This result was proven in the 1960s by David Kazhdan and Grigory Margulis.
The main technical result of Kazhdan–Margulis, which is interesting in its own right and from which the better-known statement above follows immediately, is the following.
[5] One of the motivations of Kazhdan–Margulis was to prove the following statement, known at the time as Selberg's hypothesis (recall that a lattice is called uniform if its quotient space is compact): This result follows from the more technical version of the Kazhdan–Margulis theorem and the fact that only unipotent elements can be conjugated arbitrarily close (for a given element) to the identity.
A corollary of the theorem is that the locally symmetric spaces and orbifolds associated to lattices in a semisimple Lie group cannot have arbitrarily small volume (given a normalisation for the Haar measure).
For hyperbolic surfaces this is due to Siegel, and there is an explicit lower bound of
for the smallest covolume of a quotient of the hyperbolic plane by a lattice in
For hyperbolic three-manifolds the lattice of minimal volume is known and its covolume is about 0.0390.
[6] In higher dimensions the problem of finding the lattice of minimal volume is still open, though it has been solved when restricting to the subclass of arithmetic groups.