Combable functions, quasimorphisms, and the central limit theorem

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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 (i) homomorphisms to Z (ii) word length with respect to a finite generating set (iii) 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 ϕₙ is the value of ϕon a random element of word length n (in a certain sense), there are E and σfor which there is convergence in the sense of distribution n⁻¹/²(ϕₙ - nE) → N(0,σ), 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 lambda₁,₂ 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(√n).