Rational and transcendental growth series for the higher Heisenberg groups
Rational and transcendental growth series for the higher Heisenberg groups
复制标题
高海森堡群的理性和超越增长级数
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
M. Stoll
中科院分区：
文献类型：
--
作者：
M. Stoll
抽象。本文考虑具有无限循环导子群的二步幂零群的增长级数。每一个这样的群G都有一个子群，其指数为Hn× m，其中Hn是长度为2n+1的离散海森堡群。我们称n为G的Heisenberg秩。我们证明了这类群都有有限生成集，使得相应的增长序列是有理的。另一方面，我们证明了如果G的Heisenberg秩为n <$2，则G有一个有限生成集，使得相应的增长级数是超越幂级数。
Abstract. This paper considers growth series of 2-step nilpotent groups with infinite cyclic derived subgroup. Every such group G has a subgroup of finite index of the form Hn×ℤm, where Hn is the discrete Heisenberg group of length 2n+1. We call n the Heisenberg rank of G. We show that every group of this type has some finite generating set such that the corresponding growth series is rational. On the other hand, we prove that if G has Heisenberg rank n ≧ 2, then G possesses a finite generating set such that the corresponding growth series is a transcendental power series.