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Remarks on minimal sets and conjectures of Cassels, Swinnerton-Dyer, and Margulis. (English) Zbl 1343.37009

Summary: We prove that a hypothesis of J. W. S. Cassels and H. P. F. Swinnerton-Dyer [Philos. Trans. R. Soc. Lond., Ser. A 248, 73–96 (1955; Zbl 0065.27905)], recast by Margulis as statement on the action of the diagonal group \(A\) on the space of unimodular lattices, is equivalent to several assertions about minimal sets for this action. More generally, for a maximal \(\mathbb{R}\)-diagonalizable subgroup \(A\) of a reductive group \(G\) and a lattice \(\Gamma\) in \(G\), we give a sufficient condition for a compact \(A\)-minimal subset \(Y\) of \(G/\Gamma\) to be of a simple form, which is also necessary if \(G\) is \(\mathbb{R}\)-split. We also show that the stabilizer of \(Y\) has no nontrivial connected unipotent subgroups.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
22E40 Discrete subgroups of Lie groups
11H46 Products of linear forms

Citations:

Zbl 0065.27905
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