Automatic groups: A guided tour
This paper is a useful survey on the theory of automatic groups, which has been developed by Thurston et al. It contains self-contained definitions and a few historical remarks. The author explains how that theory grew out of the desire to understand the structure of a finitely generated group by studying the geometry of its Cayley graph. The paper contains also a list of examples and non-examples, with sometimes ideas of proofs. In particular, there is a detailed proof of a result due to Cannon, which asserts that the fundamental group of a compact manifold with strictly negative curvature is automatic. Finally, the paper contains a list of 12 interesting open problems. The reader should be aware of the fact that problem number 1, conjecturing that mapping class groups of surfaces of negative Euler characteristic are automatic (a conjecture due to Thurston) has been solved a couple of years ago by Lee Mosher.
scientific article; zbMATH DE number 221265