Mathematics – Group Theory
Scientific paper
2008-05-13
Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1343-1369
Mathematics
Group Theory
26 pages; version 3: typos corrected, referee's comments incorporated
Scientific paper
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}).
Calegari Danny
Fujiwara Koji
No associations
LandOfFree
Combable functions, quasimorphisms, and the central limit theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Combable functions, quasimorphisms, and the central limit theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Combable functions, quasimorphisms, and the central limit theorem will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-471834