SL(n,Z) como reticulado de SL(n,R) via conjuntos de Siegel

Abstract

In this work, we will present a reconstruction of the proof that the group SL(n, Z) forms a lattice in SL(n, R). With the aim of providing a geometric perspective on the definition of lattice and fundamental domain, we will introduce the concept of lattices in R^n. Additionally, we will explore essential concepts related to Lie groups. We will also briefly delve into the Haar measure. Drawing from various reference works, we will define Siegel sets as a suitable fundamental domain for the action of SL(n, Z) on SL(n, R). We will demonstrate that SL(n, Z) is a discrete subgroup of SL(n, R), and that the Siegel sets, properly defined, constitute a slightly larger and simpler fundamental domain for this action, with finite volume. Thus, SL(n, Z) forms a lattice in SL(n, R).