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On the topology of relative orbits for actions of algebraic groups over complete fields. (English) Zbl 1217.14031

Suppose \(k\) is a field which is complete with respect to a non-trivial valuation of real rank \(1\) (for example, \(k\) could be a \(\mathfrak{p}\)-adic field). Let \(G\) be a linear algebraic \(k\)-group acting on an affine \(k\)-variety \(V\). Suppose \(x \in V(k)\) is a closed \(k\)-point. The main aim of this paper is to investigate connections between the Zariski closedness of the \(G\)-orbit \(G\cdot x\) in \(V\) and the Hausdorff closedness of the so-called relative orbit \(G(k)\cdot x\) in \(V(k)\). The methods involve finding ways of equipping certain cohomology groups with a topology.
The paper contains a review of many of the existing results in the literature, in particular results by D. Birkes [Ann. Math. (2) 93, 459–475 (1971; Zbl 0198.35001)] and R. J. Bremigan [J. Reine Angew. Math. 453, 21–47 (1994; Zbl 0808.14040)], and also presents some new results. The authors state that proofs of their results will appear elsewhere.

MSC:

14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
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[1] D. Birkes, Orbits of linear algebraic groups, Ann. of Math. (2) 93 (1971), 459-475. · Zbl 0212.36402
[2] A. Borel, Introduction aux groupes arithmétiques , Hermann, Paris, 1969. · Zbl 0186.33202
[3] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485-535. · Zbl 0107.14804
[4] R. J. Bremigan, Quotients for algebraic group actions over non-algebraically closed fields, J. Reine Angew. Math. 453 (1994), 21-47. · Zbl 0808.14040
[5] M. Demazure and P. Gabriel, Groupes algébriques. Tome I , Masson & Cie, Éditeur, Paris, 1970. · Zbl 0203.23401
[6] P. Gille and L. Moret-Bailly, Action algébriques des groupes arithmétiques. Appendice to the article by Ullmo-Yafaev “Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture”. (Preprint). http://www.math.ens.fr/ gille/ · Zbl 1076.14531
[7] T. Kambayashi, M. Miyanishi and M. Takeuchi, Unipotent algebraic groups , Lecture Notes in Math., 414, Springer, Berlin, 1974. · Zbl 0294.14022
[8] G. R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299-316. · Zbl 0406.14031
[9] L. Lifschitz, Superrigidity theorems in positive characteristic, J. Algebra 229 (2000), no. 1, 375-404. · Zbl 0959.22006
[10] G. A. Margulis, Discrete subgroups of semisimple Lie groups , Springer, Berlin, 1991.
[11] J. S. Milne, Arithmetic duality theorems , Second edition, BookSurge, LLC, Charleston, SC, 2006. · Zbl 1127.14001
[12] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory , Third edition, Springer, Berlin, 1994. · Zbl 0797.14004
[13] J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique \(p\), Invent. Math. 78 (1984), no. 1, 13-88. · Zbl 0542.20024
[14] M. S. Raghunathan, A note on orbits of reductive groups, J. Indian Math. Soc. (N.S.) 38 (1974), no. 1-4, 65-70 (1975). · Zbl 0365.20048
[15] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2) 36 (1984), no. 2, 269-291. · Zbl 0567.14027
[16] J.-P. Serre, Cohomologie galoisienne , Fifth edition, Springer, Berlin, 1994. MR1324577 (96b:12010)
[17] R. Steinberg, Conjugacy classes in algebraic groups , Springer, Berlin, 1974. · Zbl 0281.20037
[18] N. Q. Thǎńg and N. D. Tan, On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I, J. Algebra 319 (2008), no. 10, 4288-4324. · Zbl 1149.11020
[19] J. Tits, Lectures on algebraic groups , Yale Univ., 1967.
[20] T. N. Venkataramana, On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Invent. Math. 92 (1988), no. 2, 255-306. · Zbl 0649.22008
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