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Stable quasiconformal mapping class groups and asymptotic Teichmüller spaces
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 3, June 2011
- pp. 637-675
- 10.1353/ajm.2011.0017
- Article
- Additional Information
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The stable quasiconformal
mapping class group is a group of quasiconformal mapping classes of a Riemann surface
that are homotopic to the identity outside some topologically finite subsurface.
Its analytic counterpart
is a group of mapping classes that act
on the asymptotic Teich\-m\"ul\-ler space trivially.
We prove that the stable quasiconformal
mapping class group is coincident with the
asymptotically trivial mapping class group for
every Riemann surface satisfying a certain geometric condition.
Consequently, the intermediate Teich\-m\"ul\-ler space, which is
the quotient space of the Teich\-m\"ul\-ler space
by the asymptotically trivial mapping class group, has a complex manifold structure,
and its automorphism group is geometrically isomorphic to
the asymptotic Teich\-m\"ul\-ler modular group.
The proof utilizes a condition for an asymptotic Teich\-m\"ul\-ler modular
transformation to be of finite order, and this is given by the consideration of
hyperbolic geometry of topologically infinite surfaces and its deformation under
quasiconformal homeomorphisms. Also these arguments enable us to show that
every asymptotic Teich\-m\"ul\-ler modular
transformation of finite order has a fixed point on the
asymptotic Teich\-m\"ul\-ler space, which can be regarded as
an asymptotic version of the Nielsen theorem.