论文标题
花圈产品和灯塔组的死端
Dead ends on wreath products and lamplighter groups
论文作者
论文摘要
对于任何有限的组$ a $和任何有限生成的组$ b $，我们证明相应的lamplighter组$ a \ wr b $允许具有无限深度的标准生成集，如果$ b $是abelian，那么上面的是每个标准生成集合。这概括了$ b = \ mathbb {z} $以及由于Cleary和Taback而引起的循环发生器。当$ b = h*k $是两个有限组$ h $和$ k $的免费产品时，我们表明哪些相关灯塔组的标准发电机在与$ h $和$ k $的Cayley图相关的几何常数方面具有无限的深度。特别是，我们发现与一维情况的差异：在两个足够大的有限循环基团的自由产物上的lamplighter组相对于某些标准生成集具有统一的界限深度。
For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=\mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.
