GGD/302**

Published online: 2014-12-31

Pseudo-Anosov subgroups of fibered 3-manifold groups

Spencer Dowdall^{[1]}, Richard P. Kent IV

^{[2]}and Christopher J. Leininger

^{[3]}(1) University of Illinois at Urbana-Champaign, USA

(2) University of Wisconsin, Madison, USA

(3) University of Illinois at Urbana-Champaign, USA

Let $S$ be a hyperbolic surface and let $\mathring{S}$ be the surface obtained from $S$ by removing a point. The mapping class groups $\mathrm {Mod}(S)$ and $\mathrm {Mod}(\mathring{S})$ fit into a short exact sequence \[ 1 \to \pi_1(S) \to \mathrm {Mod}(\mathring{S}) \to \mathrm {Mod}(S) \to 1. \] 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 \[ 1 \to \pi_1(S) \to \pi_1(M) \to \mathbb Z \to 1 \] that injects into the one above. We show that, when viewed as subgroups of $\mathrm {Mod} (\mathring{S})$, finitely generated purely pseudo-Anosov subgroups of $\pi_1(M)$ are convex cocompact in the sense of Farb and Mosher. More generally, if we have a $\delta$-hyperbolic surface group extension \[ 1 \to \pi_1(S) \to \Gamma_\Theta \to \Theta \to 1, \] any quasiisometrically embedded purely pseudo-Anosov subgroup of $\Gamma_\Theta$ is convex cocompact in $\mathrm {Mod}(\mathring{S})$. We also obtain a generalization of a theorem of Scott and Swarup by showing that finitely generated subgroups of $\pi_1(S)$ are quasiisometrically embedded in hyperbolic extensions $\Gamma_\Theta$.

*Keywords: *Mapping class group, pseudo-Anosov, fibered 3-manifold, hyperbolic, Gromov hyperbolic, convex cocompact

Dowdall Spencer, Kent IV Richard, Leininger Christopher: Pseudo-Anosov subgroups of fibered 3-manifold groups. *Groups Geom. Dyn.* 8 (2014), 1247-1282. doi: 10.4171/GGD/302