  In this paper we obtain uniform positive lower bounds on stable commutator
length 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 the Bass-Serre tree). If G is a word
hyperbolic group which is delta hyperbolic with respect to a symmetric
generating set S, then there is a positive constant C depending only on delta
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 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, 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".
