Corpus ID: 17784418

Introduction to Arithmetic Groups

  title={Introduction to Arithmetic Groups},
  author={Dave Witte Morris},
This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice… Expand
Arithmeticity of discrete subgroups
  • Y. Benoist
  • Mathematics
  • Ergodic Theory and Dynamical Systems
  • 2020
Abstract The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work withExpand
Vi Arithmetic Subgroups
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A brief introduction to Quillen conjecture
Introduction: In 1971, Quillen stated a conjecture that on cohomology of arithmetic groups, a certain module structure over the Chern classes of the containing general linear group is free. OverExpand
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In this article, we compute the mean number of 2-torsion elements in class groups of monogenized cubic orders, when such orders are enumerated by height. In particular, we show that the average sizeExpand
On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one
In the first paper of this series ( we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groupsExpand
A tour through the proof of Margulis Superrigidity
Margulis’ Superrigidity theorem tells us that for a higher rank semisimple Lie groupG, any representation of a lattice Γ extends to a continuous representation ofG (under reasonably mild conditions).Expand
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We compute L²-invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel-Serre compactification of arithmetically defined locally symmetric spaces. The main results giveExpand
Constructing thin subgroups of SL(n + 1, ℝ) via bending
In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups ofExpand


A note on generators for arithmetic subgroups of algebraic groups
In this paper we construct systems of generators for arithmetic subgroups of algebraic groups. 1.1. Let k be a global field and G an absolutely almost simple simply connected (connected) ^-algebraicExpand
The quasi-isometry classification of lattices in semisimple Lie groups
This paper is a report on the recently completed quasi-isometry classification of lattices in semisimple 1 Lie groups. The main theorems stated here are a summary of work of several people over aExpand
Discrete Subgroups of Semisimple Lie Groups
1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1.Expand
On the compactness of arith-metically defined homogeneous spaces
Let G be an algebraic matric group defined over the field Q of rational numbers. Let GR denote the subgroup of elements in G with real coefficients, and let G, denote the subgroup of elements in GExpand
Non-vanishing theorems for the cohomology of certain arithmetic quotients.
This paper is concerned with constructions of automorphic forms on classical groups. Our main tool is a very old one, namely that of theta series. Suppose (G, G') is a real reductive dual pair [14].Expand
Tits Geometry, Arithmetic Groups, and the Proof of a Conjecture of Siegel
Let X = G/K be a Riemannian symmetric space of non- compact type and of rank 2. An irreducible, non-uniform lattice G in the isometry group of X is arithmetic and gives rise to a locally symmet- ricExpand
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The Dual Space of Semi-Simple Lie Groups
Introduction. This paper is inspired by Kazhdan's work [8]. In [8], he has studied the structure of lattices, i.e., discrete subgroups with finite invariant measure on the factor space, of a LieExpand
Isotropic nonarchimedean S-arithmetic groups are not left orderable
Abstract If O S is the ring of S-integers of an algebraic number field F, and O S has infinitely many units, we show that no finite-index subgroup of SL ( 2 , O S ) is left orderable. (Equivalently,Expand
Finite Factor Groups of the Unimodular Group
1. Let SL(n, Z) be the group of all n x n-matrices with rationali nteger coefficients and det = + 1. In the present paper, we shall study the finite factor groups of SL(n, Z). For n = 2, every finiteExpand