#### Volume 12, issue 4 (2008)

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Surface subgroups from homology

### Danny Calegari

Geometry & Topology 12 (2008) 1995–2007
##### Abstract

Let $G$ be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If ${H}_{2}\left(G;ℚ\right)$ 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.

##### Keywords
hyperbolic group, surface subgroup, graph of groups, Thurston norm, rational polyhedron
##### Mathematical Subject Classification 2000
Primary: 20F65, 20F67
Secondary: 57M07