Tessellations and Speiser graphs arising from meromorphic functions on simply connected Riemann surfaces

Abstract

Motivated by W. P. Thurston, we ask: What is the shape of a meromorphic function on a simply connected Riemann surface ${\Omega }_z$? We consider Speiser functions, i.e. meromorphic functions on a simply connected Riemann surface, that have a finite number $q$ at least 2 of singular (critical or asymptotic) values. As a first result, we make precise the correspondence between: Speiser functions $w(z)$, Speiser Riemann surfaces $R_w(z)$, Speiser $q$-tessellation, and analytic Speiser graphs of index $q$. As the second main result, we characterize tessellations with alternating colors (equivalently abstract pre-Speiser graphs) that are realized by Speiser functions on ${\Omega }_z$. The characterization is in terms of the $q$-regular extension problem of bipartite planar graphs. As third main results, the Speiser Riemann surface $R_w(z)$ can be constructed by isometric glueing of a finite number of types of sheets, where each sheet is a maximal domain of single-valuedness of the inverse of $w(z)$. Furthermore, a unique decomposition of $R_w(z)$ into maximal logarithmic towers and a soul is provided. Using vector fields we recognize that logarithmic towers come in two flavors: exponential or $h$-tangent blocks, directly related to the exponential or the hyperbolic tangent functions on the upper half plane. The surface $R_w(z)$ of a finite Speiser function is characterized by surgery of a rational block and a finite number of exponential or $h$-tangent blocks.