An Invitation to Coarse Groups

Preface: About this Book
Contents
List of Symbols
1 Introduction
	1.1 Background and Motivation
	1.2 List of Findings I: Basic Theory
	1.3 List of Findings II: Selected Topics
Part I Basic Theory
	2 Introduction to the Coarse Category
		2.1 Some Notation for Subsets and Relations
		2.2 Coarse Structures
		2.3 Coarse Structures via Partial Coverings
		2.4 Controlled Maps
		2.5 The Category of Coarse Spaces
		2.6 Binary Products
		2.7 Equi Controlled Maps
	3 Properties of the Category of Coarse Spaces
		3.1 Pull-Back and Push-Forward
		3.2 Controlled Thickenings and Asymptoticity
		3.3 Coarse Subspaces, Restrictions, Images and Quotients
		3.4 Containments and Intersections of Coarse Subspaces
	4 Coarse Groups
		4.1 Preliminary: Group Objects in a Category
		4.2 Coarse Groups: Definition, Notation and Examples
		4.3 Equi Invariant Coarse Structures and Automatic Control
		4.4 Making Sets into Coarse Groups
		4.5 Coarse Groups are Determined Locally
		4.6 Summary: A Concrete Description of Coarse Groups
	5 Coarse Homomorphisms, Subgroups and Quotients
		5.1 Definitions and Examples of Coarse Homomorphisms
		5.2 Properties of Coarse Homomorphisms
		5.3 Coarse Subgroups
		5.4 Coarse Quotients
	6 Coarse Actions
		6.1 Definition and Examples
		6.2 Coarse Invariance and Equivariance
		6.3 Coarse Action by Conjugation
		6.4 Coarse Orbits
		6.5 The Fundamental Observation of Geometric Group Theory
		6.6 Quotient Coarse Actions
		6.7 Coarse Quotient Actions of the Action by Left Multiplication
		6.8 Coarse Cosets Spaces: Subsets
		6.9 Coarse Cosets Spaces: Subgroups
	7 Coarse Kernels
		7.1 Coarse Preimages and Kernels
		7.2 The Isomorphism Theorems
		7.3 Short Exact Sequences of Coarse Groups
		7.4 A Criterion for the Existence of Coarse Kernels
		7.5 Some Comments and Questions
Part II Selected Topics
	8 Coarse Structures on Set-Groups
		8.1 Connected Coarsifications of Set-Groups
		8.2 Metric Coarsifications and Bi-Invariant Metrics
		8.3 Some Examples of (Un)bounded Set-Groups
	9 Coarse Structures on Z
		9.1 Coarse Structures Generated by Cayley Graphs
		9.2 Coarse Structures Generated by Topologies
		9.3 Some Questions
	10 On Bi-Invariant Word Metrics
		10.1 Computing the Cancellation Metric on Free Groups
		10.2 The Canonical Coarsification of Finitely Generated Set-Groups and their Subgroups
	11 A Quest for Coarse Groups that are not Coarsified Set-Groups
		11.1 General Observations
		11.2 A Conjecture on Coarse Groups that are not Coarsified Set-Groups
		11.3 A Few More Questions
	12 On Coarse Homomorphisms and Coarse Automorphisms
		12.1 Elementary Constructions of Coarse Automorphisms
		12.2 Coarse Homomorphisms into Banach Spaces
		12.3 Hartnick-Schweitzer Quasimorphisms
		12.4 Coarse Homomorphisms of Finitely Normally GeneratedGroups
	13 Spaces of Controlled Maps
		13.1 Fragmentary Coarse Structures
		13.2 The Fragmentary Coarse Space of Controlled Maps
		13.3 Coarse Actions Revisited
		13.4 Frag-Coarse Groups of Controlled Transformations
		13.5 Coarse Groups as Coarse Subgroups
A Categorical Aspects of Coarse
	A.1 Limits and Colimits
	A.2 Subobjects and Quotients
	A.3 The Category of Coarse Groups
	A.4 The Category of Fragmentary Coarse Spaces
	A.5 Enriched Coarse Category
	A.6 The Pre-Coarse Category
B Metric Groups and Quasifications
	B.1 Quasifications of Metric Spaces
	B.2 Groups in Metric Categories
	B.3 Quasi-Metric Groups
C Extra Topics
	C.1 Computing Cancellation Metrics
	C.2 Proper Coarse Actions
References
Index