no. 2
Proper affine actions on semisimple Lie algebras
[Actions affines propres sur les algèbres de Lie semisimples]
Smilga, Ilia1
1 Department of Mathematics Yale University P.O. Box 208283 New Haven, CT 06520-8283 (USA)
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 785-831.

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.

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.

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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
Mot clés : Sous-groupes discrets de groups de Lie, groupes affines, conjecture d’Auslander, conjecture de Milnor, variétés affines plates, représentation adjointe, invariant de Margulis, quasi-translation, groupe libre, groupe de Schottky

Smilga, Ilia 1

1 Department of Mathematics Yale University P.O. Box 208283 New Haven, CT 06520-8283 (USA) 
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Smilga, Ilia. Proper affine actions on semisimple Lie algebras. Annales de l'Institut Fourier, Tome 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., Volume 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, Volume 153 (2011), pp. 1-46 | DOI

[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, Volume 87 (2001) no. 1-3, pp. 309-333 | DOI

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

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

[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, Volume 31 (1992) no. 4, pp. 677-683 | DOI

[10] Drumm, Todd A. Linear holonomy of Margulis space-times, J. Differential Geom., Volume 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., Volume 47 (1983) no. 1, pp. 1-49 | DOI

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

[14] Margulis, G. A. Free completely discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR, Volume 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, Volume 36 (1987) no. 1, pp. 129-139 | DOI

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

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

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

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