1998/11/02 by Indranil Biswas, Biswas, Indranil, Subhashis Nag +1
Mathematics · #Algebraic Geometry (math.AG) #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.math/9811005
latex2e, 40 pages
arxiv created 1998/11/02 · openalex publication_date 1998/11/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To any compact hyperbolic Riemann surface X, we associate a new type of automorphism group -- called its *commensurability automorphism group*, ComAut(X). The members of ComAut(X) arise from closed circuits, starting and ending at X, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by X (in T∞), for the action of the universal commensurability modular group on the universal direct limit of Teichmüller spaces, T∞. Now, each point of T∞ represents a complex structure on the universal hyperbolic solenoid. We notice that ComAut(X) acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing X is *arithmetic*. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over T∞, are studied by us.