Combable functions, quasimorphisms, and the central limit theorem

TL;DR

This paper proves a central limit theorem for bicombable functions on word-hyperbolic groups, showing their values on random elements follow a normal distribution and establishing a relation between word lengths in different generating sets.

Contribution

It introduces the concept of bicombable functions, proves a central limit theorem for them on hyperbolic groups, and relates word lengths across different generating sets.

Findings

Bicombable functions satisfy a central limit theorem with normal distribution convergence.

Word lengths in different generating sets are asymptotically proportional with error O(√n).

Examples include homomorphisms, word length, and quasimorphisms.

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 (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 \bar{\phi}_n is the value of \phi on a random element of word length n (in a certain sense), there are E and \sigma for which there is convergence in the sense of distribution n^{-1/2}(\bar{\phi}_n - nE) \to N(0,\sigma), where N(0,\sigma) denotes the normal distribution with standard deviation \sigma. 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 lambda_{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\lambda_{1,2} in the S_2 metric, with error of size O(\sqrt{n}).