Proper affine actions on semisimple Lie algebras

TL;DR

This paper constructs a free, nonabelian group of affine transformations acting properly discontinuously on the Lie algebra of any noncompact semisimple real Lie group, with a Zariski-dense linear part.

Contribution

It introduces a new class of affine actions on Lie algebras with dense linear parts, expanding understanding of proper affine actions in Lie theory.

Findings

Constructed a free, nonabelian affine group acting properly discontinuously.

Linear part of the group is Zariski-dense in the adjoint group.

Applicable to all noncompact semisimple real Lie groups.

Abstract

For any noncompact semisimple real Lie group $G$, we construct a group of affine transformations of its Lie algebra $\mathfrak{g}$ whose linear part is Zariski-dense in $\operatorname{Ad} G$ and which is free, nonabelian and acts properly discontinuously on $\mathfrak{g}$.