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 $α\in [0,1]$, we construct a branch group $G_α$ such that $G_α$ has a finitely generated subgroup $H$ where $H$ has Hausdorff dimension $α$ in $G$. Using results by Barnea, Shalev and Klopsch we further deduce that the finitely generated Hausdorff spectrum of this group $G_α$ contains $\mathcal{L}_α\cup ([0, 1] \cap \mathcal{L})$, where $\mathcal{L}$ is a countable subset of $\mathbb{Q}$ and $\mathcal{L}_α$ is a certain set of countably many irrational numbers in the interval $[0,α]$. This answers a question of Benjamin Klopsch.