Arithmetic groups
Lattices of minimal covolume in SL_n
A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\mathrm{SL}_2(\mathbb{R})$. 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 the characteristic p-modular group 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.
If time permits, we will sketch future directions, namely for the p-adic groups SL(n,Q_p).
