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

TL;DR

The paper addresses the problem of describing the shape of a meromorphic function on a simply connected Riemann surface by establishing a complete dictionary among analytic and combinatorial objects: Speiser functions with $q\ge2$ singular values, their Speiser Riemann surfaces ${\mathcal{R}}_{w(z)}$, Speiser $q$-tessellations, and analytic Speiser graphs of index $q$. It introduces the Schwarz–Klein–Speiser tessellation algorithm and proves a 1-1 correspondence with the associated Speiser graphs, linking geometric tessellations to analytic data. A key advance is the Hall-type balance characterization for when a pre-Speiser graph extends to a Speiser graph, solved via a bipartite transportation framework. The work also provides a geometrical decomposition of Speiser surfaces into maximal logarithmic towers and a soul, with towers corresponding to exponential or $h$-tangent blocks, enabling a constructive surgery description: finite Speiser functions arise from rational blocks plus a finite collection of towers, while $N$-functions correspond to a rational block with specific tower configurations. Overall, the framework unifies combinatorial, topological, and analytic perspectives and yields practical criteria for recognizing and constructing Speiser functions and their finite variants.

Abstract

Motivated by W. P. Thurston, we ask: What is the shape of a meromorphic function on a simply connected Riemann surface $Ω_z$? 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 $Ω_z$. 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.

Figures (13)

• Figure 1: Examples of $\tt t$--graphs $\Gamma$, whose corresponding tessellations $\mathscr{T}(\Gamma)$ do not represent Speiser functions.
• Figure 2: Affine view of the tessellation of the non generic rational function $R(z)={z(z^2-1)(z^2-4)}/{(z-3)}$ of degree 5. It has 6 critical points, one of them being $\infty\in{\widehat{\mathbb C}}_z$ and 6 critical values $\mathcal{SP}_R=\{ {\tt w}_1, {\tt w}_2, {\tt w}_3, {\tt w}_4, {\tt w}_5, {\tt w}_6=\infty \}$ lying on $\gamma={\mathbb R}\cup\{\infty\}$. a) The ${\tt t}$--graph $\Gamma= R^{-1} ({\mathbb R} \cup \{ \infty\})$ and its non homogeneous tessellation $\mathscr{T}(\Gamma)$. b) The ${\tt t}$--graph $\Gamma$ with consistent $6$--labelling $\mathcal{L}_{\mathcal{W}_6}: V(\Gamma) \longrightarrow \mathcal{W}_6$, where $\mathcal{L}_{\mathcal{W}_6}(\infty)= {\tt w}_6$. c) The ${\tt A}$--map $\widehat{\Gamma}= R^* \gamma$, its homogeneous tessellation $\mathscr{T}(\widehat{\Gamma}_6)$, and its consistent $6$--labelling $\mathcal{L}_{\mathcal{W}_6}$; each tile is a $6$--gon, with vertices at the (red) critical points $\mathcal{SP}_R$, the point $\infty\in{\widehat{\mathbb C}}_z$ (which has label ${\tt w}_6$), and the (green) cocritical points $\mathcal{CS}_R$. This figure appears as figure 1 of AlvarezGutierrezMucino with slightly different notation.
• Figure 3: ${\tt A}$--map with 4 tiles of each color, where each tile is a 6--gon, and a non consistent 6--labelling with labels $\mathcal{W}_6=[1,2,3,4,5,6]$; label '5' is only assigned to vertices of valence 2. This is figure 10 of Koch-Lei and is attributed to W. P. Thurston.
• Figure 4: Analytic Speiser graphs of index ${\tt q}$ that represent $N$--functions $w(z)$; (a)--(c) with ${\tt p}={\tt q}=4$, (d) with ${\tt p}=4, {\tt q}=3$. The nuclei are colored red and the ${\tt p}$ logarithmic ends are colored black. (e) is a labelled Speiser graph that does not have a consistent 4--labelling, since every $4$--face is a digon. (f) The tessellation corresponding to the Speiser graph of (e), does not have a consistent 4--labelling since the label '4' only appears on vertices of valence 2. The labels follow the convention of Remark \ref{['rem:notation-of-labels']}.
• Figure 5: (a) Tessellation $\mathscr{T}(\Gamma)$ of the sphere ${\mathbb S}^2$, its $\tt t$--graph $\Gamma$ has $k$--gons, $k=2,3,4,5$, as tiles. (b) The dual of (a) is a planar bipartite graph on ${\mathbb S}^2$, the pre--Speiser graph $\mathfrak{S}$, with vertices of valence $k$.

Theorems & Definitions (128)

• Remark 2.1: Natural boundary of $w(z)$
• Definition 2.2: Singularities of $w^{-1}(z)$; Iversen, BergweilerEremenko, EremenkoReview
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 3.1
• Example 3.1: Speiser functions and finite Speiser functions
• Definition 3.2
• Proposition 3.3