On the self-similarity index of p-adic analytic pro-p groups

2020/12/02 by Francesco Noseda, Noseda, Francesco, Ilir Snopce +1
Mathematics · #20E18 #22E20 (Primary) 22E60 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2012.00919

openalex publication_date 2020/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Related

• On hereditarily self-similar p-adic analytic pro-p groups
• Self-similarity of the classical p-adic Lie groups and Lie algebras
• On the self-similarity of the norm one group of p-adic division algebras
• On self-similar finite p-groups
• Self-similarity of p-adic groups