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 ∈ [0, 1], we construct a branch group Gα such that Gα 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 Gα contains Lα ∪ ([0, 1] ∩ L), where L is a countable subset of Q and Lα is a certain set of countably many irrational numbers in the interval [0, α]. This answers a question of Benjamin Klopsch [8].