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.



Lattices in $SL(n,\mathbb R)$
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:
<http://www.math.bgu.ac.il/~barakw/papers/clorbit.pdf>


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). 


