Einführung in die kombinatorische und die geometrische Gruppentheorie
This textbook is based on a course on combinatorial and geometric group theory given by the authors at the University of Dortmund. The book is aimed at students of mathematics and computer science of an undergraduate level. The first seven chapters present the basics. It focuses on the Dehn problems and discusses the Nielsen and Reidemeister-Schreier rewriting methods, the Todd-Coxeter coset enumeration method, considering basic constructions as free products and free products with amalgamation, HNN-extensions and groups with one defining relation. This part culminates with the Magnus theorem that every group with one defining relation has a solvable word problem. The following twelve chapters present foundations of geometric group theory. It starts with Lyndon-van-Kampen diagrams and small cancellation theory, followed by the Dehn algorithm in small cancellation groups and the conjugation problem in these groups. These foundations are followed by six chapters on hyperbolic groups. The final short chapter introduces automatic groups.