ALGB Seminar: François Thilmany (University of California, San Diego)

Title: Lattices of minimal covolume in SL(n,R)

Abstract: 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 \mathrm{SL}_2(F) with F=\mathbb{F}_q((t)) is given by the so-called p-adic modular group \mathrm{SL}_2(\mathbb{F}_q[1/t]). He noted that, in contrast with Siegel’s lattice, the quotient by \mathrm{SL}_2(\mathbb{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 briefly review some of these known results, and then discuss the case of \mathrm{SL}_n(\mathbb{R}) for n > 2. It turns out that, up to automorphism, the unique lattice of minimal covolume in \mathrm{SL}_n(\mathbb{R}) is \mathrm{SL}_n(\mathbb{Z}). In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case.