Symmetries in Graphs, Maps, and Polytopes
5th SIGMAP Workshop, West Malvern, UK, July 2014
1st Edition
Published on 26. March 2016
X, 332 pages
978-3-319-30451-9 (ISBN)
Description
This volume contains seventeen of the best papers delivered at the SIGMAP Workshop 2014, representing the most recent advances in the field of symmetries of discrete objects and structures, with a particular emphasis on connections between maps, Riemann surfaces and dessins d'enfant.Providing the global community of researchers in the field with the opportunity to gather, converse and present their newest findings and advances, the Symmetries In Graphs, Maps, and Polytopes Workshop 2014 was the fifth in a series of workshops. The initial workshop, organized by Steve Wilson in Flagstaff, Arizona, in 1998, was followed in 2002 and 2006 by two meetings held in Aveiro, Portugal, organized by Antonio Breda d'Azevedo, and a fourth workshop held in Oaxaca, Mexico, organized by Isabel Hubard in 2010.This book should appeal to both specialists and those seeking a broad overview of what is happening in the area of symmetries of discrete objects and structures.iv>
More details
Series
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Product notice
Reflowable
Illustrations
X, 332 p. 46 illus.
File size
9,42 MB
ISBN-13
978-3-319-30451-9 (9783319304519)
DOI
10.1007/978-3-319-30451-9
Schweitzer Classification
Other editions
Additional editions
Jozef Sirán | Robert Jajcay
Symmetries in Graphs, Maps, and Polytopes
5th SIGMAP Workshop, West Malvern, UK, July 2014
Book
03/2016
Springer
€160.49
Shipment within 10-15 days
Content
- Intro
- Preface
- Acknowledgements
- Contents
- Powers of Skew-Morphisms
- 1 Introduction
- 2 Basic Properties of Skew-Morphisms and Their Relation to Cayley Maps
- 3 Generalization of t-Balanced Skew-Morphisms
- 4 Powers of Skew-Morphisms
- 5 Coset-Preserving Powers
- 6 Coset-Preserving Skew-Morphisms of Cyclic Groups
- References
- Census of Quadrangle Groups Inclusions
- 1 Introduction
- 2 Generalised Quadrangle Groups and Constellations
- 3 How to Read the Census
- References
- Some Unexpected Consequences of Symmetry Computations
- 1 Introduction
- 2 Arc-Transitive Cubic Graphs and SL(3,mathbbZ)
- 3 Sierpinski's Gasket and Binary Gray Codes
- 4 Regular Maps
- References
- A 3D Spinorial View of 4D Exceptional Phenomena
- 1 Introduction
- 2 Root Systems and Reflection Groups
- 3 Clifford Versor Framework
- 4 H4 as a Rotation Rather Than Reflection Group I: From E8
- 5 H4 as a Rotation Rather Than Reflection Group II: From H3
- 6 The General Construction: Spinor Induction and the 4D Platonic Solids, Trinities and McKay Correspondence
- 7 Group and Representation Theory with Clifford Multivectors
- 8 Conclusion
- References
- Möbius Inversion in Suzuki Groups and Enumeration of Regular Objects
- 1 Introduction
- 2 Categories and Groups
- 2.1 Maps, Hypermaps and Groups
- 2.2 Reflexibility
- 2.3 Covering Spaces
- 3 Counting Homomorphisms
- 4 The Suzuki Groups and Their Subgroups
- 4.1 The Definition of the Suzuki Groups
- 4.2 Basic Properties of Suzuki Groups
- 4.3 Some Particular Subgroups
- 4.4 The Möbius Function of a Suzuki Group
- 5 Subgroups H with G (H)
- 5.1 Maxint Subgroups
- 5.2 Maximal Subgroups
- 5.3 Point-Stabilisers in Maximal Subgroups
- 5.4 Subgroups H of F
- 5.5 Subgroups H of Bi
- 6 Size of Conjugacy Classes
- 7 Calculating Values of muG
- 8 Enumerations
- 8.1 Orientably Regular Hypermaps
- 8.2 Regular Hypermaps
- 8.3 Orientably Regular Maps
- 8.4 Regular Maps
- 8.5 Surface Coverings
- 9 The Smallest Simple Suzuki Group
- 10 Postscript
- References
- More on Strongly Real Beauville Groups
- 1 Introduction
- 2 Preliminaries
- 3 The Finite Simple Groups
- 4 Characteristically Simple Groups
- 5 Almost Simple Groups
- 6 The Groups G:langlegrangletimesG:langlegrangle
- 7 Abelian and Nilpotent Groups
- References
- On Pentagonal Geometries with Block Size 3, 4 or 5
- 1 Introduction
- 2 Constructions
- 3 Block Size 3
- 4 Block Sizes 4 and 5
- 5 The Case r=2k+1
- 6 Concluding Remarks
- References
- The Grothendieck-Teichmüller Group of a Finite Group and G-Dessins d'enfants
- 1 Introduction
- 2 Generalities
- 2.1 The Group gb
- 2.2 The Group gts(G)
- 2.3 Inverse Limits
- 2.4 The Galois Group of mathbbQ
- 2.5 p-Groups and Nilpotent Groups
- 3 An Elementary Example: Dihedral Groups
- 4 The Case of Simple Groups
- 4.1 Notation
- 4.2 An Action of outgb on pc
- 4.3 The Group mathscrS(G)
- 4.4 Properties of mathscrS(G)
- 4.5 A Complete Example
- 5 Computing Explicitly
- 5.1 Computing gts1G
- 5.2 Computing sG
- 5.3 Simple Groups of Small Order
- 5.4 p-Groups
- 6 Dessins d'enfants
- 6.1 The Category of Dessins
- 6.2 G-Dessins
- 6.3 G-Objects in mathfrakSetsxy
- 6.4 A Complete Example
- Cyclic Dessins
- 7 Concluding Comments
- References
- Discrete Groups and Surface Automorphisms: A Theorem of A.M. Macbeath
- 1 Introduction
- 2 Hurwitz's Theorem Revisited
- 3 Automorphisms and Geometry in Three Dimensional Hyperbolic Space
- References
- Isometric Point-Circle Configurations on Surfaces from Uniform Maps
- 1 Introduction
- 2 Constructing Configurations of Points and Isometric Circles on Surfaces
- 3 Point-Circle Configurations from Some Classical Regular Maps
- 3.1 The Bolza Curve
- 3.2 Klein's Quartic
- 3.3 Bring's Curve
- 4 Pentagonal Geometries as Point-Circle Configurations from Moore Graphs
- 5 Point-Circle Configurations on p-gonal Surfaces with Two Cyclic p-gonal Morphisms
- References
- Dessins, Their Delta-Matroids and Partial Duals
- 1 Introduction
- 2 Dessins and Bipartite Maps
- 2.1 A Permutation Representation of Dessins
- 3 Belyi's Theorem and the Galois Action on Dessins
- 3.1 Galois Action on Dessins
- 3.2 Galois Invariants
- 4 Matroids and Delta-Matroids
- 4.1 Delta-Matroids
- 5 Galois Action on the Delta-Matroids of Maps
- 5.1 Trivial Delta-Matroidal Galois Invariants
- 5.2 Self-Duality of Maps and Matroids
- 6 Partial Duals and Twists of Delta-Matroids
- 6.1 Partial Duals, Delta-Matroids and the Galois Action
- 7 Maps, Their Partial Duals and Tropical Curves
- References
- Faithful Embeddings of Planar Graphs on Orientable Closed Surfaces
- 1 Introduction
- 2 Faithful Rotation Schemes
- 3 The 3-Regular Case
- 4 The 4-Regular Case
- 5 Conclusion
- References
- The Higher Dimensional Hemicuboctahedron
- 1 Introduction
- 2 Regular Polytopes
- 3 2-orbit Polytopes
- 4 The Higher Dimensional Hemicuboctahedron
- 5 Conclusions
- References
- Groups of Order at Most 6,000 Generated by Two Elements, One of Which Is an Involution, and Related Structures
- 1 Introduction
- 2 Constructing the Census of Small (2,*)-Groups
- 2.1 Using a Database of Small Groups
- 2.2 Using the Magma LowIndexNormalSubgroups Algorithm
- 2.3 Using Group Extensions
- 3 Proof of Theorem 2
- 4 (2,*)-Groups and Graphs
- 4.1 Cubic Cayley Graphs
- 4.2 Arc-Transitive Digraphs of Out-Valency Two
- 5 (2,*)-groups and Maps
- 5.1 Monodromy Groups of Maps
- 5.2 Oriented Rotary Maps
- 5.3 Regular Maps
- References
- Even-Integer Continued Fractions and the Farey Tree
- 1 Introduction
- 2 Infinite Continued Fractions
- 3 Representing Real Numbers by Even-Integer Continued Fractions
- 4 Serret's Theorem on Continued Fractions
- 5 An Alternative Characterisation of Convergents of Even-Integer Continued Fractions
- 6 Approximating Irrationals by Rationals
- 7 Concluding Remark
- References
- Triangle Groups and Maps
- 1 Triangle Groups
- 2 Some Personal History
- 3 Basic Concepts
- 4 Belyi's Theorem
- 5 Other Triangle Groups
- 6 Klein Surfaces and Real Belyi Functions
- 7 Hypermaps
- 8 Triangle Groups with Infinite Periods
- 9 Away from Triangle Groups
- References
- Nilpotent Symmetric Dessins of Class Two
- 1 Introduction
- 2 Algebraic Dessins
- 3 External Symmetries
- 4 Nilpotent Regular Dessins
- 5 Simple Symmetric p-Dessins of Class Two
- 6 Symmetric p-dessins of Class Two with Multiple Edges
- References
