Hausdorff dimension in a family of self-similar groups

Abstract

For every prime p and every monic polynomial f, invertible over p, we define a group G p, 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 \(G_{2, x^{2}+x+1}\). 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).