论文标题
用无限循环顶部组的花圈产品的通勤程度
The degree of commutativity of wreath products with infinite cyclic top group
论文作者
论文摘要
有限基团的通勤程度是两个均匀和随机选择的通勤元素的概率。这个概念自然地扩展到有限生成的$ g $：通勤性$ \ text {dc} _s（g）$，就给定有限生成的套装$ s $而言，考虑到cayley Graph $ \ nathcal $ \ nath $ \ mathcal $ \ c}的通勤对元素的通勤成对的分数所产生的。我们专注于限制的花环产品表格$ g = h \ wr \ langle t \ rangle $，其中$ h \ ne 1 $有限地生成，而顶级组$ \ langle t \ rangle $是无限环保的。根据更一般的猜想，我们表明，对于此类组$ g $，无论选择$ s $，$ \ text {dc} _s（g）= 0 $。 这扩展了Cox的结果，后者考虑了某些类型的生成集的载板组。我们还得出了Cox的主要辅助结果的概括：在上面的“合理大型”同构图像$ g $中，基本组的图像相对于某些类型的生成集，基本组的密度为零。
The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups $G$: the degree of commutativity $\text{dc}_S(G)$, with respect to a given finite generating set $S$, results from considering the fractions of commuting pairs of elements in increasing balls around $1_G$ in the Cayley graph $\mathcal{C}(G,S)$. We focus on restricted wreath products the form $G = H \wr \langle t \rangle$, where $H \ne 1$ is finitely generated and the top group $\langle t \rangle$ is infinite cyclic. In accordance with a more general conjecture, we show that $\text{dc}_S(G) = 0$ for such groups $G$, regardless of the choice of $S$. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox's main auxiliary result: in `reasonably large' homomorphic images of wreath products $G$ as above, the image of the base group has density zero, with respect to certain types of generating sets.
