2012/08/13 by Spencer Dowdall, Dowdall, Spencer, Richard P. Kent IV +3 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT
paper · pdf · doi:10.48550/arxiv.1208.2495
v2. Revised according to referee's suggestions. To appear in Groups, Geometry, and Dynamics. 29 pages, no figures. v1. 27 pages, no figures
arxiv created 2013/04/11 · arxiv updated 2013/04/12
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.