Surface subgroups from homology. (English) Zbl 1185.20046

Calegari, Danny

Geom. Topol. 12, No. 4, 1995-2007 (2008).

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.

Reviewer: Athanase Papadopoulos (Strasbourg)

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

Keywords:

word-hyperbolic groups; Gromov-Thurston norm; stable commutator lengths; surface subgroups

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