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

• Main Author: Montoya, Laura Daniela Bermúdez
• Publication Date: 2023
• Format: Master thesis
• Language: por
• Source: Repositório Institucional da UFU
• Download full: https://repositorio.ufu.br/handle/123456789/39012; http://doi.org/10.14393/ufu.di.2023.396
• topic: Conjuntos de Siegel; Grupos aritméticos; Reticulados em grupos de Lie; Siegel sets; Arithmetic groups; Lattices in Lie groups; CNPQ::CIENCIAS EXATAS E DA TERRA; Matemática; Grupos aritméticos; Geometria; Topologia algébrica

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).