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Divisible convex sets. II. (Convexes divisibles. II.) (French) Zbl 1037.22022
Summary: “A properly convex open cone in \(\mathbb{R}^m\) is called divisible if there exists a discrete subgroup \(\Gamma\) of \(\text{GL} (\mathbb{R}^m)\) preserving \(C\) such that the quotient \(\Gamma\setminus C\) is compact. We describe the Zariski closure of such a group \(\Gamma\). We show that if \(C\) is divisible but is neither a product nor a symmetric cone, then \(\Gamma\) is Zariski dense in \(\text{GL} (\mathbb{R}^m)\).” (Part I in C. R. Acad. Sci., Paris, Ser. I, Math. 332, 387–390 (2001; Zbl 1010.37014)].

MSC:
22E40 Discrete subgroups of Lie groups
20H15 Other geometric groups, including crystallographic groups
53A20 Projective differential geometry
57S30 Discontinuous groups of transformations
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[1] H. Abels, Parallelizability of proper actions, global \(K\)-slices and maximal compact subgroups , Math. Ann. 212 (1974/75), 1–19. · Zbl 0276.57019 · doi:10.1007/BF01343976 · eudml:182752
[2] –. –. –. –., Some topological aspects of proper group actions; noncompact dimension of groups , J. London Math. Soc. (2) 25 (1982), 525–538. · Zbl 0458.54029 · doi:10.1112/jlms/s2-25.3.525
[3] R. Benedetti \et J.-J. Risler, Real Algebraic and Semi-Algebraic Sets , Actualités Math., Hermann, Paris, 1990. · Zbl 0694.14006
[4] Y. Benoist, Automorphismes des cônes convexes , Invent. Math. 141 (2000), 149–193. · Zbl 0957.22008 · doi:10.1007/s002220000067
[5] –. –. –. –., Convexes divisibles , C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 387–390. · Zbl 1010.37014 · doi:10.1016/S0764-4442(01)01878-X
[6] ——–, Convexes divisibles I , preprint, 2001, · Zbl 1010.37014 · doi:10.1016/S0764-4442(01)01878-X · dma.ens.fr
[7] J.-P. Benzécri, Sur les variétés localement affines et localement projectives , Bull. Soc. Math. France 88 (1960), 229–332. · Zbl 0098.35204 · numdam:BSMF_1960__88__229_0 · eudml:86987
[8] A. Borel, Compact Clifford-Klein forms of symmetric spaces , Topology 2 (1963), 111–122. · Zbl 0116.38603 · doi:10.1016/0040-9383(63)90026-0
[9] H. Braun \et M. Koecher, Jordan-Algebren , Grundlehren Math. Wiss. 128 , Springer, Berlin, 1966. · Zbl 0145.26001
[10] S. Choi \et W. M. Goldman, Convex real projective structures on closed surfaces are closed , Proc. Amer. Math. Soc. 118 (1993), 657–661. · Zbl 0810.57005 · doi:10.2307/2160352
[11] J. Faraut \et A. Korányi, Analysis on symmetric cones , Oxford Math. Monogr., Oxford Univ. Press, New York, 1994. · Zbl 0841.43002
[12] W. M. Goldman, Convex real projective structures on compact surfaces , J. Differential Geom. 31 (1990), 791–845. · Zbl 0711.53033
[13] K. Jo, Quasi-homogeneous domains and convex affine manifolds , · Zbl 1039.57006 · doi:10.1016/S0166-8641(03)00106-8 · arxiv.org
[14] D. Johnson \et J. J. Millson, “Deformation spaces associated to compact hyperbolic manifolds” dans Discrete Groups in Geometry and Analysis , Progr. Math. 67 , Birkhaüser, Boston, 1984, 48–106. · Zbl 0664.53023
[15] M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications , Lecture Notes in Math. 1710 , Springer, Berlin, 1999. · Zbl 1072.17513
[16] J.-L. Koszul, Déformations des connexions localement plates , Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 1, 103–114. · Zbl 0167.50103 · doi:10.5802/aif.279 · numdam:AIF_1968__18_1_103_0 · eudml:73941
[17] H. Kraft, P. Slodowy, \et T. A. Springer, eds., Algebraic Transformation Groups and Invariant Theory , DMV Sem. 13 , Birkhaüser, Basel, 1989. · Zbl 0682.00008
[18] N. H. Kuiper, “On convex locally-projective spaces” dans Convegno Internazionale di Geometria Differenziale (Italie, 1953) , Edizione Cremonese, Rome, 1954, 200–213. · Zbl 0057.14303
[19] G. A. Margulis \et E. B. Vinberg, Some linear groups virtually having a free quotient , J. Lie Theory 10 (2000), 171–180. · Zbl 0958.22008 · eudml:120869
[20] H. Saito, On a classification of prehomogeneous vector spaces over local and global fields , J. Algebra 187 (1997), 510–536. · Zbl 0874.14046 · doi:10.1006/jabr.1996.6807
[21] M. Sato \et T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariant , Nagoya Math. J. 65 (1977), 1–155. · Zbl 0321.14030
[22] J. Vey, Sur les automorphismes affines des ouverts convexes dans les espaces numériques , C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A249–A251. · Zbl 0186.55701
[23] –. –. –. –., Sur les automorphismes affines des ouverts convexes saillants , Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 641–665. · Zbl 0206.51302 · numdam:ASNSP_1970_3_24_4_641_0 · eudml:83543
[24] E. B. Vinberg, The theory of convex homogeneous cones , Transl. Mosc. Math. Soc. (1963), 340–403. · Zbl 0138.43301
[25] –. –. –. –., The structure of the group of automorphisms of a homogeneous convex cone , Transl. Mosc. Math. Soc. (1965), 63–93. · Zbl 0311.17008
[26] E. Vinberg \et V. Kac, Quasi-homogeneous cones , Math. Notes 1 (1967), 231–235. · Zbl 0163.16902 · doi:10.1007/BF01098890
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