Abstract:
A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices SL(2,R). In general, given a semisimple Lie group G over some local field F, one may ask which lattices in G attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel’s result, Lubotzky determined that a lattice of minimal covolume in SL(2,F) with F=F_q((t)) is given by the so-called characteristic p modular group SL(2,F_q[1/t]). He noted that, in contrast with Siegel’s lattice, the quotient by SL(2,F_q[1/t]) was not compact, and asked what the typical situation should be: « for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice? ».
In the talk, we will review some of the known results, and then discuss the case of SL(n,R}) for n > 2. It turns out that, up to automorphism, the unique lattice of minimal covolume in SL(n,R) (n > 2) is SL(n,Z). In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case.