GGS-groups: order of congruence quotients and Hausdorff dimension

TL;DR

This paper calculates the order of congruence quotients for GGS-groups over p-adic trees and derives their Hausdorff dimension, providing formulas based on the defining vector and its properties.

Contribution

It introduces explicit formulas for the order of congruence quotients of GGS-groups depending on the defining vector's symmetry and computes their Hausdorff dimension.

Findings

Formulas for the order of GGS-group quotients depending on vector properties

Classification into three cases based on symmetry of the defining vector

Determination of Hausdorff dimension for closures of GGS-groups

Abstract

If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n. If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.