Rational growth and degree of commutativity of graph products
Rational growth and degree of commutativity of graph products
复制标题
图积的有理增长与交换度
DOI：
10.1016/j.jalgebra.2019.01.001
复制
发表时间：
2017
影响因子：
0.9
通讯作者：
Motiejus Valiunas
中科院分区：
文献类型：
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作者：
Motiejus Valiunas
设G是无限群，X是G的有限生成集，使得G关于X的增长级数是有理函数；本文得到了关于Cayley图Γ(G，X)中球体(或球)大小的一个结果：即，对于某个整数αλ0和某个α≥1，半径为n的球体的大小上下有界于nλ≥n的正常数倍.作为这一结果的一个应用，给出了一个交换度(DC)的计算：对于有限群F，其DC定义为F中两个随机选择的元素与Antolín交换的概率.Martino和Ventura最近将这一概念推广到所有有限生成的群。有人猜想指数增长的群G的DC为零。本文证明了当G是直角Artin群时的猜想(对于某些生成集)，或者更一般地，当G是有理增长群的图积时，其中非平凡元素的中心子是“一致小的”。
Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ (G, X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of n α λ n for some integer α≥ 0 and some λ≥ 1. As an application of this result, a calculation of degree of commutativity (dc) is provided: for a finite group F, its dc is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the dc of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are “uniformly small”.