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Combable functions, quasimorphisms, and the central limit theorem

Calegari, Danny and Fujiwara, Koji (2010) Combable functions, quasimorphisms, and the central limit theorem. Ergodic Theory and Dynamical Systems, 30 (5). pp. 1343-1369. ISSN 0143-3857.

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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: (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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Additional Information:© 2010 Cambridge University Press. Received 17 May 2009 and accepted in revised form 23 June 2009. We would like to thank Ian Agol, Ilya Kapovich, Shigenori Matsumoto, Curt McMullen, Mark Pollicott, Richard Sharp, Amie Wilkinson and the anonymous referees for useful comments and corrections. This research was largely carried out while the first author was at the Tokyo Institute of Technology (TIT), hosted by Sadayoshi Kojima. He thanks TIT and Professor Kojima for their hospitality. The second author would like to thank Caltech for their hospitality in hosting him while some of this paper was written. The paper T19U was a catalyst for discussions which led to the content of this paper, and we are pleased to acknowledge it. Danny Calegari was supported by NSF grant DMS 0707130. Koji Fujiwara is supported by JSPS-KAKENHI-19340013.
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NSFDMS 0707130
Japan Society for the Promotion of Science (JSPS)JSPS-KAKENHI-19340013
Issue or Number:5
Record Number:CaltechAUTHORS:20101108-100431663
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20709
Deposited By: Jason Perez
Deposited On:24 Nov 2010 00:08
Last Modified:03 Oct 2019 02:13

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