On the finitely generated Hausdorff spectrum of spinal groups

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.