Introduction to Arithmetic Groups

TL;DR

This book offers an accessible introduction to arithmetic subgroups of semisimple Lie groups, covering key theorems and background concepts, suitable for graduate students and self-study.

Contribution

It provides a comprehensive, beginner-friendly overview of arithmetic groups and major theorems, with exercises and background primers for learning and research.

Findings

Discussion of the Mostow Rigidity Theorem

Explanation of the Margulis Superrigidity Theorem

Classification of arithmetic subgroups of classical groups

Abstract

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 subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background.