On the finitely generated Hausdorff spectrum of spinal groups

Abstract

We study the finitely generated Hausdorff spectrum of spinal automorphism groups acting on rooted trees. Given any $\alpha \in [0,1]$, we construct a branch group $G_{\alpha }$ such that $G_{\alpha }$ has a finitely generated subgroup $H$ where $H$ has Hausdorff dimension $\alpha$ in $G$. Using results by Barnea, Shalev and Klopsch we further deduce that the finitely generated Hausdorff spectrum of this group $G_{\alpha }$ contains ${\mathcal{L}}_{\alpha }\cup ([0,1]\cap \mathcal{L})$, where $\mathcal{L}$ is a countable subset of $\mathbb{Q}$ and ${\mathcal{L}}_{\alpha }$ is a certain set of countably many irrational numbers in the interval $[0,\alpha ]$. This answers a question of Benjamin Klopsch.

11 pages, 2 figures