论文标题
有限nilpotent组的随机cayley图直径
Diameters of random Cayley graphs of finite nilpotent groups
论文作者
论文摘要
我们证明存在有限无方向的无方向性组和尼尔肽类别的有限nilpotent群的随机无方向图的限制分布，从而扩展了处理亚伯利亚案例的莎皮拉和扎克的结果。限制分布定义在单型晶格的空间上，例如阿贝尔组的随机cayley图。当我们专门针对某个Unitriangular群体的家族时，我们的结果建立了最近的Hermon和Thomas的猜想。我们得出这是由于普遍不平等的结果，表明nilpotent群的Cayley图的直径受其阿贝里亚化的直径控制。
We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.
