Richard Thompson's famous group \(F\) has the striking property that it can be realized as a dense subgroup of the group of all orientation-preserving homeomorphisms of the unit interval, but it can also be given by a simple 2-generator-2-relator presentation, in fact as the fundamental group of an aspherical complex with only two cells in each dimension.

This monograph studies a natural generalization of \(F\) that also includes Melanie Stein's generalized \(F\)-groups. The main aims of this monograph are the determination of isomorphisms among the generalized \(F\)-groups and the study of their automorphism groups.

This book is aimed at graduate students (or teachers of graduate students) interested in a class of examples of torsion-free infinite groups with elements and composition that are easy to describe and work with, but have unusual properties and surprisingly small presentations in terms of generators and defining relations.