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Office Hours with a Geometric Group Theorist 1st Edition by Matt Clay, Dan Margalit ISBN 9781400885398 1400885396

Name: Office Hours with a Geometric Group Theorist 1st Edition by Matt Clay, Dan Margalit ISBN 9781400885398 1400885396
SKU: EB-51950168
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Office Hours with a Geometric Group Theorist Matt Clay (Editor) Digital Instant Download

Author(s): Matt Clay (editor); Dan Margalit (editor)

ISBN(s): 9781400885398, 1400885396

Edition: Pilot project, eBook available to selected US libraries only

File Details: PDF, 2.69 MB

Year: 2017

Language: English

SKU: EB-51950168 Category: Mathematics Tags: Dan Margalit (editor), Matt Clay (editor)

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Office Hours with a Geometric Group Theorist 1st Edition by Matt Clay, Dan Margalit – Ebook PDF Instant Download/Delivery: 9781400885398 ,1400885396
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ISBN 10: 1400885396
ISBN 13: 9781400885398
Author: Matt Clay, Dan Margalit

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It’s like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson’s groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Office Hours with a Geometric Group Theorist 1st Edition Table of contents:

PART 1. GROUPS AND SPACES

1. Groups

1.1 Groups

1.2 Infinite groups

1.3 Homomorphisms and normal subgroups

1.4 Group presentations

2. . . . and Spaces

2.1 Graphs

2.2 Metric spaces

2.3 Geometric group theory: groups and their spaces

PART 2. FREE GROUPS

3. Groups Acting on Trees

3.1 The Farey tree

3.2 Free actions on trees

3.3 Non-free actions on trees

4. Free Groups and Folding

4.1 Topological model for the free group

4.2 Subgroups via graphs

4.3 Applications of folding

5. The Ping-Pong Lemma

5.1 Statement, proof, and first examples using ping-pong

5.2 Ping-pong with Möbius transformations

5.3 Hyperbolic geometry

5.4 Final remarks

6. Automorphisms of Free Groups

6.1 Automorphisms of groups: first examples

6.2 Automorphisms of free groups: a first look

6.3 Train tracks

PART 3. LARGE SCALE GEOMETRY

7. Quasi-isometries

7.1 Example: the integers

7.2 Bi-Lipschitz equivalence of word metrics

7.3 Quasi-isometric equivalence of Cayley graphs

7.4 Quasi-isometries between groups and spaces

7.5 Quasi-isometric rigidity

8. Dehn Functions

8.1 Jigsaw puzzles reimagined

8.2 A complexity measure for the word problem

8.3 Isoperimetry

8.4 A large-scale geometric invariant

8.5 The Dehn function landscape

9. Hyperbolic Groups

9.1 Definition of hyperbolicity

9.2 Examples and nonexamples

9.3 Surface groups

9.4 Geometric properties

9.5 Hyperbolic groups have solvable word problem

10. Ends of Groups

10.1 An example

10.2 The number of ends of a group

10.3 Semidirect products

10.4 Calculating the number of ends of the braid groups

10.5 Moving beyond counting

11. Asymptotic Dimension

11.1 Dimension

11.2 Motivating examples

11.3 Large-scale geometry

11.4 Topology and dimension

11.5 Large-scale dimension

11.6 Motivating examples revisited

11.7 Three questions

11.8 Other examples

12. Growth of Groups

12.1 Growth series

12.2 Cone types

12.3 Formal languages and context-free grammars

12.4 The DSV method

PART 4. EXAMPLES

13. Coxeter Groups

13.1 Groups generated by reflections

13.2 Discrete groups generated by reflections

13.3 Relations in finite groups generated by reflections

13.4 Coxeter groups

14. Right-Angled Artin Groups

14.1 Right-angled Artin groups as subgroups

14.2 Connections with other classes of groups

14.3 Subgroups of right-angled Artin groups

14.4 The word problem for right-angled Artin groups

15. Lamplighter Groups

15.1 Generators and relators

15.2 Computing word length

15.3 Dead end elements

15.4 Geometry of the Cayley graph

15.5 Generalizations

16. Thompson’s Group

16.1 Analytic definition and basic properties

16.2 Combinatorial definition

16.3 Presentations

16.4 Algebraic structure

16.5 Geometric properties

17. Mapping Class Groups

17.1 A brief user’s guide to surfaces

17.2 Homeomorphisms of surfaces

17.3 Mapping class groups

17.4 Dehn twists in the mapping class group

17.5 Generating the mapping class group by Dehn twists

18. Braids

18.1 Getting started

18.2 Some group theory

18.3 Some topology: configuration spaces

18.4 More topology: punctured disks

18.5 Connection: knot theory

18.6 Connection: robotics

18.7 Connection: hyperplane arrangements

18.8 A stylish and practical finale

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Tags: Matt Clay, Dan Margalit, Office Hours, Geometric Group