GGS-groups: Order of congruence quotients and Hausdorff dimension

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 $G_n=G/\mathrm {Stab}_G(n)$ for every $n$. If $G$ is defined by the vector $\mathbf {e}=(e_1,\ldots ,e_{p-1})\in \mathbb {F}_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according to whether $\mathbf {e}$ 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 $\mathbf {e}$. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the $p$-adic tree.