GEOMETRY AND GROUPS (C) 24 lectures, Michaelmas term
Analysis II and Geometry are useful, but any concepts required from these courses will be introduced and developed in the
course.
A brief discussion of the Platonic solids. The identification of the symmetry groups of the Platonic
solids with the finite subgroups of SO(3). [4]
The upper half-plane and disc models of the hyperbolic plane. The hyperbolic metric; a comparison
of the growth of circumference and area in Euclidean and hyperbolic geometry. The identification of
isometries of the upper half-plane with real M¨obius maps, and the classification of isometries of the
hyperbolic plane. The Modular group SL(2, Z) and its action on the upper half-plane. The tesselation
of hyperbolic plane by ideal triangles. [6]
Group actions: discrete groups, properly discontinuous group actions. Quotients by group actions and
fundamental domains. Brief discussion of Escher’s work as illustrations of tesselations of the hyperbolic
plane, and of Euclidean crystallographic groups. [5]
Examples of fractals, including Cantor’s middle-third set and similar constructions. Cantor sets, and
non-rectifiable curves, as limit sets of discrete M¨obius groups. Fractals obtained dynamically. The
Hausdorff dimension of a set, computations of examples. [5]
The upper half-space of R
3 as hyperbolic three-space; the identification of the group of M¨obius maps
as the isometries of hyperbolic three-space, and of the finite groups of M¨obius maps with the the finite
subgroups of SO(3). Discussion of Euclidean crystallographic groups in higher dimensions. [4]
Appropriate books
A.F. Beardon The geometry of discrete groups. Springer-Verlag, Graduate Texts No. 91, 1983 (£?)
K. Falconer Fractal Geometry. John Wiley & Sons, 1990 (£85.00 Hardback)
R.C. Lyndon Groups and Geometry, LMS Lecture Notes 101. Cambridge University Press, 1985 (£?)
V.V. Nikulin and I.R. Shafarevich Geometries and Groups. Springer-Verlag, 1987 (£?)
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