Minicourse:Discrete subgroups of SL(2,R)
This is an introductory level course in pure mathematics with an
emphasis on geometry (broadly defined) and algebra.
Topics:
1. Review of fractional linear transformations from complex analysis.
2. The most elementary aspects of hyperbolic geometry (Poincar'e metric
on upper half-plane) (geodesics, areas of geodesic triangles, isometry groups, etc.)
3. Properly discontinuous group actions and construction of quotient spaces.
4. Fundamental domains for discrete subgroups.
5. Areas of fundamental domains, elliptic points, and genus of the quotient.
6. Examples of discrete groups and computations of the genus.
There are many connections to many different areas of pure mathematics ranging
from pure algebra through geometry to certain areas of analysis. As the
opportunity arises these will be pointed out. Unfortunately, in the context
of a minicourse, we will not be able to pursue most of these connections in earnest.
Prerequisites: Basic algebra (Math 200-201) or Math 251 or the equivalent.
Basic analysis (Math 203) and at least concurrent enrollment in Math 204.
Some familiarity with complex numbers. (Fractional linear transformations will
be reviewed rapidly in the first 15 minutes. Students not familiar with these
transformations will need to begin to learn this material immediately.) Additional
courses in algebra, complex analysis, topology, and differential geometry would
add perspective, but should not be crucial.
Recommended Text: Fuchsian Groups, by Svetlana Katok. This text is not
required, but it is very relevant to the course and very reader friendly.
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