Growth in free groups (and other stories)—twelve years later
Growth in free groups (and other stories)—twelve years later
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自由团体（和其他故事）的增长——十二年后
DOI：
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发表时间：
2010
影响因子：
0.6
通讯作者：
Igor Rivin
中科院分区：
文献类型：
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作者：
Igor Rivin
.我们首先研究自由群Fn的（循环约化的）元素相对于它们的阿贝尔化（或者等价地，它们在H1（Fn，Z）中的类）的分布。我们推导出一个明确的生成函数，和一个极限分布，通过某些结果（独立的兴趣）切比雪夫多项式，我们还证明了这些类的减少模p（p -任意素数）是渐近等分布的，我们研究了偏离等分布。我们将我们的技术扩展到一个更一般的设置，并使用它们来研究定期（有向和无向）图的长周期（和路径）的统计特性。我们回到自由群来研究一些共轭类的数目的增长函数作为其循环缩减长度的函数。
. We start by studying the distribution of (cyclically reduced) elements of the free groups F n with respect to their Abelianization (or equivalently, their class in H 1 ( F n , Z )). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p ( p — an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.