2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165277Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.9 pages; version 2: typos correctedGroup TheoryGeometric Topology20F65; 20F67; 57M07text