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

TL;DR

This paper explores the geometric and combinatorial structures of Speiser functions on simply connected Riemann surfaces, establishing correspondences with tessellations, graphs, and surface decompositions, and characterizing their construction via isometric gluing.

Contribution

It provides a detailed characterization of Speiser functions, their associated tessellations, and Riemann surfaces, including new methods for constructing and decomposing these surfaces.

Findings

Established correspondence between Speiser functions, tessellations, and graphs.

Characterized tessellations realizable by Speiser functions via bipartite planar graphs.

Described the construction of Speiser Riemann surfaces through isometric gluing and surface decomposition.

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.