Pseudo-Anosov subgroups of fibered 3-manifold groups

TL;DR

This paper characterizes convex cocompact subgroups of the mapping class group arising from fibered hyperbolic 3-manifold groups, establishing conditions for when such subgroups are purely pseudo-Anosov and finitely generated.

Contribution

It proves a new criterion linking convex cocompactness in the mapping class group to purely pseudo-Anosov, finitely generated subgroups within fibered 3-manifold groups, and generalizes existing theorems.

Findings

Convex cocompact subgroups are exactly the finitely generated, purely pseudo-Anosov subgroups.

Generalization to Gromov hyperbolic extensions of fibered 3-manifold groups.

Extension of Scott and Swarup's theorem on geometric finiteness.

Abstract

Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing a point from X. We prove that a subgroup G in H is convex cocompact in Mod(Y) if and only if G is finitely generated and purely pseudo-Anosov. We also prove a generalization of this theorem with H replaced by an arbitrary Gromov hyperbolic extension of the fundamental group of X, and an additional hypothesis of quasi-convexity of G in H. Along the way, we obtain a generalization of a theorem of Scott and Swarup on the geometric finiteness of subgroups of fibered 3-manifold groups.