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Generic Coarse Geometry of Leaves 2018 ed. [Minkštas viršelis]

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  • Formatas: Paperback / softback, 173 pages, aukštis x plotis: 235x155 mm, weight: 2993 g, 16 Illustrations, black and white; XV, 173 p. 16 illus., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2223
  • Išleidimo metai: 29-Jul-2018
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319941313
  • ISBN-13: 9783319941318
Kitos knygos pagal šią temą:
Generic Coarse Geometry of Leaves 2018 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 173 pages, aukštis x plotis: 235x155 mm, weight: 2993 g, 16 Illustrations, black and white; XV, 173 p. 16 illus., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2223
  • Išleidimo metai: 29-Jul-2018
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319941313
  • ISBN-13: 9783319941318
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319941318.html
Raktažodžiai:
This book provides a detailed introduction to the coarse quasi-isometry of leaves of a foliated space and describes the cases where the generic leaves have the same quasi-isometric invariants.







Every leaf of a compact foliated space has an induced coarse quasi-isometry type, represented by the coarse metric defined by the length of plaque chains given by any finite foliated atlas.  When there are dense leaves either all dense leaves without holonomy are uniformly coarsely quasi-isometric to each other, or else every leaf is coarsely quasi-isometric to just meagerly many other leaves.  Moreover, if all leaves are dense, the first alternative is characterized by a condition on the leaves called coarse quasi-symmetry.  Similar results are proved for more specific coarse invariants, like growth type, asymptotic dimension, and amenability. The Higson corona of the leaves is also studied. All the results are richly illustrated with examples.







The book is primarily aimed at researchers on foliated spaces. More generally, specialists in geometric analysis, topological dynamics, or metric geometry may also benefit from it.
1 Introduction
1(12)
Part I Coarse Geometry of Metric Spaces
2 Coarse Quasi-Isometries
13(16)
2.1 Notation, Conventions and Terminology
13(1)
2.2 Coarse Quasi-Isometries
14(2)
2.3 Coarse Composites
16(3)
2.4 A Coarsely Quasi-Isometric Version of Arzela-Ascoli Theorem
19(1)
2.5 Large Scale Lipschitz Maps
19(5)
2.6 Coarse and Rough Maps
24(5)
3 Some Classes of Metric Spaces
29(8)
3.1 Graphs
29(3)
3.2 Metric Spaces of Coarse Bounded Geometry
32(2)
3.3 Coarsely Quasi-Symmetric Metric Spaces
34(1)
3.4 Coarsely Quasi-Convex Metric Spaces
35(2)
4 Growth of Metric Spaces
37(6)
4.1 Growth of Non-decreasing Functions
37(1)
4.2 Growth of Metric Spaces
38(3)
4.3 Growth Symmetry
41(2)
5 Amenability of Metric Spaces
43(8)
5.1 Amenability
43(4)
5.2 Amenable Symmetry
47(4)
6 Coarse Ends
51(14)
6.1 Ends
51(1)
6.2 Coarse Connectivity
52(2)
6.3 Coarse Ends
54(4)
6.4 Functoriality of the Space of Coarse Ends
58(3)
6.5 Coarse End Space of a Class of Metric Spaces
61(4)
7 Higson Corona and Asymptotic Dimension
65(12)
7.1 Compactifications
65(3)
7.2 Higson Compactification
68(5)
7.3 Asymptotic Dimension
73(4)
Part II Coarse Geometry of Orbits and Leaves
8 Pseudogroups
77(14)
8.1 Pseudogroups
77(2)
8.2 Coarse Quasi-Isometry Type of Orbits
79(2)
8.3 A Coarse Version of Local Reeb Stability
81(4)
8.4 Topological Dynamics
85(6)
8.4.1 Preliminaries on Baire Category
85(1)
8.4.2 Saturated Sets
86(1)
8.4.3 The Property of Being Recurrent on Baire Sets
87(1)
8.4.4 Pseudogroups Versus Group Actions
88(3)
9 Generic Coarse Geometry of Orbits
91(24)
9.1 Coarsely Quasi-Isometric Orbits
92(3)
9.2 Growth of the Orbits
95(7)
9.2.1 Orbits with the Same Growth Type
95(2)
9.2.2 Some Growth Classes of the Orbits
97(5)
9.3 Amenable Orbits
102(3)
9.4 Asymptotic Dimension of the Orbits
105(3)
9.5 Highson Corona of the Orbits
108(5)
9.5.1 Limit Sets
108(4)
9.5.2 Semi Weak Homogeneity of the Higson Corona
112(1)
9.6 Measure Theoretic Versions
113(2)
10 Generic Coarse Geometry of Leaves
115(18)
10.1 Foliated Spaces
115(3)
10.2 Saturated Sets
118(1)
10.3 Coarse Quasi-Isometry Type of the Leaves
119(5)
10.4 Higson Corona of the Leaves
124(3)
10.4.1 Higson Compactification
124(1)
10.4.2 Limit Sets
125(2)
10.5 Algebraic Asymptotic Invariants
127(2)
10.6 Versions with Quasi-Invariant Currents
129(2)
10.7 There Is No Measure Theoretic Version of Recurrence
131(2)
11 Examples and Open Problems
133(30)
11.1 Foliated Spaces Defined by Suspensions
133(1)
11.2 Foliated Spaces Defined by Locally Free Actions of Lie Groups
134(1)
11.3 Inverse Limits of Covering Spaces
135(1)
11.4 Bounded Geometry and Leaves
135(3)
11.5 Graph Spaces
138(2)
11.6 Case of Cayley Graphs
140(2)
11.7 Construction of Limit-Aperiodic Functions
142(1)
11.8 Graph Matchbox Manifolds
143(2)
11.9 Concrete Examples of Graph Matchbox Manifolds
145(5)
11.9.1 The Ghys-Kenyon Matchbox Manifold
145(3)
11.9.2 An Example with Uncountably Many Growth Types of Leaves
148(1)
11.9.3 An Example with Equi-Amenable Dense Leaves and Other Non-amenable Leaves
149(1)
11.9.4 An Example with Leaves of Infinite Asymptotic Dimension
150(1)
11.10 Foliations of Codimension One
150(11)
11.10.1 Hector's Example
151(2)
11.10.2 Generic Ergodicity of Equivalence Relations
153(1)
11.10.3 Generic Ergodicity of Metric Equivalence Relations
154(1)
11.10.4 Generic Ergodicity of the Growth Type Relation in Hector's Example
155(4)
11.10.5 The Theory of Levels
159(2)
11.11 Open Problems
161(2)
References 163(6)
Index 169
Jesśs A. Įlvarez López, was born in 1962, and studied Mathematics at the University of Santiago de Compostela (Spain), obtaining the PhD in 1987. He visited the Department of Mathematics of the University of Illinois at Urbana-Champaign during 1988-1990, with a Fulbright grant. Nowadays, he is a professor at the University of Santiago de Compostela. His research is about several topics, like cohomology of foliations, description of equicontinuous foliated spaces, quasi-isometric types of leaves, leafwise heat flow, a trace formula for foliated flows, and Wittens complex on stratified spaces.  Alberto Candel did his undergraduate work in Mathematics at the University of Santiago de Compostela, worked briefly at the Universidad de Oviedo (Spain), and then moved on to the USA do his graduate work at Washington University in St. Louis, obtaining his PhD in Mathematics in 1992 under the direction of L. Conlon. After postdoctoral work at several places(IAS, U of Chicago, and Caltech), he settled at California State University, Northridge in 2000. His research is in geometric analysis and dynamical systems, with particular emphasis foliations.