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El. knyga: Geometric Group Theory: An Introduction

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  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 19-Dec-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319722542
Kitos knygos pagal šią temą:
Geometric Group Theory: An IntroductionDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 19-Dec-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319722542
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Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology.

Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability.

This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.

Recenzijos

The structure of the chapters can make the reader independent, thus the book can be used outside of the classroom for self-teaching by both young researchers and experienced scholars. The book is well written . it is ready to fill a gap in the literature for such an interesting and active branch of mathematics. (Dimitrios Varsos, zbMATH 1426.20001, 2020)

1 Introduction
1(6)
Part I Groups
7(44)
2 Generating groups
9(42)
2.1 Review of the category of groups
10(9)
2.1.1 Abstract groups: axioms
10(2)
2.1.2 Concrete groups: automorphism groups
12(4)
2.1.3 Normal subgroups and quotients
16(3)
2.2 Groups via generators and relations
19(12)
2.2.1 Generating sets of groups
19(1)
2.2.2 Free groups
20(5)
2.2.3 Generators and relations
25(4)
2.2.4 Finitely presented groups
29(2)
2.3 New groups out of old
31(20)
2.3.1 Products and extensions
32(2)
2.3.2 Free products and amalgamated free products
34(5)
2.E Exercises
39(12)
Part II Groups → Geometry
51(114)
3 Cayley graphs
53(22)
3.1 Review of graph notation
54(3)
3.2 Cayley graphs
57(4)
3.3 Cayley graphs of free groups
61(14)
3.3.1 Free groups and reduced words
62(3)
3.3.2 Free groups → trees
65(1)
3.3.3 Trees → free groups
66(2)
3.E Exercises
68(7)
4 Group actions
75(40)
4.1 Review of group actions
76(10)
4.1.1 Free actions
77(3)
4.1.2 Orbits and stabilisers
80(3)
4.1.3 Application: Counting via group actions
83(1)
4.1.4 Transitive actions
84(2)
4.2 Free groups and actions on trees
86(9)
4.2.1 Spanning trees for group actions
87(1)
4.2.2 Reconstructing a Cayley tree
88(4)
4.2.3 Application: Subgroups of free groups are free
92(3)
4.3 The ping-pong lemma
95(2)
4.4 Free subgroups of matrix groups
97(18)
4.4.1 Application: The group SL(2, Z) is virtually free
97(3)
4.4.2 Application: Regular graphs of large girth
100(2)
4.4.3 Application: The Tits alternative
102(3)
4.E Exercises
105(10)
5 Quasi-isometry
115(50)
5.1 Quasi-isometry types of metric spaces
116(6)
5.2 Quasi-isometry types of groups
122(5)
5.2.1 First examples
125(2)
5.3 Quasi-geodesics and quasi-geodesic spaces
127(5)
5.3.1 (Quasi-)Geodesic spaces
127(1)
5.3.2 Geodesification via geometric realisation of graphs
128(4)
5.4 The Svarc--Milnor lemma
132(9)
5.4.1 Application: (Weak) commensurability
137(2)
5.4.2 Application: Geometric structures on manifolds
139(2)
5.5 The dynamic criterion for quasi-isometry
141(7)
5.5.1 Application: Comparing uniform lattices
146(2)
5.6 Quasi-isometry invariants
148(17)
5.6.1 Quasi-isometry invariants
148(2)
5.6.2 Geometric properties of groups and rigidity
150(1)
5.6.3 Functorial quasi-isometry invariants
151(5)
5.E Exercises
156(9)
Part III Geometry of groups
165(152)
6 Growth types of groups
167(36)
6.1 Growth functions of finitely generated groups
168(2)
6.2 Growth types of groups
170(9)
6.2.1 Growth types
171(1)
6.2.2 Growth types and quasi-isometry
172(4)
6.2.3 Application: Volume growth of manifolds
176(3)
6.3 Groups of polynomial growth
179(9)
6.3.1 Nilpotent groups
180(1)
6.3.2 Growth of nilpotent groups
181(1)
6.3.3 Polynomial growth implies virtual nilpotence
182(2)
6.3.4 Application: Virtual nilpotence is geometric
184(1)
6.3.5 More on polynomial growth
185(1)
6.3.6 Quasi-isometry rigidity of free Abelian groups
186(1)
6.3.7 Application: Expanding maps of manifolds
187(1)
6.4 Groups of uniform exponential growth
188(15)
6.4.1 Uniform exponential growth
188(2)
6.4.2 Uniform uniform exponential growth
190(1)
6.4.3 The uniform Tits alternative
190(2)
6.4.4 Application: The Lehmer conjecture
192(2)
6.E Exercises
194(9)
7 Hyperbolic groups
203(54)
7.1 Classical curvature, intuitively
204(4)
7.1.1 Curvature of plane curves
204(1)
7.1.2 Curvature of surfaces in R3
205(3)
7.2 (Quasi-)Hyperbolic spaces
208(12)
7.2.1 Hyperbolic spaces
208(2)
7.2.2 Quasi-hyperbolic spaces
210(3)
7.2.3 Quasi-geodesics in hyperbolic spaces
213(6)
7.2.4 Hyperbolic graphs
219(1)
7.3 Hyperbolic groups
220(4)
7.4 The word problem in hyperbolic groups
224(5)
7.4.1 Application: "Solving" the word problem
225(4)
7.5 Elements of infinite order in hyperbolic groups
229(17)
7.5.1 Existence
229(6)
7.5.2 Centralisers
235(6)
7.5.3 Quasi-convexity
241(4)
7.5.4 Application: Products and negative curvature
245(1)
7.6 Non-positively curved groups
246(11)
7.E Exercises
250(7)
8 Ends and boundaries
257(32)
8.1 Geometry at infinity
258(1)
8.2 Ends
259(8)
8.2.1 Ends of geodesic spaces
259(3)
8.2.2 Ends of quasi-geodesic spaces
262(2)
8.2.3 Ends of groups
264(3)
8.3 The Gromov boundary
267(10)
8.3.1 The Gromov boundary of quasi-geodesic spaces
267(2)
8.3.2 The Gromov boundary of hyperbolic spaces
269(1)
8.3.3 The Gromov boundary of groups
270(1)
8.3.4 Application: Free subgroups of hyperbolic groups
271(6)
8.4 Application: Mostow rigidity
277(3)
8.5 Exercises
280(9)
9 Amenable groups
289(28)
9.1 Amenability via means
290(5)
9.1.1 First examples of amenable groups
290(2)
9.1.2 Inheritance properties
292(3)
9.2 Further characterisations of amenability
295(9)
9.2.1 Følner sequences
295(3)
9.2.2 Paradoxical decompositions
298(2)
9.2.3 Application: The Banach-Tarski paradox
300(2)
9.2.4 (Co)Homological characterisations of amenability
302(2)
9.3 Quasi-isometry invariance of amenability
304(1)
9.4 Quasi-isometry vs. bilipschitz equivalence
305(12)
9.E Exercises
309(8)
Part IV Reference material
317(2)
A Appendix
319(34)
A.1 The fundamental group
320(5)
A.1.1 Construction and examples
320(2)
A.1.2 Covering theory
322(3)
A.2 Group (co)homology
325(4)
A.2.1 Construction
325(2)
A.2.2 Applications
327(2)
A.3 The hyperbolic plane
329(20)
A.3.1 Construction of the hyperbolic plane
329(1)
A.3.2 Length of curves
330(2)
A.3.3 Symmetry and geodesies
332(9)
A.3.4 Hyperbolic triangles
341(5)
A.3.5 Curvature
346(1)
A.3.6 Other models
347(2)
A.4 An invitation to programming
349(4)
Bibliography 353(14)
Index of notation 367(6)
Index 373
Clara Löh is Professor of Mathematics at the University of Regensburg, Germany. Her research focuses on the interaction between geometric topology, geometric group theory, and measurable group theory. This includes cohomological, geometric, and combinatorial methods. 
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