Proper affine actions on semisimple Lie algebras
Annales de l'Institut Fourier, Volume 66 (2016) no. 2, pp. 785-831.

For any noncompact semisimple real Lie group G, we construct a group of affine transformations of its Lie algebra 𝔤 whose linear part is Zariski-dense in AdG and which is free, nonabelian and acts properly discontinuously on 𝔤.

Pour tout groupe de Lie réel semisimple non compact G, on construit un groupe discret de transformations affines de son algèbre de Lie 𝔤 dont la partie linéaire est Zariski-dense dans AdG et qui est libre, non abélien et agit proprement sur 𝔤.

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DOI: 10.5802/aif.3026
Classification: 20G20,  22E40,  20H15
Keywords: Discrete subgroups of Lie groups, Affine groups, Auslander conjecture, Milnor conjecture, Flat affine manifolds, Adjoint representation, Margulis invariant, Quasi-translation, Free group, Schottky group
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Smilga, Ilia. Proper affine actions on semisimple Lie algebras. Annales de l'Institut Fourier, Volume 66 (2016) no. 2, pp. 785-831. doi : 10.5802/aif.3026. https://aif.centre-mersenne.org/articles/10.5802/aif.3026/

[1] Abels, H.; Margulis, G. A.; Soifer, G. A. On the Zariski closure of the linear part of a properly discontinuous group of affine transformations, J. Differential Geom., Tome 60 (2002) no. 2, pp. 315-344 http://projecteuclid.org/euclid.jdg/1090351104

[2] Abels, H.; Margulis, G. A.; Soifer, G. A. The linear part of an affine group acting properly discontinuously and leaving a quadratic form invariant, Geom. Dedicata, Tome 153 (2011), pp. 1-46 | Article

[3] Abels, H.; Margulis, G. A.; Soifer, G. A. The Auslander conjecture for dimension less than 7 (2013) (http://arxiv.org/abs/1211.2525)

[4] Abels, Herbert Properly discontinuous groups of affine transformations: a survey, Geom. Dedicata, Tome 87 (2001) no. 1-3, pp. 309-333 | Article

[5] Auslander, Louis The structure of complete locally affine manifolds, Topology, Tome 3 (1964) no. suppl. 1, pp. 131-139 | Article

[6] Benoist, Yves Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2), Tome 144 (1996) no. 2, pp. 315-347 | Article

[7] Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny Geometry and topology of complete Lorentz spacetimes of constant curvature (To appear in Annales Scientifiques de l’École Normale Supérieure.)

[8] Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny Margulis spacetimes via the arc complex (2014) (http://arxiv.org/abs/1407.5422.)

[9] Drumm, Todd A. Fundamental polyhedra for Margulis space-times, Topology, Tome 31 (1992) no. 4, pp. 677-683 | Article

[10] Drumm, Todd A. Linear holonomy of Margulis space-times, J. Differential Geom., Tome 38 (1993) no. 3, pp. 679-690 http://projecteuclid.org/euclid.jdg/1214454487

[11] Eberlein, Patrick B. Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, vii+449 pages

[12] Fried, David; Goldman, William M. Three-dimensional affine crystallographic groups, Adv. in Math., Tome 47 (1983) no. 1, pp. 1-49 | Article

[13] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, Tome 140, Birkhäuser Boston, Inc., Boston, MA, 1996, xvi+604 pages | Article

[14] Margulis, G. A. Free completely discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR, Tome 272 (1983) no. 4, pp. 785-788

[15] Margulis, G.A. Complete affine locally flat manifolds with a free fundamental group, Journal of Soviet Mathematics, Tome 36 (1987) no. 1, pp. 129-139 | Article

[16] Milnor, John On fundamental groups of complete affinely flat manifolds, Advances in Math., Tome 25 (1977) no. 2, pp. 178-187 | Article

[17] Smilga, Ilia Fundamental domains for properly discontinuous affine groups, Geom. Dedicata, Tome 171 (2014), pp. 203-229 | Article

[18] Tits, J. Free subgroups in linear groups, J. Algebra, Tome 20 (1972), pp. 250-270 | Article

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