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# 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