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If $\Gamma\subseteq SL(n,\mathbb{R})$ is a lattice (i.e. discrete and finite covolume), does $\Gamma$ necessarily contain some $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$?

I know that the answer is yes if the lattice is cocompact, and that the answer is also yes in the case $\Gamma=SL(n,\mathbb Z)$. So I wonder if every lattice satisfies this property.

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What exactly does your second sentence mean??? – Igor Rivin Jul 3 '12 at 23:46
I've edited the question to make it clearer. Hopefully I didn't change what the OP is asking. – John Pardon Jul 4 '12 at 0:16
Ah, much better! – Igor Rivin Jul 4 '12 at 0:37
That is what I meant, thanks for the correction. – ALB Jul 4 '12 at 3:24
@ALB: I am curious as to why the proof is easier for uniform lattices?! – Igor Rivin Jul 4 '12 at 4:38
up vote 11 down vote accepted

The answer is yes. It is theorem [2.13] of the following paper of Prasad and Raghunathan:

Prasad, Gopal; Raghunathan, M. S. Cartan subgroups and lattices in semi-simple groups. Ann. of Math. (2) 96 (1972), 296–317.

There is also a lot of information in this paper:

Note that diagonalizable copies of $\mathbb{Z}^{n-1}$ in $\Gamma$ correspond to closed orbits for the action of the full diagonal subgroup of $SL(n,\mathbb{R})$ on $SL(n,\mathbb{R})/\Gamma$.

This is related to the Margulis conjecture which (with some caveats) states that the closure of any orbit of the full diagonal on $SL(n,\mathbb{R})/\Gamma$ is algebraic, i.e. is itself the closed orbit of some subgroup. This conjecture is the biggest open problem in homogeneous dynamics (and in particular implies the Littlewood conjecture in number theory).

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This is a theorem of G. Prasad and M.S. Raghunathan. See Theorem 7.2 in this paper of Steve Hurder's (rigidity of Anosov actions) -- the original reference is a bit less friendly.

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@Alex Eskin and Igor Rivin: Ok, thank you for your precise answers and additionnal information. – ALB Jul 4 '12 at 16:22

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