CAT(0) cube complexes : an introduction / Petra Schwer.

Format:

Book

Author/Creator:

Schwer, Petra, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 2324.

Lecture Notes in Mathematics, 0075-8434 ; v. 2324

Language:

English

Subjects (All):

Geometric group theory.

Physical Description:

xii, 186 pages : illustrations ; 24 cm

Place of Publication:

Cham : Springer, [2023]

Summary:

In recent years cube complexes have become a cornerstone topic of geometric group theory and have proven to be a powerful tool in other areas, such as low dimensional topology, phylogenetic trees or in the context of optimization problems. This book covers a wide variety of algebraic and geometric properties of cube complexes and the groups acting on them. The content ranges from basic properties of metric spaces, notions of non-positive curvature, Gromov's link condition and the varcMilnor theorem to advanced material such as the cubulation of half-space systems and the Roller boundary, the construction of cube complexes associated with Coxeter groups, and the Tits alternative for cubical groups. Being the first self-contained, comprehensive introduction to cube complexes this book serves as an entry point for researchers interested in the subject. The material is accessible to advanced undergraduate and graduate students. The text is illustrated with many figures and examples and comes with a large collection of exercises.

Contents:

Intro

Preface

This book ...

Contents

1 Introduction

1.1 What Is a Cube Complex?

1.2 What Is the CAT(0) Property?

1.3 Which Cube Complexes Are CAT(0)?

1.4 Why Should One Study CAT(0) Cube Complexes?

1.5 Where to Look for Cube Complexes?

1.6 What's in This Book?

1.7 Some (of the Many) Things I Have Omitted

1.8 Final Remarks

2 Metric Spaces Meet Groups

2.1 A Tiny Bit of Metric Geometry

2.2 Groups as Metric Spaces

2.3 Exercises

3 Non-positive Curvature

3.1 Metric Spaces of Non-positive Curvature

3.1.1 Model Spaces of Constant Sectional Curvature

3.1.2 Definition of CAT(0) Spaces

3.1.3 Properties of CAT(0) Spaces

3.2 Angles in CAT(0) Spaces

3.3 Flat Cones

3.4 Exercises

4 Cube Complexes and Gromov's Link Condition

4.1 Polyhedral Complexes and the Link Condition

4.1.1 The Local Structure of Cell Complexes

4.2 CAT(0) Cube Complexes

4.3 Cube Completions

4.4 Right Angled Artin Groups

4.5 Exercises

5 Hyperplanes and Half-Spaces

5.1 Hyperplanes

5.2 Half-Space Systems

5.3 Roller Duality

5.4 Exercises

6 Cubulating Coxeter Groups

6.1 Coxeter Groups: Definition and Examples

6.2 The Deletion Condition

6.3 Walls in Coxeter Groups

6.4 Half-Space System for Coxeter Groups

6.5 Exercises

7 A Panoramic Tour

7.1 Fixed Point Properties of Group Actions

7.1.1 Actions on Trees

7.1.2 Helly's Theorem for CAT(0) Cube Complexes

7.2 The Tits Alternative

7.2.1 HNN-Extensions and Free Amalgamations

7.2.2 Ends of Groups

7.2.3 Fuchsian Groups

7.2.4 The Tits Alternative

7.3 Special Cube Complexes

7.3.1 What Makes a Cube Complex Special?

7.3.2 Special Cubulated Groups Embed in RAAGs

7.3.3 Cube Complexes Meet 3-Dimensional Manifolds

7.4 Phylogenetic Trees

7.4.1 Modeling Mutation Using Trees

7.4.2 Exploring the Structure of n-Trees

7.4.3 The Space of Phylogenetic Trees

7.5 Exercises

References

Index

Notes:

Includes bibliographical references (pages 173-178) and index.

ISBN:

3031436210

9783031436215

OCLC:

1393079626