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Surface subgroups from homology. (English) Zbl 1185.20046
The author shows that if a group \(G\) is a graph of free groups amalgamated along cyclic subgroups, and if \(A\in H_2(G;\mathbb{Q})\) is a homology class with nonzero Gromov-Thurston norm, then some map of a surface to a \(K(G,1)\) realizes the Gromov-Thurston norm in the projective class of \(A\), and therefore \(G\) contains a closed hyperbolic surface subgroup. The paper contains some motivation of this result, in particular from three-manifold topoogy. The author also makes the connection between this result and Gromov’s famous question asking whether every one-ended non-elementary word-hyperbolic group contains a closed surface subgroup.
The author also shows that the Gromov-Thurston norm on \(H_2(G;\mathbb{Q})\) is piecewise rational linear, and that if \(G\) is word-hyperbolic, then the unit ball of the Gromov-Thurston norm on \(H_2(G;\mathbb{Q})\) is a finite-sided rational polyhedron.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M60 Group actions on manifolds and cell complexes in low dimensions
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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