Geometry & Topology

Surface subgroups from homology

Danny Calegari

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Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H2(G;) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on H2(G;) is a finite-sided rational polyhedron.

Article information

Geom. Topol., Volume 12, Number 4 (2008), 1995-2007.

Received: 4 April 2008
Revised: 9 June 2008
Accepted: 8 June 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 57M07: Topological methods in group theory

hyperbolic group surface subgroup graph of groups Thurston norm rational polyhedron


Calegari, Danny. Surface subgroups from homology. Geom. Topol. 12 (2008), no. 4, 1995--2007. doi:10.2140/gt.2008.12.1995.

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  • C Bavard, Longueur stable des commutateurs, Enseign. Math. $(2)$ 37 (1991) 109–150
  • M Bestvina, Questions in geometric group theory, available at\char'176bestvina
  • D Calegari, scl, monograph, available at\char'176dannyc
  • D Calegari, Stable commutator length is rational in free groups
  • D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
  • S M Gersten, Cohomological lower bounds for isoperimetric functions on groups, Topology 37 (1998) 1031–1072
  • C M Gordon, D D Long, A W Reid, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004) 135–148
  • M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) 5–99
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, Math.. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
  • A Hatcher, Algebraic topology, Cambridge University Press (2002)
  • J Hempel, $3$–Manifolds, Ann. of Math. Studies 86, Princeton University Press (1976)
  • S Mac Lane, Homology, Die Grund. der math. Wissenschaften 114, Academic Press Publishers, New York (1963)
  • P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. $(2)$ 17 (1978) 555–565
  • J-P Serre, Trees, Springer Monographs in Math., Springer, Berlin (2003) Translated from the French original by J Stillwell, Corrected 2nd printing of the 1980 English translation
  • W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130