Proper affine actions: a sufficient criterion
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arXiv
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| Format: | Preprint |
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2016
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| _version_ | 1866909120428244992 |
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| author | Smilga, Ilia |
| author_facet | Smilga, Ilia |
| contents | For a semisimple real Lie group $G$ with an irreducible representation $ρ$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $ρ$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $ρ(G)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1612_08942 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Proper affine actions: a sufficient criterion Smilga, Ilia Group Theory 20G20, 20G05, 22E40, 20H15 For a semisimple real Lie group $G$ with an irreducible representation $ρ$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $ρ$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $ρ(G)$ 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. |
| title | Proper affine actions: a sufficient criterion |
| topic | Group Theory 20G20, 20G05, 22E40, 20H15 |
| url | https://arxiv.org/abs/1612.08942 |