Profinite Graphs and Groups This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject. The author begins by carefully developing relevant notions in topology profinite groups and homology including free products of profinite groups cohomological methods in profinite groups and fixed points of automorphisms of free pro p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups with sections on finitely generated subgroups of free groups separability conditions in free and amalgamated products and algorithms in free groups and finite monoids. Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups geometric group theory graphs and connections with the theory of formal languages. A complete reference on the subject the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading 1 Preliminaries Part I Basic Theory 2 Profinite Graphs 3 The Fundamental Group of a Profinite Graph 4 Profinite Groups Acting on C Trees 5 Free Products of Pro C Groups 6 Graphs of Pro C Groups Part II Applications to Profinite Groups 7 Subgroups of Fundamental Groups of Graphs of Groups 8 Minimal Subtrees 9 Homology and Graphs of Pro C Groups 10 The Virtual Cohomological Dimension of Profinite Groups Part III Applications to Abstract Groups 11 Separability Conditions in Free and Polycyclic Groups 12 Algorithms in Free Groups and Monoids 13 Abstract Groups vs their Profinite Completions 14 Conjugacy in Free Products and in Free by finite Groups 15 Conjugacy Separability in Amalgamated Products Appendix A Abstract Graphs Appendix B Rational Sets in Free Groups and Automata Bibliography Index