Commensurability automorphism groups and infinite constructions in Teichmüller theory

Abstract

To a compact Riemann surface X, we associate its commensurability automorphism group, ComAut( X), consisting of finite holomorphic self-correspondences of X. The group ComAut( X) is the stabilizer of the point [ X] for the action of the universal commensurability modular group, CM ∞( X), on the universal direct limit of Teichmüller spaces, τ ∞( X). Also, ComAut( X) acts by holomorphic automorphisms on the Riemann surface lamination H ∞(X). This action is ergodic precisely when the Fuchsian group uniformizing X is arithmetic. The action of CM ∞( X), and of its isotropy subgroups, on some natural vector bundles over τ ∞( X) are studied.