On hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

TL;DR

This paper demonstrates that specific inverse limits of iterated wreath products possess a complete Hausdorff dimension spectrum and provides explicit subgroups with predetermined dimensions.

Contribution

It establishes the complete Hausdorff dimension spectrum for certain hereditarily just infinite profinite groups and constructs subgroups with specific dimensions.

Findings

Inverse limits of iterated wreath products have complete Hausdorff dimension spectrum.

Explicit construction of subgroups with given Hausdorff dimensions.

Results apply to groups with a unique maximal filtration of open normal subgroups.

Abstract

We prove that the inverse limit of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.