# Séminaires : Séminaire d'Algèbres d'Opérateurs

 Equipe(s) : ao, Responsables : Pierre Fima, François Le Maître, Romain Tessera Email des responsables : Salle : 1013 Adresse : Sophie Germain Description

 Orateur(s) François Thilmany - FNRS, Titre Réseaux de covolume minimal dans SL_n(R) Date 09/01/2020 Horaire 14:00 à 15:00 Diffusion Résume 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 characteristic $p$ 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 review some of the 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 $\mathrm{SL}_n(\mathbb{R})$ ($n > 2$) is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case. Salle Adresse Sophie Germain
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