Proper affine actions: a sufficient criterion
Abstract
For a semisimple real Lie group with an irreducible representation on a finite-dimensional real vector space , we give a sufficient criterion on for existence of a group of affine transformations of whose linear part is Zariski-dense in and that is free, nonabelian and acts properly discontinuously on . 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.
Cite
@article{arxiv.1612.08942,
title = {Proper affine actions: a sufficient criterion},
author = {Ilia Smilga},
journal= {arXiv preprint arXiv:1612.08942},
year = {2024}
}
Comments
This paper generalizes the author's previous papers arXiv:1406.5906 and arXiv:1605.03833 . The structure of the proof is similar; a few passages are borrowed from the earlier papers. In this version, I corrected a mathematical mistake in Definition 6.12.(iii). This mistake remains present in the published version; an erratum has been published to address it