Stable quasiconformal mapping class groups and asymptotic teichmü ller spaces

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 Teichm üller 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 Teichmüller space, which is the quotient space of the Teichmüller space by the asymptotically trivial mapping class group, has a complex manifold structure, and its automorphism group is geometrically isomorphic to the asymptotic Teichmüllermodular group. The proof utilizes a condition for an asymptotic Teichmüller 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 Teichmüller modular transformation of finite order has a fixed point on the asymptotic Teichmüller space, which can be regarded as an asymptotic version of the Nielsen theorem.