Stable quasiconformal mapping class groups and asymptotic Teichmüller spaces

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.

The Nielsen realization problem for asymptotic Teichmüller modular groups

“…In this section, we give a rather direct proof for it by summarizing necessary results proved in [11]. Actually, it is easy to verify that Lemma 4.9 in [11], which is used in the proof of Proposition 3.3 below, is still valid under this upper bound condition. This is because we avoided inessential complexity in the entire argument.…”

“…The next proposition, which has been essentially proved in [11], ensures that we can always take a Θ-graph that is not fixed by a non-trivial quasiconformal mapping class. The next proposition, which has been essentially proved in [11], ensures that we can always take a Θ-graph that is not fixed by a non-trivial quasiconformal mapping class.…”

“…In [11], we have proved that every element of Mod AT (R) of finite order has a fixed point in AT (R) if R satisfies a certain stronger bounded geometry condition. Our main theorem in this paper is an extension of this result to the following asymptotic version of the fixed point theorem.…”

“…In [11], this theorem was proved in an indirect way. In this section, we give a rather direct proof for it by summarizing necessary results proved in [11].…”

“…(In [11], we have used a similar object to a Θ-graph but a more general one called thetaframe.) (In [11], we have used a similar object to a Θ-graph but a more general one called thetaframe.)…”