Groups are algebraic objects, consisting of a set with a binary operation that satisfies a short list of required properties: the binary operation must be associative; there is an identity element; and every element has an inverse. Presenting groups in this formal, abstract algebraic manner is both useful and powerful. Yet it avoids a wonderful geometric perspective on group theory that is also useful and powerful, particularly in the study of infinite groups. This perspective is hinted at in the combinatorial approach to finite groups that is often seen in a first course in abstract algebra. It is my intention to bring the geometric perspective forward, to establish some elementary results that indicate the utility of this perspective, and to highlight some interesting examples of particular infinite groups along the way. My own bias is that these groups are just as interesting as the theorems.

The topics covered in this book fit inside of “geometric group theory,” a field that sits in the impressively large intersection of abstract algebra, geometry, topology, formal language theory, and many other fields. I hope that this book will provide an introduction to geometric group theory at a broadly accessible level, requiring nothing more than a single-semester exposure to groups and a naive familiarity with the combinatorial theory of graphs.

The chapters alternate between those devoted to general techniques and theorems (odd numbers) and brief chapters introducing some of the standard examples of infinite groups (even numbers).