A nonhomogeneous orbit closure of a diagonal subgroup. (English) Zbl 1192.22006

In an unpublished paper, M. Rees exhibited a lattice \(\Gamma\) of \(\text{SL}(3, \mathbb R)\) and a point \(x\in G/\Gamma\) such that for the full diagonal group \(A\) the orbit closure \(\overline{Ax}\) is not homogeneous. This shows that factor actions of 1-parameter non-Ad-unipotent groups are obstructions to the topological rigidity of diagonal subgroups. Margulis’s conjecture essentially states that these are the only ones [G. A. Margulis and N. Qian, Ergodic Theory Dyn. Syst. 21, No.1, 121–164 (2001; Zbl 0976.22009)].
Here the author exhibits some counterexamples when \(G=\text{SL}(n,\mathbb R)\) for \(n\geq 6\) and \(A\) is some strict subgroup of the diagonal group of matrices with nonnegative entries.
The idea behind the construction can also be used to yield examples of “nonhomogeneous” orbits for diagonal toral endomorphisms. It turns out that a proper closed invariant set is not necessarily a subset of a finite union of rational affine tori.


22E40 Discrete subgroups of Lie groups
37B99 Topological dynamics


Zbl 0976.22009
Full Text: DOI Link


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