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Stable quasiconformal mapping class groups and asymptotic Teichmüller spaces

From: American Journal of Mathematics
Volume 133, Number 3, June 2011
pp. 637-675 | 10.1353/ajm.2011.0017

Abstract

Abstract:

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.



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