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Combable functions, quasimorphisms, and the central limit theorem (joint with K. Fujiwara)

#### Danny Calegari, California Institute of Technology

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}).