Higher rank arithmetic groups which are not invariably generated

TL;DR

This paper investigates the invariable generation property of higher rank arithmetic groups, providing counterexamples that challenge previous conjectures linking it to the Congruence Subgroup Property.

Contribution

It demonstrates that some higher rank arithmetic groups are not invariably generated, countering the conjecture that they always possess this property.

Findings

Certain higher rank arithmetic groups are not invariably generated.

The results challenge the conjecture linking invariable generation to the Congruence Subgroup Property.

Provides new insights into the structure of arithmetic groups.

Abstract

It was conjectured in [KLS14] that for arithmetic groups, Invariable Generation is equivalent to the Congruence Subgroup Property. In view of the famous Serre conjecture this would imply that higher rank arithmetic groups are invariably generated. In this paper we prove that some higher rank arithmetic groups are not invariably generated.