Freely indecomposable groups acting on hyperbolic spaces

Abstract

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-free locally quasiconvex hyperbolic groups (even though it is unsolvable for the class of all torsion-free hyperbolic groups). We apply our results to 3-manifold groups. Namely, suppose $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over a circle and suppose that all finitely generated subgroups of $G$ are topologically tame. We prove that for any $k\ge 2$ the group $G$ has only finitely many conjugacy classes of non-elementary freely indecomposable $k$-generated subgroups of infinite index in $G$.