Remarks on minimal sets and conjectures of Cassels, Swinnerton-Dyer, and Margulis

Jinpeng An, Barak Weiss

Research output: Contribution to journalArticlepeer-review


We prove that a hypothesis of Cassels, Swìnnerton-Dyer, recast by Margulis as a 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 R-diagonaliz-able subgroup A of a reductive group G and a lattice r in G, we give a sufficient condition for a compact A-minimal subset Y of G/T to be of a simple form, which is also necessary if G is R-split. We also show that the stabilizer of Y has no nontrivial connected unipotent subgroups.

Original languageEnglish
Pages (from-to)260-279
Number of pages20
JournalMoscow Journal of Combinatorics and Number Theory
Issue number3-4
StatePublished - 2013
Externally publishedYes


  • bounded orbit
  • discrete subgroup
  • minimal set
  • product of linear forms

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics


Dive into the research topics of 'Remarks on minimal sets and conjectures of Cassels, Swinnerton-Dyer, and Margulis'. Together they form a unique fingerprint.

Cite this