On the finitely generated Hausdorff spectrum of spinal groups

TL;DR

This paper constructs specific spinal automorphism groups acting on rooted trees to analyze their finitely generated subgroups' Hausdorff dimensions, revealing the structure of their Hausdorff spectrum and answering a question by Klopsch.

Contribution

It introduces a method to realize any Hausdorff dimension in [0,1] within finitely generated subgroups of certain spinal groups, expanding understanding of their spectral properties.

Findings

Constructed groups with prescribed Hausdorff dimensions for subgroups

Determined the structure of the finitely generated Hausdorff spectrum

Answered a question of Benjamin Klopsch about spectrum structure

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.