Lattices of minimal covolume in SL(n,R)

Venue:
Geb. 20.30, SR 2.058

Date:
16.11.2023
 Speaker:

Time:
15:45 Uhr

Abstract: A classical result of Siegel asserts that the (2,3,7)triangle group attains the smallest covolume among lattices of 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 socalled 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 the basic notions at hand, 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.