GGS-groups: Order of congruence quotients and Hausdorff dimension

By G. A. Fern'andez-Alcober, Amaia Zugadi-Reizabal. 2011

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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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c6475a6ff3c2dea7c08b6263a48933bf32147614