An Introduction to Right-angled Artin Groups

Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey article is meant to introduce readers to these groups and to give an overview of the relevant literature. Artin groups span a wide range of groups from braid groups to free groups to free abelian groups and have strong connections with geometry. They are defined by presentations of specific form and are closely related to Coxeter groups. Right-angled Artin groups are those Artin groups for which all relators are commutators between specified generators. On first glance the most elementary class of Artin groups, right-angled Artin groups turn out to have a surprising richness and flexibility that has led to some remarkable applications. Right-angled Artin groups were first introduced in the 1970’s by A. Baudisch [5] and further developed in the 1980’s by C. Droms [45] [46] [47] under the name “graph groups”. They have been studied extensively since that time (as evidenced by the long bibliography to this article). Recently, these groups have attracted much interest in geometric group theory due to their actions on CAT(0) cube complexes. In this survey article, we introduce the reader to the basic algebraic and geometric properties of right-angled Artin groups. While we have tried to touch on all the major developments and to give references to related work where appropriate, the author apologizes for any (no doubt many) omissions. The first section of the paper gives a brief introduction to more general Artin groups. The second section reviews essential properties of right-angled Artin (and Coxeter) groups and the cube complexes associated to them. The third section introduces some areas of current research on automorphisms and quasiisometries of right-angled Artin groups, and the final section discusses a few of our favorite applications. This article grew out of a series of talks given at the Pacific Geometry Conference at Oregon State University in June 2006. It is written for a broad audience. We hope that the content is interesting to geometric group theorists, but still accessible to those outside the field. The author would like to thank John Meier for offering some excellent suggestions during the preparation of this article.