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An Invitation to Coarse Groups / / by Arielle Leitner, Federico Vigolo
| An Invitation to Coarse Groups / / by Arielle Leitner, Federico Vigolo |
| Autore | Leitner Arielle |
| Edizione | [1st ed. 2023.] |
| Pubbl/distr/stampa | Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023 |
| Descrizione fisica | 1 online resource (249 pages) |
| Disciplina | 512.55 |
| Altri autori (Persone) | VigoloFederico |
| Collana | Lecture Notes in Mathematics |
| Soggetto topico |
Group theory
Topological groups Lie groups Group Theory and Generalizations Topological Groups and Lie Groups Teoria de grups Grups topològics Grups de Lie |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-42760-2 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- 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. |
| Record Nr. | UNISA-996587869503316 |
Leitner Arielle
|
||
| Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
An Invitation to Coarse Groups / / by Arielle Leitner, Federico Vigolo
| An Invitation to Coarse Groups / / by Arielle Leitner, Federico Vigolo |
| Autore | Leitner Arielle |
| Edizione | [1st ed. 2023.] |
| Pubbl/distr/stampa | Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023 |
| Descrizione fisica | 1 online resource (249 pages) |
| Disciplina | 512.55 |
| Altri autori (Persone) | VigoloFederico |
| Collana | Lecture Notes in Mathematics |
| Soggetto topico |
Group theory
Topological groups Lie groups Group Theory and Generalizations Topological Groups and Lie Groups Teoria de grups Grups topològics Grups de Lie |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-42760-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- 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. |
| Record Nr. | UNINA-9910770247003321 |
|
Leitner Arielle
|
||
| Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||