Surface subgroups from homology

Abstract

Let $G$ be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If $H_2\left(G;\mathbb{Q}\right)$ is nonzero, then $G$ contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on $H_2\left(G;ℝ\right)$ is a finite-sided rational polyhedron.