Properly discontinuous group actions on affine homogeneous spaces

Tomanov, Georges

doi:10.1134/s008154381601017x

Cited by 7 publications

(4 citation statements)

References 27 publications

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“…On the other hand, Fried and Goldman [13] proved the Auslander conjecture in dimension 3 (the cases n = 1 and 2 are easy). Recently, Tomanov [25] proved it in dimension n ≤ 5, and independently Abels, Margulis and Soifer [4] proved it in dimension n ≤ 6. See [1] for a survey of already known results.…”

Section: Background and Motivationmentioning

confidence: 99%

Proper affine actions: a sufficient criterion

Smilga

2021

Math. Ann.

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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. KeywordsDiscrete subgroups of Lie groups • Auslander conjecture • Flat affine manifolds • Margulis invariant

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Section: Background and Motivationmentioning

confidence: 99%

Proper affine actions: a sufficient criterion

Smilga

2021

Math. Ann.

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“…This conjecture still remains unproved in the general case but in lower dimensions (see Fried-Goldman [FG83], Abels-Margulis-Soifer [AMS20], Tomanov [Tom16]) results confirming the conjecture has been attained. Recently, an excellent survey of this topic has appeared in Danciger-Drumm-Goldman-Smilga [DDGS22].…”

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confidence: 82%

Affine Anosov Representations and Proper Actions

Ghosh

Treib

2022

International Mathematics Research Notices

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We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\textsf {SO}^0(n+1,n)\ltimes {\mathbb {R}}^{2n+1}$. We then show that a representation $\rho $ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt {L}_\rho $ is Anosov in $\textsf {SO}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $\rho (\Gamma )$ acts properly on $\mathbb {R}^{2n+1}$.

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“…On the other hand, Fried and Goldman [FG83] proved that the Auslander conjecture does hold in dimension 3 (the cases n = 1 and 2 are easy). Recently, Abels, Margulis and Soifer [AMS] proved it in dimension n ≤ 6, and independently Tomanov [Tom16] proved it in dimension n ≤ 5. See [Abe01] for a survey of already known results.…”

Section: Background and Motivationmentioning

confidence: 99%

Construction of Milnorian representations

Smilga

2019

Geom Dedicata

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We prove a partial converse to the main theorem of the author's previous paper Proper affine actions: a sufficient criterion (submitted; available at arXiv:1612.08942). More precisely, let G be a semisimple real Lie group with a representation ρ on a finite-dimensional real vector space V , that does not satisfy the criterion from the previous paper. Assuming that ρ is irreducible and under some additional assumptions on G and ρ, we then prove that there does not exist a group of affine transformations acting properly discontinuously on V whose linear part is Zariski-dense in ρ(G).

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Properly discontinuous group actions on affine homogeneous spaces

Abstract: 1 Our result [To1] can be also found in [So1].

Cited by 7 publications

References 27 publications

Proper affine actions: a sufficient criterion

Proper affine actions: a sufficient criterion

Affine Anosov Representations and Proper Actions

Construction of Milnorian representations

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