Homogeneous number of free generators

TL;DR

This paper demonstrates the existence of profinitely dense free subgroups within certain arithmetic groups, providing explicit bounds and showing the uncountability of such subgroups, advancing understanding of their algebraic and topological properties.

Contribution

It constructs explicit examples of profinitely dense free subgroups in SLn(Z) and other arithmetic groups with the congruence subgroup property, and proves their abundance.

Findings

Existence of 4-generated profinitely dense free subgroups in SLn(Z) for n>2

Explicit bounds on the rank of such free subgroups in arithmetic groups

Uncountability of the set of profinitely dense, locally free subgroups

Abstract

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group \Gamma which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group \Gamma is uncountable.