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The modern theory of Kleinian groups starts with the work of Lars
Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness
theorem, and Bers' observation that their joint work on the
Beltrami equation has deep implications for the theory of Kleinian
groups and their deformations. From the point of view of
uniformizations of Riemann surfaces, Bers' observation has the
consequence that the question of understanding the different
uniformizations of a finite Riemann surface poses a purely
topological problem; it is independent of the conformal structure
on the surface. The last two chapters here give a topological
description of the set of all (geometrically finite)
uniformizations of finite Riemann surfaces. We carefully skirt
Ahlfors' finiteness theorem. For groups which uniformize a finite
Riemann surface; that is, groups with an invariant component, one
can either start with the assumption that the group is finitely
generated, and then use the finiteness theorem to conclude that the
group represents only finitely many finite Riemann surfaces, or, as
we do here, one can start with the assumption that, in the
invariant component, the group represents a finite Riemann surface,
and then, using essentially topological techniques, reach the same
conclusion. More recently, Bill Thurston wrought a revolution in
the field by showing that one could analyze Kleinian groups using
3-dimensional hyperbolic geome try, and there is now an active
school of research using these methods."
The modern theory of Kleinian groups starts with the work of Lars
Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness
theorem, and Bers' observation that their joint work on the
Beltrami equation has deep implications for the theory of Kleinian
groups and their deformations. From the point of view of
uniformizations of Riemann surfaces, Bers' observation has the
consequence that the question of understanding the different
uniformizations of a finite Riemann surface poses a purely
topological problem; it is independent of the conformal structure
on the surface. The last two chapters here give a topological
description of the set of all (geometrically finite)
uniformizations of finite Riemann surfaces. We carefully skirt
Ahlfors' finiteness theorem. For groups which uniformize a finite
Riemann surface; that is, groups with an invariant component, one
can either start with the assumption that the group is finitely
generated, and then use the finiteness theorem to conclude that the
group represents only finitely many finite Riemann surfaces, or, as
we do here, one can start with the assumption that, in the
invariant component, the group represents a finite Riemann surface,
and then, using essentially topological techniques, reach the same
conclusion. More recently, Bill Thurston wrought a revolution in
the field by showing that one could analyze Kleinian groups using
3-dimensional hyperbolic geome try, and there is now an active
school of research using these methods."
This volume contains papers and abstracts by participants of the
Conference on Riemann Surfaces and Related Topics, which was held
at the State University of New York at Stony Brook, June 5-9, 1978.
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