Graph of Pants Decompositions

• Graph of Pants Decompositions is a framework that models surface decompositions using dual graphs that represent complementary pairs of pants and pants graphs that record elementary moves.
• It explains how elementary shifts and mapping class group actions enable explicit distance estimates and quotient constructions in Teichmüller geometry.
• The topic links combinatorial structures with product regions and weighted models, offering practical insights into spectral limits and dynamical characteristics of hyperbolic surfaces.

In the literature on surface topology and Teichmüller geometry, the phrase graph of pants decompositions is used in two closely related senses. One is the dual graph attached to a single pants decomposition, with vertices representing complementary pairs of pants and edges representing the decomposition curves. The other is the pants graph, whose vertices are pants decompositions themselves and whose edges are elementary moves between them. For a closed orientable surface of genus $g\ge 2$, a pants decomposition has $3g-3$ curves and cuts the surface into $2g-2$ pairs of pants, so its associated dual graph is connected and trivalent with $2g-2$ vertices and $3g-3$ edges; this graph-theoretic encoding underlies distance estimates, quotient constructions modulo the mapping class group, and several later extensions to convex product regions, spectral limits, and dynamical criteria (Moser, 2011, Sultan, 2011).

1. Basic definitions and competing graph constructions

For a finite-type orientable surface, the standard complexity notation is either $\xi(\Sigma)=3g+b-3$ or $\kappa(\Sigma)=3g-3+r$, depending on the source. A pants decomposition is a maximal multicurve: a maximal collection of pairwise disjoint essential simple closed curves. For a closed surface of genus $g$, such a decomposition consists of exactly $3g-3$ curves and cuts the surface into $2g-2$ pairs of pants (Taylor et al., 2013, Moser, 2011).

The pants graph has one vertex for each pants decomposition. Two vertices are adjacent when the decompositions differ by one elementary move, supported on a unique complexity-one subsurface. In a once-punctured torus the replacement curve intersects the original one once, and in a four-times-punctured sphere it intersects it twice. The graph metric assigns length $3g-3$0 to each edge. In the usage of Hatcher–Lochak–Schneps adopted by Moser, the pants complex is obtained by adding $3g-3$1-cells to this $3g-3$2-skeleton, but the distance considered there is still the path metric on the $3g-3$3-skeleton, so “distance in the pants complex” means the same metric as on the pants graph (Moser, 2011, Taylor et al., 2013).

The pants decomposition graph or dual graph is different. It records one fixed decomposition rather than all decompositions. This distinction is a recurring source of terminological ambiguity: one graph encodes a single decomposition combinatorially, while the other organizes all decompositions into a move graph (Moser, 2011).

2. Dual graphs of individual decompositions

For a closed genus-$3g-3$4 surface and a pants decomposition $3g-3$5, Moser defines the graph $3g-3$6 by taking one vertex for each of the $3g-3$7 complementary pairs of pants and one edge for each of the $3g-3$8 curves in $3g-3$9. If a curve is adjacent to two distinct pairs of pants, its edge joins the corresponding vertices; if it lies on the same pair of pants on both sides, it becomes a loop edge. Every vertex has valence $2g-2$0, with a loop counted as contributing two half-edges, so $2g-2$1 is a connected trivalent graph with $2g-2$2 vertices and $2g-2$3 edges (Moser, 2011).

For a general finite-type surface $2g-2$4, the graph $2g-2$5 attached to a pants decomposition has $2g-2$6 vertices, $2g-2$7 edges, and is connected and at most cubic. Conversely, every connected at-most-cubic graph with those counts arises as $2g-2$8 for some pants decomposition of $2g-2$9. In particular, the graph model is exact at the level of topological type rather than merely approximate (Sultan, 2011).

This dual graph also records the genus. For $2g-2$0, $2g-2$1 is the free group of rank $2g-2$2. Thus genus $2g-2$3 corresponds to a tree, genus $2g-2$4 to a unicyclic graph, and for closed surfaces every vertex has valence $2g-2$5, so the graph is genuinely cubic (Sultan, 2011).

A central classification fact is that two pants decompositions have the same pants decomposition graph if and only if they divide the surface in the same topological manner; equivalently, they differ by an element of the mapping class group. In Moser’s formulation for closed surfaces, $2g-2$6 and $2g-2$7 are isomorphic if and only if there exists $2g-2$8 with $2g-2$9 (Sultan, 2011, Moser, 2011).

Dual-graph realizability also appears in representation theory. For a non-elementary $3g-3$0-representation of a closed surface group, any trivalent graph with $3g-3$1 edges that has at least one one-edge loop can occur as the graph of a compatible pants decomposition. For dense-image $3g-3$2-representations, every trivalent graph with $3g-3$3 edges can occur (Detcherry et al., 2022).

3. Elementary shifts, quotient graphs, and explicit distance bounds

An elementary move on a pants decomposition induces a local operation on the dual graph. In the one-holed torus case, the corresponding edge is a loop before and after the move, so the pants decomposition graphs are isomorphic. In the four-holed sphere case, the move becomes a collapse-and-expand operation: collapse the relevant edge, create a $3g-3$4-valent intermediate vertex, and expand again with a new pairing of the four incident half-edges. Moser calls this graph operation an elementary shift (Moser, 2011).

Because graph isomorphism classes coincide with mapping class group orbits, Moser defines the orbit graph $3g-3$5, the quotient of the pants graph by the action of $3g-3$6. Its vertices are isomorphism classes of pants decomposition graphs, and its edges are elementary shifts between those classes. This turns the infinite pants graph into a finite graph of trivalent-graph isomorphism classes, while preserving path length in the sense that any path in $3g-3$7 lifts to a path of the same length in the pants graph (Moser, 2011).

For a closed surface of genus $3g-3$8, the diameter $3g-3$9 of the orbit graph satisfies

$\xi(\Sigma)=3g+b-3$0

The proof proceeds by creating loops using girth estimates, performing loop surgery to reduce genus, and then standardizing to a distinguished $\xi(\Sigma)=3g+b-3$1-loop graph $\xi(\Sigma)=3g+b-3$2. The orbit contribution therefore grows like $\xi(\Sigma)=3g+b-3$3 (Moser, 2011).

Within the distinguished orbit, Moser uses the standard Lickorish generating curves $\xi(\Sigma)=3g+b-3$4 and constructs explicit paths from $\xi(\Sigma)=3g+b-3$5 to $\xi(\Sigma)=3g+b-3$6. The path lengths are

$\xi(\Sigma)=3g+b-3$7

Hence a word in these twists gives a quantitative within-orbit bound, and the full distance estimate in the pants complex is the sum of the orbit-graph term and the twist-word term (Moser, 2011).

4. Separating curves, cut-edges, and logarithmic obstructions

For the dual graph $\xi(\Sigma)=3g+b-3$8, separating curves become graph cut data. A curve in a pants decomposition is separating if and only if the corresponding edge in $\xi(\Sigma)=3g+b-3$9 is a non-trivial cut-edge, meaning that deleting it disconnects the graph in a non-trivial way. The paper further distinguishes curves that cut off genus from curves that cut off boundary components: the former are exactly those whose edge deletion yields two cyclic components, whereas if one component is acyclic then the curve cuts off boundary (Sultan, 2011).

This translation makes the distance to the separating locus a graph-theoretic problem. If $\kappa(\Sigma)=3g-3+r$0 is the cardinality of a minimal non-trivial connected cut-set in $\kappa(\Sigma)=3g-3+r$1, then any pants decomposition $\kappa(\Sigma)=3g-3+r$2 containing a separating curve that cuts off genus satisfies

$\kappa(\Sigma)=3g-3+r$3

The lower bound comes from the fact that a separating curve cutting off genus must cross enough pants pieces to see either the girth or the smallest connected cut-set (Sultan, 2011).

There is a complementary upper bound: $\kappa(\Sigma)=3g-3+r$4 A short cycle can be shortened by successive elementary moves until it becomes a loop, and a loop already corresponds to a separating curve. This yields asymptotically sharp estimates for the maximum distance $\kappa(\Sigma)=3g-3+r$5 from any pants decomposition of $\kappa(\Sigma)=3g-3+r$6 to one containing a separating curve. The explicit theorem gives

$\kappa(\Sigma)=3g-3+r$7

and for $\kappa(\Sigma)=3g-3+r$8,

$\kappa(\Sigma)=3g-3+r$9

while for $g$0,

$g$1

For fixed $g$2, $g$3; for fixed $g$4 and all $g$5, $g$6 (Sultan, 2011).

5. Product regions, Farey factors, and large flats

A different graph-theoretic theme concerns subgraphs of the pants graph obtained by fixing a multicurve $g$7. If the complementary subsurface $g$8 is a disjoint union $g$9 of complexity-one components, then every pants decomposition containing $3g-3$0 is determined by choosing one curve in each $3g-3$1. The resulting full subgraph

$3g-3$2

identifies naturally with

$3g-3$3

a product of Farey graphs (Taylor et al., 2013).

The principal theorem in this setting states that if $3g-3$4 is an $3g-3$5-multicurve, then $3g-3$6 is totally geodesic in $3g-3$7. Equivalently, every geodesic in the ambient pants graph joining two vertices of this product region remains entirely inside it. As a consequence, any embedded product of Farey graphs in a pants graph is totally geodesic, and $3g-3$8 contains an isometric embedding

$3g-3$9

if and only if

$2g-2$0

This is the paper’s convex-flat analogue of the rank theorem (Taylor et al., 2013).

A closely related result is formulated for $2g-2$1-handle multicurves. If $2g-2$2 is an $2g-2$3-handle multicurve, then the subgraph $2g-2$4 is also totally geodesic and isomorphic to a product of $2g-2$5 Farey graphs. Choosing a bi-infinite geodesic in each Farey factor yields an isometric embedding of $2g-2$6 into the pants graph. The maximal rank obtained in this way is

$2g-2$7

with notation $2g-2$8 for the number of boundary components (Estévez, 2013).

These product regions give an exact combinatorial counterpart of the product strata in the Weil–Petersson completion: the geometry is not merely coarse, but totally geodesic.

6. Special settings, weighted models, and extensions

For nonorientable surfaces, the pants graph has additional local move types. Besides the orientable-type moves on $2g-2$9 and $3g-3$00, there are moves on $3g-3$01 and $3g-3$02. The local subgraphs through an edge are accordingly of three combinatorial types: Farey subgraphs, fan subgraphs, and thin edges. Using this local structure, the automorphism group of the pants graph is shown to be the mapping class group except in the low-complexity cases $3g-3$03, $3g-3$04, and $3g-3$05; in particular,

$3g-3$06

This is an Ivanov-type rigidity theorem in the nonorientable setting (Stukow et al., 16 Jul 2025).

For surfaces of infinite type, the usual pants graph becomes badly disconnected: two pants decompositions lie in the same connected component if and only if they differ by finitely many curves. The graph therefore has infinitely many, in fact uncountably many, components, and the extended mapping class group embeds as a proper subgroup of its automorphism group. Branman repairs this by placing a coarser topology on the same geometric realization, producing a pants space that is path-connected and whose automorphism group is isomorphic to $3g-3$07 (Branman, 2020).

Weighted dual graphs also appear in analytic and spectral applications. For a closed genus-$3g-3$08 surface, the graph of a pants decomposition has four vertices, and only five simple graph types occur: the star $3g-3$09, the cycle $3g-3$10, the path $3g-3$11, the kite, and the complete graph $3g-3$12. When edge weights are the sums of the relevant cuff lengths, the positive eigenvalues of the associated weighted Laplacian govern the first three small nonzero eigenvalues of a degenerating genus-$3g-3$13 hyperbolic surface (Erchenko et al., 29 Apr 2026).

For infinite Riemann surfaces with an upper-bounded geodesic pants decomposition, the dual graph becomes a conductance network $3g-3$14, with conductances equal to cuff lengths. The Brownian motion on the surface is recurrent if and only if the random walk on $3g-3$15 is recurrent, and therefore the geodesic flow on the surface is ergodic if and only if the random walk on the weighted dual graph is recurrent. In this setting the graph of the pants decomposition is not merely a combinatorial shadow: it is the electrical skeleton controlling recurrence, type, and ergodicity (Bordenave et al., 13 Apr 2026).

These variants show that the graph of a pants decomposition is not a single object but a family of closely related combinatorial models. In finite type it classifies decomposition type and supports explicit distance estimates; in quotient form it organizes mapping class orbits; in product regions it detects flats and rank; and in weighted form it governs spectral and dynamical phenomena.