Ergodic Theory and Dynamical Systems

Research Article

Combable functions, quasimorphisms, and the central limit theorem


a1 Department of Mathematics, Caltech, Pasadena CA, 91125, USA (email:

a2 Graduate School of Information Science, Tohoku University, Sendai, Japan (email:


A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:

  • (1) homomorphisms to ℤ;
  • (2) word length with respect to a finite generating set;
  • (3) most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).
We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if $\overline {\phi }_n$ is the value of ϕ on a random element of word length n (in a certain sense), there are E and σ for which there is convergence in the sense of distribution $n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, where N(0,σ) denotes the normal distribution with standard deviation σ. As a corollary, we show that if S1 and S2 are any two finite generating sets for G, there is an algebraic number λ1,2 depending on S1 and S2 such that almost every word of length n in the S1 metric has word length nλ1,2 in the S2 metric, with error of size $O(\sqrt {n})$.

(Received May 17 2009)

(Revised June 23 2009)