标题： 可组合函数，拟同态，以及中心极限定理 显示英文标题

标题： Combable functions, quasimorphisms, and the central limit theorem

摘要： 一个离散群上的函数，如果其相对于一个梳状结构的离散导数可以通过有限状态自动机计算，则该函数是弱可梳的。 一个弱可梳函数如果是左右不变字度量中的Lipschitz函数，则它是双可梳的。 在字双曲群上的双可梳函数的例子包括（i）到Z的同态（ii）相对于有限生成集的字长（iii）大多数已知的显式拟同态构造（例如 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 . 作为推论，我们证明如果 S_1 和 S_2 是 G 的任意两个有限生成集，则存在一个代数数 lambda_{1,2}，它依赖于 S_1 和 S_2，使得几乎所有的长度为 n 的 S_1 度量下的字在 S_2 度量下的字长为 n\lambda _{1,2}，误差大小为 O(\sqrt{n})。 显示英文摘要

摘要： 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}).