Arithmetic Groups and Their Generalizations : What, Why, and How

In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n,\mathbf{Z})$.

Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known.

Indeed, arithmetic groups admit many natural and important generalizations.

The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry.

It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics.