Do uniform lattices in simple lie groups contain free abelian subgroups?

Question

Let $\Gamma$ be a cocompact lattice in a simple Lie group $G$ such that $G$ has rank at least $2$. Does $\Gamma$ contain a rank $2$ free abelian group?

Answer 1

This, in fact, also holds for all lattices in all noncompact connected semisimple Lie groups $G$: Every such lattice $\Gamma$ is Zariski dense (this is due to Borel) and, hence, is not virtually solvable. Now, using Tits' Alternative, conclude that $\Gamma$ contains $Z * Z$.

As for free abelian subgroups of rank 2, again, all lattices in semisimple Lie groups of rank $\ge 2$ (uniform and not) contain $Z^2$: Theorem 2.8 in

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