Stable commutator length in word-hyperbolic groups

TL;DR

This paper establishes uniform positive lower bounds for stable commutator length in word-hyperbolic groups and related hyperbolic space actions, constructing explicit quasimorphisms and analyzing accumulation points for these bounds.

Contribution

It provides new uniform bounds and explicit constructions of quasimorphisms in hyperbolic groups, extending understanding of stable commutator length and its geometric implications.

Findings

Positive lower bounds depend only on hyperbolicity and generating set size

Explicit construction of quasimorphisms with uniform defect control

First accumulation point for stable commutator length is between 1/12 and 1/2

Abstract

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".