Groups, Geometry, and Dynamics

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Volume 4, Issue 1, 2010, pp. 59–90
DOI: 10.4171/GGD/75

Published online: 2009-12-23

Stable commutator length in word-hyperbolic groups

Danny Calegari[1] and Koji Fujiwara[2]

(1) California Institute of Technology, Pasadena, United States
(2) Kyoto University, Japan

In this paper we obtain uniform positive lower bounds on the stable commutator length of elements in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on an associated Bass–Serre tree). If G is a word-hyperbolic group that is δ-hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on δ and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length of the element is at least equal to C. By Bavard’s theorem, these lower bounds on stable commutator length imply the existence of quasimorphisms with uniform control on the defects; however, we show how to construct such quasimorphisms directly.
  We also prove various separation theorems on families of elements in such groups, constructing homogeneous quasimorphisms (again with uniform estimates) which are positive on some prescribed element while vanishing on some family of independent elements whose translation lengths are uniformly bounded.
  Finally, we prove that the first accumulation point for stable commutator length in a torsion-free word-hyperbolic group is contained between 1/12 and 1/2. This gives a universal sense of what it means for a conjugacy class in a hyperbolic group to have a small stable commutator length, and can be thought of as a kind of “homological Margulis lemma”.

Keywords: Quasimorphism, stable commutator length, scl, hyperbolic group, pleated surface, Mineyev’s flow space, mapping class group, defect

Calegari Danny, Fujiwara Koji: Stable commutator length in word-hyperbolic groups. Groups Geom. Dyn. 4 (2010), 59-90. doi: 10.4171/GGD/75