论文标题
关于$ p $ -Adic Analytic Pro- $ P $组的自相似指数
On the self-similarity index of $p$-adic analytic pro-$p$ groups
论文作者
论文摘要
令$ p $为素数。我们说，如果在$ p^k $ - yar的常规生根树上承认忠实的自相似动作，那么该索引$ p^k $是自相似的。自相似于$ p $ g $ g $的自相似指数定义为$ p $的最小功率，例如$ p^k $，因此$ g $是索引$ p^k $的自相似的。我们表明，对于每个Prime $ p \ geqslant 3 $，所有整数$ d $都存在于无限的许多成对的非晶状体自相似的3维遗传性遗传上的纯粹的统一统一亲$ $ p $ $ p $ $ p $组的自相似性索引索引，大于$ d $。这通常意味着，对于自相似$ p $ - 亚种分析pro- $ p $组，一个人不能仅取决于组的尺寸，就不能绑定自相似性索引。
Let $p$ be a prime. We say that a pro-$p$ group is self-similar of index $p^k$ if it admits a faithful self-similar action on a $p^k$-ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-$p$ group $G$ is defined to be the least power of $p$, say $p^k$, such that $G$ is self-similar of index $p^k$. We show that for every prime $p\geqslant 3$ and all integers $d$ there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-$p$ groups of self-similarity index greater than $d$. This implies that, in general, for self-similar $p$-adic analytic pro-$p$ groups one cannot bound the self-similarity index by a function that depends only on the dimension of the group.
