TITLE: Uniform lattices in semisimple Lie groups
QUESTION [6 upvotes]: Let $\Gamma$ be a uniform lattice in a semisimple Lie group $G$.

Must $\Gamma$ be virtually torsion-free?
If (1) is true, then does this work more generally if $G$ is reductive?

I am motivated by a prima facie knowledge of Theorem B of Armand Borel's paper, "Compact Clifford--Klein forms of symmetric spaces" (1963).

REPLY [9 votes]: If $G$ is linear (which would be the case, for instance, if it is centerless) then it is special case of the more general fact that any finitely generated subgroup of $GL_n(F)$ for a field $F$ of characteristic zero is virtually torsion-free.
So all you need to know is that the lattices are finitely generated. This, for cocompact lattices, is easy, and for instance follows from what is usually called Milnor-Schwarz lemma, which is a very general lemma about cocompact isometric actions of groups on spaces. You can find a version of it in Pierre de la Harpe's book.
For general lattices, in higher rank, this follows from property T and in rank 1 by some more geometric methods.
If not, this may fail. See this for instance:
http://people.uleth.ca/~dave.morris/talks/deligne-torsion.pdf