# Lattices of minimal covolume in SL_n(R)

## TL;DR

This paper proves that for dimensions greater than 3, the minimal covolume lattices in SL_n(R) are uniquely given by SL_n(Z), showing they are non-uniform and answering a key question about their structure.

## Contribution

It establishes the uniqueness of minimal covolume lattices in SL_n(R) for n > 3, identifying SL_n(Z) as the sole such lattice up to automorphism.

## Key findings

- SL_n(Z) is the unique minimal covolume lattice in SL_n(R) for n > 3
- Lattices of minimal covolume are non-uniform for n > 3
- Answers Lubotzky's question on the typical nature of minimal covolume lattices

## Abstract

The objective of this paper is to determine the lattices of minimal covolume in SL_n(R), for n greater than 3. The answer turns out to be the simplest one: SL_n(Z) is, up to automorphism, the unique lattice of minimal covolume in SL_n(R). In particular, lattices of minimal covolume in SL_n(R) are non-uniform when n is greater than 3, contrasting with Siegel's result for SL_2(R). This answers for SL_n(R) the question of Lubotzky: is a lattice of minimal covolume typically uniform or not?