2006/08/19 by Zoran Šunić, Sunic, Zoran · 3 citations
Mathematics · #20E08 #20E18 #20F50 #20F69 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.math/0608488
openalex publication_date 2006/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each prime p and a monic polynomial f, invertible over p, we define a group Gp,f of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group G2,x2+x+1. We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further ``finiteness'' properties are also discussed (amenability, torsion and intermediate growth).