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.


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
Full Text: DOI arXiv


[1] C Bavard, Longueur stable des commutateurs, Enseign. Math. \((2)\) 37 (1991) 109 · Zbl 0810.20026
[2] M Bestvina, Questions in geometric group theory
[3] D Calegari, scl, monograph · Zbl 1187.20035
[4] D Calegari, Stable commutator length is rational in free groups · Zbl 1225.57002
[5] D Gabai, Foliations and the topology of \(3\)-manifolds, J. Differential Geom. 18 (1983) 445 · Zbl 0539.57013
[6] S M Gersten, Cohomological lower bounds for isoperimetric functions on groups, Topology 37 (1998) 1031 · Zbl 0933.20026
[7] C M Gordon, D D Long, A W Reid, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004) 135 · Zbl 1057.20031
[8] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) 5 · Zbl 0516.53046
[9] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015
[10] A Hatcher, Algebraic topology, Cambridge University Press (2002) · Zbl 1044.55001
[11] J Hempel, \(3\)-Manifolds, Ann. of Math. Studies 86, Princeton University Press (1976) · Zbl 0345.57001
[12] S Mac Lane, Homology, Die Grund. der math. Wissenschaften 114, Academic Press Publishers (1963)
[13] P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. \((2)\) 17 (1978) 555 · Zbl 0412.57006
[14] J P Serre, Trees, Springer Monographs in Math., Springer (2003) · Zbl 1013.20001
[15] W P Thurston, A norm for the homology of \(3\)-manifolds, Mem. Amer. Math. Soc. 59 (1986) · Zbl 0585.57006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.