Combable functions, quasimorphisms, and the central limit theorem

Abstract

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(overbar)_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(overbar)_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 S1 metric has word length n∙λ_(1,2) in the S_2 metric, with error of size O (√n).

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