Borelian subgroups of simple Lie groups

TL;DR

This paper proves that simple real Lie groups do not contain Borel measurable dense subgroups with intermediate Hausdorff dimension, highlighting a rigidity property of their subgroup structure.

Contribution

It establishes a new restriction on the size and measure-theoretic complexity of dense subgroups in simple real Lie groups.

Findings

No Borel measurable dense subgroup of intermediate Hausdorff dimension exists in simple real Lie groups.

The result constrains the possible measure-theoretic properties of subgroups in these groups.

Provides insights into the structure and measure theory of Lie group subgroups.

Abstract

We prove that in a simple real Lie group, there is no Borel measurable dense subgroup of intermediate Hausdorff dimension.