Metric Ribboned Graphs: Structures & Recursions

• Metric Ribboned Graphs are combinatorial ribbon graphs equipped with positive edge lengths, enabling explicit piecewise-linear modeling of moduli spaces.
• Directed versions introduce orientation maps that impose residue conditions and dualities with bipartite maps, linking Abelian differentials with topological recursion.
• These structures facilitate precise volume computations and recursive decompositions in moduli spaces, underpinning enumerative techniques for Hurwitz numbers and dessins d’enfants.

A metric ribboned graph is a combinatorial structure vital to the paper of moduli spaces of surfaces, mapping class groups, and enumeration of objects such as dessins d’enfants. A metric ribboned graph, or metric ribbon graph, is a combinatorial ribbon graph equipped with a positive real length on each edge. These objects admit additional orientations and can be endowed with directed structures relevant to moduli of Abelian differentials and volume recursion schemes for spaces parameterizing Riemann surfaces with prescribed boundary lengths. Directed metric ribbon graphs occupy a central role in topological recursion, the theory of Hurwitz numbers, and connections to algebraic geometry via bijections with bipartite maps and dessins d’enfants (Barazer, 2021).

1. Definition and Combinatorial Structure

A combinatorial ribbon graph $R$ is governed by a finite set of half-edges $X_R$ together with an involutive pairing $s_1: X_R \to X_R$ (without fixed points) pairing half-edges into edges, and a cyclic permutation %%%%3%%%% encoding the cyclic ordering at each vertex. Given these, one defines $s_2 = s_1 \circ s_0^{-1}$, and the respective orbits identify the sets $X_0R = X_R/\langle s_0\rangle$ (vertices), $X_1R = X_R/\langle s_1\rangle$ (edges), and $X_2R = X_R/\langle s_2\rangle$ (faces or boundaries).

A metric ribbon graph is a pair $(R, m)$ where $m : X_1R \to \mathbb{R}_{>0}$ assigns each edge a positive real length, so the space of metrics is $Met(R) = \mathbb{R}_{>0}^{X_1R}$. This additional structure enables the explicit parameterization of piecewise-linear models of decorated moduli spaces. The notion of boundary length for a face $\beta$ is given by $\ell_\beta(R, m) = \sum_{[e]_2=\beta} m_e$.

The metric ribboned graph framework leads to a cell decomposition of moduli space: fixing the combinatorial type $R$ and constraining the boundary lengths to a prescribed vector $L \in \mathbb{R}_+^{X_2R}$ defines affine cells in $M^{comb}(g, n)$, the combinatorial moduli space of genus $g$ surfaces with $n$ boundaries.

2. Oriented and Directed Metric Ribbon Graphs

A directed metric ribbon graph is an oriented metric ribbon graph, denoted $R^\circ = (R, \epsilon)$, where the orientation ("flow-structure") is specified by a map $\epsilon : X_R \to \{\pm 1\}$ satisfying $\epsilon \circ s_2 = \epsilon$ and $\epsilon \circ s_1 = -\epsilon$. This endows faces $X_2R$ with signs, interpreted as "in-flows" ($+1$) and "out-flows" ($-1$). Ribbon graphs permitting such an $\epsilon$ are orientable; together with a metric, this data defines a directed metric ribbon graph.

The boundary lengths in this setting must satisfy the residue condition: $$\sum_{\beta \in X_2R} \epsilon(\beta)\, \ell_\beta(R, m) = 0,$$ so the admissible length vectors reside in the hyperplane of total signed length zero.

The dual of a directed metric ribbon graph is realized as a bipartite map: after capping each boundary of the thickened surface $M_R$ and coloring each face via $\epsilon$, adjacent faces receive opposite colors, yielding a canonical bipartite structure. In the context of the theory of Abelian differentials, this dual map encodes the horizontal foliation, where black (respectively white) faces index zeros (respectively poles) of the differential.

3. Canonical Acyclic Decomposition and Multi-Curves

The canonical acyclic decomposition theorem addresses the decomposition of an oriented metric ribbon graph $R^\circ$ (with at least two vertices) via admissible multi-curves. An admissible multi-curve $\Gamma$ is a collection of simple closed curves on $M_R$ such that cutting along $\Gamma$ does not fragment any vertex.

Theorem (Barazer, Canonical Acyclic Decomposition): For each vertex $v$ of $R^\circ$, there is a unique admissible multi-curve $\Gamma_v^+$ such that:

• Its associated directed stable graph $G_v$ contains a distinguished component $c_0$ with exactly vertex $v$;
• Every curve in $\Gamma_v^+$ is a boundary curve of $c_0$;
• Each such boundary is incoming, i.e., for $\beta$ a boundary of $c_0$, $\epsilon(\beta) = -1$.

The directed stable graph $G_v$ corresponding to $\Gamma_v^+$ is acyclic. The union $\Gamma = \bigcup_{v \in X_0R} \Gamma_v^+$ is the unique maximal admissible multi-curve yielding an acyclic stable graph compatible with any vertex ordering.

Cutting along an admissible multi-curve $\Gamma$ produces a new ribbon graph $R_\Gamma$ on the cut surface $M_{R, \Gamma}$, with the induced metric $m_\Gamma$ determined by summing the edge-lengths of parallel arcs generated by the cut. This structure underpins precise surgeries and recursive decompositions of the moduli space.

4. Recursion Schemes for Volumes of Combinatorial Moduli Spaces

The cell decomposition facilitated by metric ribbon graphs allows the explicit calculation of volumes for the moduli spaces $M_{g,n^+,n^-}^{comb}(L^+|L^-)$ of directed metric ribbon graphs with genus $g$, $n^+$ positive and $n^-$ negative boundaries, and boundary-length vectors $(L^+, L^-)$, with the residue condition $\sum_i L_i^+ = \sum_j L_j^-$. The natural volume form is the Lebesgue measure normalized by integer-length graphs.

A key result is a Mirzakhani-type topological recursion. By strategically cutting off a canonical acyclic multi-curve of "pair-of-pants" type around each positive boundary, one can recursively express the volume $Z_{g,n^+,n^-}(L^+|L^-)$ in terms of analogous volumes of lower complexity. The recursion encompasses four combinatorial types of pair-of-pants subtraction, captured by summations, integration, and compositions over genus- and boundary-partitions.

For the case of four-valent (trivalent for faces) directed metric ribbon graphs, the moduli cells are top-dimensional and the volumes become piecewise homogeneous polynomials of total degree $4g - 3 + n^+ + n^-$, with initial values given for small genera and boundary counts. The recursion then admits a concise polynomial formulation.

5. Enumeration and Applications to Dessins d’Enfants

Specializing to the case of one negative boundary ($n^- = 1$), the residue condition reduces the number of free parameters, and the volume recursion yields a polynomial recursion for

$$F_{g,n}(L_1, \ldots, L_n) = Z_{g,n,1}(L_1, \ldots, L_n \mid L^-), \quad L^- = \sum_i L_i,$$

which can be explicitly solved via Laplace transforms or direct polynomial ansatz: $$F_{g,n}(L) = \sum_{\alpha_1 + \cdots + \alpha_n = 4g-2+n} c(\alpha) \prod_{i=1}^n L_i^{\alpha_i},$$ with symmetric coefficients $c(\alpha)$.

The coefficients $c(\alpha)$ satisfy a cut-and-join recursion, encapsulated via a generating function $\Phi(s; \mathbf{t})$, and governed by a cut-and-join differential equation. This equation coincides with the recursion for Hurwitz numbers of three-point covers of $\mathbb{P}^1$.

Generic oriented four-valent metric ribbon graphs with one negative boundary are in bijection with dessins d’enfant: bipartite maps of genus $g$ with a single face of high degree and simple ramification over the remaining points. Thus, the combinatorial structure and recursion directly enumerate dessins with prescribed degree profiles, yielding explicit generating functions and asymptotic formulas.

6. Dualities and Relations to Bipartite Maps and Abelian Differentials

Directed metric ribbon graphs form the combinatorial backbone underlying several dualities. By capping boundaries and coloring faces via the orientation map $\epsilon$, one obtains a bipartite map on a closed surface, with vertex color classes matching the parity of the original vertex degrees. This dual map plays a central role in the combinatorial encoding of Abelian differentials: in the flat surface model, black faces correspond to zeros and white faces to poles of the differential, rendering the ribbon graph’s structure critical in the paper of foliations and translation structures on Riemann surfaces.

These combinatorial-differential correspondences elucidate the enumeration of moduli space cells, the structure of spaces of quadratic and Abelian differentials, and the counting of special algebraic curves via dessins.

7. Summary Table of Structures

Structure	Definition/Data	Role/Significance
Combinatorial Ribbon Graph	$(X_R, s_0, s_1)$	Encodes graph topology and surface cellularization
Metric Ribbon Graph	$(R, m)$	Decorates each edge with a positive real length
Oriented/Directed Ribbon Graph	$(R, \epsilon)$	Endows flow structure, distinguishes in/out boundaries
Admissible Multi-curve	Collection of non-vertex-splitting curves	Enables canonical surface decomposition
Bipartite Map (dual)	Black/white colored dual via $\epsilon$	Dual for Abelian differentials, dessins enumeration

Metric ribboned graphs, especially in their directed form, thus provide a precise and flexible framework for moduli space cell decomposition, quantification of moduli volumes, and the enumeration of algebraic structures such as dessins d’enfants (Barazer, 2021).