Aspects of the Theory of Free Groups

Abstract

Combinatorial group theory concerns itself with the theory of group presentations, that is, it deals with groups specified by a set of generators and corresponding defining relations. The theory has its origins and roots in topology and complex analysis and in particular in the theory of the fundamental groups of combinatorial cell complexes. Because of its nature and origins, combinatorial group theory comes into contact with, and uses, many different areas of mathematics: clearly algebra and topology as mentioned above, but also hyperbolic geometry via the study of the Cayley graph; pure mathematical logic through the study of various decision problems; and computer science through the study of computational aspects of group theory and rewriting systems. These last three areas have been the subject of increased interest and intense study during the 1990s. The ties with hyperbolic geometry have evolved into geometric group theory—especially the theory of the hyperbolic groups—by considering the groups themselves as geometric objects via the Cayley graph. The ties with computation and computer science have evolved into the theory of automatic groups, where rewriting and computability properties are paramount. Finally, there have been extensive studies on the nature of solutions of the various decision problems and on the logical underpinnings of the the whole theory. Relative to the latter there has been a great deal of work on the Tarski conjectures (see below) and their ties to purely group theoretical properties.KeywordsFree GroupCayley GraphFree ProductHyperbolic GroupMaximal Abelian SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.