Hausdorff dimension in $R$-analytic profinite groups

TL;DR

This paper investigates the Hausdorff dimension of R-analytic subgroups within R-analytic profinite groups, establishing that these dimensions form a finite set of rational numbers.

Contribution

It proves that the set of Hausdorff dimensions of R-analytic subgroups in such groups is finite and rational, advancing understanding of their geometric structure.

Findings

Hausdorff dimensions form a finite subset of rational numbers

The set of R-analytic subgroup dimensions is finite

Dimensions are rational numbers

Abstract

We study the Hausdorff dimension of R-analytic subgroups in an R-analytic profinite group, where R is a pro-p ring whose asso- ciated graded ring is an integral domain. In particular, we prove that the set of such Hausdorff dimensions is a finite subset of the rational numbers.