Infinite Group Actions on Polyhedra

Michael W. Davis (Autor)

Buch | Hardcover

XI, 271 Seiten

2024
Springer International Publishing (Verlag)
978-3-031-48442-1 (ISBN)

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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.

Part I: Introduction.- 1 Introduction.- Part II: Nonpositively curved cube complexes.- 2 Polyhedral preliminaries.- 3 Right-angled spaces and groups.- Part III: Coxeter groups, Artin groups, buildings.- 4 Coxeter groups, Artin groups, buildings.- Part IV: More on NPC cube complexes.- 5 General theory of cube complexes.- 6 Hyperbolization.- 7 Morse theory and Bestvina-Brady groups.- Appendix A: Complexes of groups.

The book under review provides a survey of topics in geometric group theory . This text also benefits from a large number of examples. Each chapter also contains historical background and connections to the literature. The content of the book is structured in four parts and an appendix. (Michael Dougherty, Mathematical Reviews, September, 2025)

The author made the right choice not to include proofs of the theorems stated (otherwise the size of the book would have increased by almost an order of magnitude) but only to indicate the sources where such proofs can be found. This implies a certain degree of maturity and knowledge in the reader. For this reason, the text is dedicated to advanced graduate students or to researchers. (Egle Bettio, zbMATH 1542.20002, 2024)

Erscheinungsdatum	06.03.2024
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Zusatzinfo	XI, 271 p. 9 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Artin groups • Bestvina-Brady groups • buildings • CAT (0) cube complex • Coxeter groups • Cube Complexes • Hyperbolic 3-manifolds • Hyperbolic Groups • hyperbolization • nonpositively curved polyhedra • polyhedral products • Sageev's construction • Sageev’s construction • separable subgroups • simple complexes of groups • virtual Haken conjecture • virtually Haken conjecture
ISBN-10	3-031-48442-8 / 3031484428
ISBN-13	978-3-031-48442-1 / 9783031484421
Zustand	Neuware