Proper affine actions: a sufficient criterion

TL;DR

This paper establishes a new sufficient condition for the existence of free, properly discontinuous affine actions with Zariski-dense linear parts in certain Lie group representations, extending previous criteria to more general cases.

Contribution

It introduces a more general criterion for proper affine actions that includes swinging representations, potentially serving as a necessary and sufficient condition.

Findings

Provides a sufficient criterion for affine actions with Zariski-dense linear parts.

Extends previous results to include swinging representations.

Conjectures the criterion might be necessary and sufficient for all representations.

Abstract

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $\rho(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations.