Proper affine actions: a sufficient criterion
For a semisimple real Lie group G with a 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...
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| Format: | Journal article |
| Language: | English |
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Springer
2021
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| Summary: | For a semisimple real Lie group G with a 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 Smilga (Groups Geom Dyn 12(2):449–528, 2018), insofar as it also deals with “swinging” representations. When G is split, almost all the irreducible representations of G that have 0 as a weight satisfy this criterion. We conjecture that it is actually a necessary and sufficient criterion. |
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