MT A618 Hyperbolic 3-Manifolds
This section of the course (Spring 2003) meets MWF 2:00PM-2:50PM.
Before Thurston's work in the 1970's, examples of hyperbolic 3-manifolds were exceedingly rare; only a couple of simple constructions were known. Indeed, in higher dimensions this is still the case (probably because hyperbolic 4-manifolds really are pretty rare).
Of course we now know that "most" 3-manifolds are hyperbolic and it is often a routine exercise in 3-dimensional topology to check that a particular example is hyperbolic by verifying the hypotheses of Thurston's theorem (atoroidal, acylindrical, irreducible, etc.) This approach doesn't work if one wants to know something about specific geometric properties of the resulting hyperbolic structure.
My plan is to introduce as many concrete techniques for constructing hyperbolic 3-manifolds as possible. Here "concrete" means that one should be able to recover important hyperbolic geometric data from the construction (e.g. volume, short geodesics, or generators of the fundamental group as elements of PSL(2,C)). The main examples are the "pre-Thurston" constructions: arithmetic groups and groups constructed from hyperbolic polyhedra. There are some other, more exotic, constructions if time permits.
- Arithmetic Groups
- Quaternion Algebras
- Bianchi groups
- Cocompact examples
- Volume formulae
- Higher dimensions
- Andreev's Theorem
- Coxeter groups
- Regular polyhedra
- Ideal decompositions, SnapPea
- Computing volume
- HLM examples (cone manifolds)
You can find more details on the course schedule page.
There is no required textbook. We will review some of the basics of hyperbolic 3-dimensional geometry during the first few weeks. The best references for this are the books by Ratcliffe (Foundations of Hyperbolic Manifolds) and Benedetti/Petronio (Lectures on Hyperbolic Geometry). For the arithmetic portion of the course, the new book by Colin Maclachlan and Alan Reid is highly recommended. For everything else I will hand out excerpts from papers, etc.
- For graduate students thinking of enrolling: the course is mistakenly listed in the catalog as "Topics in Algebra" (it should be, I think, "Topics in Topology").
- The image above is a hyperbolic icosahedron which serves as a fundamental domain for the five-fold cyclic branched cover of the figure-eight knot (one of the "Fibonacci manifolds"). The image was created by Kyle Cranmer, a (then) undergraduate student at Rice in 1997.