UC San Diego

The objective of the dissertation is to determine the lattices of minimal covolume in SL(n,R), for n ≥ 3. Relying on Margulis’ arithmeticity, Prasad’s volume formula, and work of Borel and Prasad, the problem will be translated in number theoretical terms. A careful analysis of the number theoretical bounds involved then leads to the identification of the lattices of minimal covolume. The answer turns out to be the simplest one: SL(n,Z) is, up to automorphism, the unique lattice of minimal covolume in SL(n,R). In particular, lattices of minimal covolume in SL(n,R) are non-uniform when n ≥ 3, contrasting with Siegel’s result for SL(2,R). This answers for SL(n,R) the question of Lubotzky: is a lattice of minimal covolume typically uniform or not?