# Introduction to Arithmetic Groups

@inproceedings{Morris2015IntroductionTA, title={Introduction to Arithmetic Groups}, author={Dave Witte Morris}, year={2015} }

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

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#### References

SHOWING 1-10 OF 388 REFERENCES

A note on generators for arithmetic subgroups of algebraic groups

- Mathematics
- 1992

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) ^-algebraic… Expand

The quasi-isometry classification of lattices in semisimple Lie groups

- Mathematics
- 1997

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 a… Expand

Discrete Subgroups of Semisimple Lie Groups

- Mathematics
- 1991

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

- Mathematics
- 1962

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 G… Expand

Non-vanishing theorems for the cohomology of certain arithmetic quotients.

- Mathematics
- 1992

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

- Mathematics
- 2002

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- ric… Expand

On systems of generators of arithmetic subgroups of higher rank groups

- Mathematics
- 1994

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of… Expand

The Dual Space of Semi-Simple Lie Groups

- Mathematics
- 1969

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 Lie… Expand

Isotropic nonarchimedean S-arithmetic groups are not left orderable

- Mathematics
- 2004

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

- Mathematics
- 1965

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 finite… Expand