Infinite Group Actions on Polyhedra

In the past fifteen years, the theory of right-angled Artin groups and special cube complexes has emerged as a central topic in geometric group theory. This monograph provides an account of this theory, along with other modern techniques in geometric group theory. Structured around the theme of group actions on contractible polyhedra, this book explores two prominent methods for constructing such actions: utilizing the group of deck transformations of the universal cover of a nonpositively curved polyhedron and leveraging the theory of simple complexes of groups. The book presents various approaches to obtaining cubical examples through CAT(0) cube complexes, including the polyhedral product construction, hyperbolization procedures, and the Sageev construction. Moreover, it offers a unified presentation of important non-cubical examples, such as Coxeter groups, Artin groups, and groups that act on buildings. Designed as a resource for graduate students and researchers specializing in geometric group theory, this book should also be of high interest to mathematicians in related areas, such as 3-manifolds.

Michael Davis received a PhD in mathematics from Princeton University in 1975. He was Professor of Mathematics at Ohio State University for thirty nine years, retiring in 2022 as Professor Emeritus. In 2015 he became a Fellow of the AMS. His research is in geometric group theory and topology. Since 1981 his work has focused on topics related to reflection groups including the construction of new examples of aspherical manifolds and the study of their properties.