Riemann surfaces and the geometrization of 3-manifolds. (English) Zbl 0765.57012
The most influential result in the theory of 3-manifolds since the characteristic submanifold theory of Jaco, Shalen, and Johannson is the geometrization theorem of Thurston. It applies to Haken 3-manifolds, the 3-manifolds which can be built up inductively from 3-balls by gluing together submanifolds of the boundary, and it asserts that they admit geometric structures. One of the key steps in the argument is to find the right geometry on a bounded 3-manifold so that the induced structures on the submanifolds match under the gluing map. Thurston’s idea to accomplish this is to use the theory of quasiconformal deformations to formulate the gluing problem as a fixed-point problem on a Teichmüller space. The fixed point is found by iteration of a map on Teichmüller space which is defined using the gluing map for the original 3-manifold.
In this survey the author outlines Thurston’s construction and sketches a new proof that the iteration converges. In the words of the author, his “argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently large covering space (e. g. the universal cover). This contraction gives an immediate estimate for the contraction of Thurston’s iteration.” The author’s work characterizes those coverings for which contraction is obtained in terms of the purely combinatorial notion of nonamenability.
This survey is very well written and well worth the time of anyone interested in 3-dimensional topology or the theory of Riemann surfaces.
In this survey the author outlines Thurston’s construction and sketches a new proof that the iteration converges. In the words of the author, his “argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently large covering space (e. g. the universal cover). This contraction gives an immediate estimate for the contraction of Thurston’s iteration.” The author’s work characterizes those coverings for which contraction is obtained in terms of the purely combinatorial notion of nonamenability.
This survey is very well written and well worth the time of anyone interested in 3-dimensional topology or the theory of Riemann surfaces.
Reviewer: D.McCullough (Norman)
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
| 30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 30F60 | Teichmüller theory for Riemann surfaces |
Keywords:
Riemann surface; skinning map; amenable; geometrization; 3-manifold; Teichmüller space; quasiconformal mapping; dilatationReferences:
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