Combable functions, quasimorphisms, and the central limit theorem

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 φ ̄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(φ ̄ₙ - nE) → N(0,σ ) , where N (0, σ ) denotes the normal distribution with standard deviation σ . As a corollary, we show that if S 1 and S 2 are any two finite generating sets for G , there is an algebraic number λ 1,2 depending on S 1 and S 2 such that almost every word of length n in the S 1 metric has word length n ⋅ λ 1,2 in the S 2 metric, with error of size O(√n) .