##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Francois Thilmany

#### UCSD

## Lattices of Minimal Covolume in ${\rm SL}_n(\mathbb{R})$

##### Abstract:

A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\rm{SL}_2(\mathbb{R})$. 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 the 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 $\mathrm{SL}_n(\mathbb{R})$ is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving an answer to Lubotzky's question in this case.

Alireza Salehi Golsefidy

### November 27, 2017

### 2:00 PM

### AP&M 7321

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