I will explain the following:
Theorem:
Let G be a semisimle Lie group without compact facors. There is a constant C=C(G) such that for any lattice L in G we have d(L) < C Vol(G/L)where d(L) is the minimal size of a generating set of L.
This in particular implies that d(L) is finite (which is well known but in general nontrivial). It also implies the classical Kazhdan–Margulis theorem, which states that there is a positive lower bound on the covolume of lattices, indeed d(L)> 2 implies vol(G/L)>2/C. It also gives bounds on the first Betti number. In a recent joint work with Belolipetsky, Lubotzky and Shalev we made use of this theorem to get estimates on the asymptotic growth of the number of arithmetic groups.
(This theorem was known before for torsion-free lattices, and is now known in general.)