Edited by Martin W. Liebeck
Edited by Jan Saxl
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:September 2010
Online ISBN:9780511629259
Paperback ISBN:9780521406857
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511629259.038
Subjects: Algebra, Discrete Mathematics Information Theory and Coding
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“Even quite ungainly objects, like chairs and tables, will become almost spherical if you wrap them in enough newspaper.”
The symmetries of any finite object, such as a chair or a table, all fix a point, say the centre of gravity of the object, and so act on the surface of a sphere, for example any sphere centred on the centre of gravity.
The symmetries of a repeating pattern on a carpet or tiled floor, or on a wall, supposed continued to infinity, will probably constitute one of the 17 plane crystallographic groups.
Among the works of the Dutch draughtsman Maurits C. Escher, one can find examples of all these 17 groups, and also some even more interesting designs such as Circle Limit I, II, …, whose symmetries are various discrete groups of isometries of the hyperbolic plane.
In this paper a surface group will be a discrete group of isometries of one of the following three surfaces:
These are all the simply-connected surfaces of constant Gaussian curvature. We shall present a simple and uniform notation that describes all three types of group. Since this notation is based on the concept of orbifold introduced by Bill Thurston, we shall call it the orbifold notation.
Roughly speaking, an orbifold is the quotient of a manifold by a discrete group acting on it. It therefore has one point for each orbit of the group on the manifold (Orbifold = Orbit-manifold).
Mirrors and mirror-boundaries
An orbifold may have boundary curves even though our three original surfaces do not. The boundary points arise from points lying on mirrors.
pp. i-iv
pp. v-vii
pp. viii-x
pp. xi-xiv
Part 1 - Sporadic groups : Read PDF
pp.
1 - Uniqueness of sporadic groups : Read PDF
pp. 1-11
2 - The study of J4 via the theory of uniqueness systems : Read PDF
pp. 12-21
3 - Y555 and all that : Read PDF
pp. 22-23
4 - Hyperbolic reflections for the Bimonster and 3Fi24 : Read PDF
pp. 24-45
5 - A geometric characterization of the Monster : Read PDF
pp. 46-62
6 - Constructing the Monster : Read PDF
pp. 63-76
pp.
7 - The signature of the normalizes of Γ0 (N) : Read PDF
pp. 77-86
8 - Completely replicahle functions : Read PDF
pp. 87-98
9 - Introduction to the Monster Lie algebra : Read PDF
pp. 99-107
10 - Remarks on Moonshine and Orbifolds : Read PDF
pp. 108-120
Part 3 - Local and geometric methods in group theory : Read PDF
pp.
11 - The classification of 3-transposition groups with trivial center : Read PDF
pp. 121-138
12 - (S3,S6,)-Amalgams : Read PDF
pp. 139-143
13 - Pushing down minimal parabolic systems : Read PDF
pp. 144-150
14 - Nonspherical spheres : Read PDF
pp. 151-158
15 - On the 2-local structure of finite groups : Read PDF
pp. 159-182
16 - Groups generated by k-root subgroups – a survey : Read PDF
pp. 183-204
Part 4 - Geometries and related groups : Read PDF
pp.
17 - Finiteness questions for geometries : Read PDF
pp. 205-217
18 - Kac-Moody groups and, their automorphisms : Read PDF
pp. 218-228
19 - Generalized hexagons as geometric hyperplanes of near hexagons : Read PDF
pp. 229-239
20 - On simplicial complexes related to the Suzuki sequence graphs : Read PDF
pp. 240-248
21 - Twin buildings and groups of Kac-Moody type : Read PDF
pp. 249-286
Part 5 - Finite and algebraic groups of Lie type : Read PDF
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22 - Some remarks on the structure of finite subgroups of simple algebraic groups : Read PDF
pp. 287-291
23 - Some (almost) multiplicity-free coset actions : Read PDF
pp. 292-310
24 - Orbits in internal Chevalley modules : Read PDF
pp. 311-315
25 - Subgroups of finite and algebraic groups : Read PDF
pp. 316-326
26 - Irreducible representations of finite Chevalley groups containing a matrix with a simple spectrum : Read PDF
pp. 327-332
27 - Overgroups of unipotent elements in simple algebraic groups : Read PDF
pp. 333-339
Part 6 - Finite permutation groups : Read PDF
pp.
28 - Some open problems on permutation groups : Read PDF
pp. 340-350
29 - The genus of a permutation group : Read PDF
pp. 351-363
30 - Primitive permutation characters : Read PDF
pp. 364-367
31 - Closures of finite permutation groups and relation algebras : Read PDF
pp. 368-379
Part 7 - Further aspects of simple groups : Read PDF
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32 - Symmetric presentations I: Introduction, with particular reference to the Mathicu groups M12 and M24 : Read PDF
pp. 380-396
33 - Finite and, locally finite groups containing a small subgroup with small centralizer : Read PDF
pp. 397-402
34 - Some topics in asymptotic group theory : Read PDF
pp. 403-421
35 - The 3-modular characters of the McLaughlin group McL and its automorphism group McL.2 : Read PDF
pp. 422-437
Part 8 - Related topics : Read PDF
pp.
36 - The orbifold notation for surface groups : Read PDF
pp. 438-447
37 - A remark on two Diophantine equations of Peter Cameron : Read PDF
pp. 448-458
38 - Testing for isomorphism between finitely presented groups : Read PDF
pp. 459-475
39 - Discrete groups and, Galois theory : Read PDF
pp. 476-479
40 - Smooth coverings of regular maps : Read PDF
pp. 480-489