Abstract.
We study the topological types of pants decompositions of a surface by associating to any pants decomposition P∈ݒ«(Sg,n)ݑƒݒ«subscriptݑ†ݑ”ݑ›P\in\mathcalP(S_g,n)italic_P ∈ caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ), in a natural way its pants decomposition graph, Γ(P).Γݑƒ\Gamma(P).roman_Γ ( italic_P ) . This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a non-trivial separating curve for all surfaces of finite type, Sg,n.subscriptݑ†ݑ”ݑ›S_g,n.italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT . In the main theorem we provide an asymptotically sharp approximation of this non-trivial distance in terms of the topology of the surface. In particular, for closed surfaces of genus gݑ”gitalic_g we show the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a separating curve grows asymptotically like the function log(g).ݑ”\log(g).roman_log ( italic_g ) .

2 Preliminaries

3 Pants Decomposition Graph

4 Proof of Theorem 1.1

5 Construction of Large Girth, Log Length Connected Graphs

6 Appendix: Low Complexity Examples

The large scale geometry of Teichmüller space has been an object of interest in recent years, especially within the circles of ideas surrounding Thurston’s Ending Lamination Conjecture. In this context, the pants complex, ݒ«(S),ݒ«ݑ†\mathcalP(S),caligraphic_P ( italic_S ) , associated to a hyperbolic surface, S,ݑ†S,italic_S , becomes relevant, as by a theorem of Jeff Brock in [Bro], the pants complex is quasi-isometric to the Teichmüller space of a surface equipped with the Weil-Petersson metric, (ݒ¯(S),dWP).ݒ¯ݑ†subscriptݑ‘ݑŠݑƒ(\mathcalT(S),d_WP).( caligraphic_T ( italic_S ) , italic_d start_POSTSUBSCRIPT italic_W italic_P end_POSTSUBSCRIPT ) . Accordingly, in order to study large scale geometric properties of Teichmüller space with the Weil-Petersson metric, it suffices to study the pants complex of a surface. For instance, significant recent results of Brock-Farb [BrF], Behrstock [Beh], Behrstock-Minsky [BeMi], and Brock-Masur [BM] among others can be viewed from this perspective.

One feature of the coarse geometry of the pants complex in common to many analyses of the subject is the existence of natural quasi-isometrically embedded product regions. These product regions, which are obstructions to δݛ¿\deltaitalic_δ-hyperbolicity, correspond to pairs of pants decompositions of the surface containing a fixed non-trivially separating (multi)curve. In fact, often in the course of studying the coarse geometry of the pants complex it proves advantageous to pass to the net of pants decompositions that contain a non-trivially separating curve, and hence lie in a natural quasi-isometrically embedded product region. See for instance work of Brock-Masur in [BM] and Behrstock-Drutu-Mosher in [BDM] in which such methods are used to prove that the pants complexes of different complexities are relatively hyperbolic or thick, respectively. Similarly, work of Masur-Schleimer [MS], relies on similar methods to prove the pants complex for large enough surfaces has one end.

In this paper, we study the net of pants decompositions of a surface that contain a non-trivially separating curve within the entire pants complex of a surface. Specifically, by graph theoretic and combinatoric considerations, we determine the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a non-trivially separating curve, for all surfaces of finite type, Sg,n.subscriptݑ†ݑ”ݑ›S_g,n.italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT . The highlight of the paper is captured by following theorem which is a slight simplification of Theorem 4.1 proven in Section 4.

Let S=Sg,nݑ†subscriptݑ†ݑ”ݑ›S=S_g,nitalic_S = italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT and set Dg,n=maxP∈ݒ«(S)(dݒ«(S)(P,ݒ«sep(S))).subscriptݐ·ݑ”ݑ›subscriptݑƒݒ«ݑ†subscriptݑ‘ݒ«ݑ†ݑƒsubscriptݒ«ݑ ݑ’ݑݑ†D_g,n=\max_{}_P\in\mathcalP(S)(d_\mathcalP(S)(P,\mathcalP_sep(% S))).italic_D start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT = roman_max start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_P ∈ caligraphic_P ( italic_S ) end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT caligraphic_P ( italic_S ) end_POSTSUBSCRIPT ( italic_P , caligraphic_P start_POSTSUBSCRIPT italic_s italic_e italic_p end_POSTSUBSCRIPT ( italic_S ) ) ) . Then, for any fixed number of boundary components (or punctures) nݑ›nitalic_n, Dg,nsubscriptݐ·ݑ”ݑ›D_g,nitalic_D start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT grows asymptotically like the function log(g),ݑ”\log(g),roman_log ( italic_g ) , that is Dg,n=Θ(log(g)).subscriptݐ·ݑ”ݑ›normal-Θݑ”D_g,n=\Theta(\log(g)).italic_D start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT = roman_Θ ( roman_log ( italic_g ) ) . On the other hand, for any fixed genus g≥2,∀n≥6g-5,formulae-sequenceݑ”2for-allݑ›6ݑ”5g\geq 2,\;\forall n\geq 6g-5,italic_g ≥ 2 , ∀ italic_n ≥ 6 italic_g - 5 , Dg,n=2.subscriptݐ·ݑ”ݑ›2D_g,n=2.italic_D start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT = 2 .

The non-trivial lower bounds in Theorem 1.1 follow from an original and explicit constructive algorithm for an infinite family of high girth at most cubic graphs with the property that the minimum cardinality of connected cutsets is a logarithmic function with respect to the vertex size of the graphs, log length connected.

It should be noted that there is a sharp contrast between the nets provided by the subcomplexes ݒžsep(S)⊂ݒž(S)subscriptݒžݑ ݑ’ݑݑ†ݒžݑ†\mathcalC_sep(S)\subset\mathcalC(S)caligraphic_C start_POSTSUBSCRIPT italic_s italic_e italic_p end_POSTSUBSCRIPT ( italic_S ) ⊂ caligraphic_C ( italic_S ) and ݒ«sep(S)⊂ݒ«(S).subscriptݒ«ݑ ݑ’ݑݑ†ݒ«ݑ†\mathcalP_sep(S)\subset\mathcalP(S).caligraphic_P start_POSTSUBSCRIPT italic_s italic_e italic_p end_POSTSUBSCRIPT ( italic_S ) ⊂ caligraphic_P ( italic_S ) . Specifically, regarding the curve complex, by topological considerations, it is immediate that the distance in the curve complex from any isotopy class of a simple closed curve to a non-trivially separating simple closed curve is bounded above by one, for all surfaces of finite type. On the other hand, in the case of the pants complex, by Theorem 1.1, the maximal distance from an arbitrary pants decomposition to any pants decompositions containing a non-trivial separating curve is a non-trivial function depending on the topology of the surface. In fact, for any infinite sequence of surfaces with a uniformly bounded number of boundary components, the function is unbounded.

A key lemma used in the course of proving the lower bounds in Theorem 1.1 and which may be of independent interest is the following:

dݒ«(S)(P,P′)≥minݑ”ݑ–ݑŸݑ¡ℎ(Γ(P)),d-1subscriptݑ‘ݒ«ݑ†ݑƒsuperscriptݑƒ′ݑ”ݑ–ݑŸݑ¡ℎΓݑƒݑ‘1d_\mathcalP(S)(P,P^\prime)\geq\min\\mboxgirth(\Gamma(P)),d\-1italic_d start_POSTSUBSCRIPT caligraphic_P ( italic_S ) end_POSTSUBSCRIPT ( italic_P , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≥ roman_min girth ( roman_Γ ( italic_P ) ) , italic_d - 1
for P′superscriptݑƒnormal-′P^\primeitalic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT any pants decomposition containing a separating curve cutting off genus.

The proof of Lemma 4.12 brings together ideas related to the topology of the surfaces and graph theory in a simple yet elegant manner.

The results of this paper have some overlap with recent results Cavendish-Parlier [CP] as well as [RT], the latter of which was posted to the arXiv subsequent to the posting of this article, regarding the asymptotics of the diameter of Moduli Space. Although similar in nature, the results of this paper are in fact distinct from the aforementioned articles. Specifically, due to the fact that the quasi-isometry constants of [Bro] between the pants complex and Teichmüller space equipped with the Weil-Petersson metric are dependent on the topology of the particular surface, the results of this paper are more properly related to complex of cubic graphs than to Moduli Space. Accordingly, while the results of this paper can be used to consider the diameter of the complex of cubic graphs, they fail to provide direct information regarding the diameter of Moduli Space. Conversely, while methods in [CP] do contain lower bounds on the diameter of entire complex of cubic graphs, this paper focuses on the finer question of the density of a natural subset inside the entire space. On the other hand, it should be noted that methods in [RT] do provide an independent and alternative (albeit non-constructive) proof of the lower bounds achieved in section 5 of this paper by considering pants decompositions whose pants decomposition graphs are expanders. Specifically, reliance on the existence of expander graphs provides for a potential alternative to the construction of log length connected graphs in Section 5. Nonetheless, the explicit and constructive nature of the family of graphs in Section 5 is a novelty of this paper as compared to [RT].

The outline of the paper is as follows. In Section 2 we introduce select concepts from graph theory. Surface topology relevant to the development in this paper. In Section 3 we consider the pants decomposition graph of a pants decomposition of a surface. The pants decomposition graph is a graph that is naturally associated to a pants decomposition of a surface which captures the topological type of the pants decomposition. In Section 4 we prove Theorem 1.1 via a sequence of lemmas and corollaries. The proof of Theorem 1.1 in Section 4 is complete modulo a construction of an infinite family of high girth, log length connected, at most cubic graphs, which is explicitly described in Section 5. Finally in Section 6, the Appendix, some low complexity examples are considered.

I want to express my gratitude to my advisors Jason Behrstock and Walter Neumann for their extremely helpful advice and insights throughout my research, and specifically with regard to this paper. I want to further thank Jason for his thorough reading and comments on this work. I would also like to acknowledge Maria Chudnovsky. Rumen Zarev for useful discussions regarding particular arguments in this paper.

2.1. Graph Theory

Let Γ=Γ(V,E)ΓΓݑ‰ݐ¸\Gamma=\Gamma(V,E)roman_Γ = roman_Γ ( italic_V , italic_E ) be an undirected graph with vertex set Vݑ‰Vitalic_V and edge set E.ݐ¸E.italic_E . The degree of a vertex v∈V,ݑ£ݑ‰v\in V,italic_v ∈ italic_V , denoted d(v),ݑ‘ݑ£d(v),italic_d ( italic_v ) , is the number of times that the vertex vݑ£vitalic_v arises as an endpoint in E.ݐ¸E.italic_E . The degree of a graph Γ,Γ\Gamma,roman_Γ , denoted d(Γ),ݑ‘Γd(\Gamma),italic_d ( roman_Γ ) , is maxv∈V.conditionalݑ‘ݑ£ݑ£ݑ‰\max\d(v).roman_max italic_d ( italic_v ) . A graph ΓΓ\Gammaroman_Γ is called k-regular if each vertex v∈Vݑ£ݑ‰v\in Vitalic_v ∈ italic_V has degree exactly k. In particular, 3-regular graphs are called cubic graphs. Furthermore, a graph ΓΓ\Gammaroman_Γ is said to be at most cubic if d(Γ)≤3.ݑ‘Γ3d(\Gamma)\leq 3.italic_d ( roman_Γ ) ≤ 3 .

Given graphs, Γ(V,E),H(V′,E′),Γݑ‰ݐ¸ݐ»superscriptݑ‰′superscriptݐ¸′\Gamma(V,E),\;H(V^\prime,E^\prime),roman_Γ ( italic_V , italic_E ) , italic_H ( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , H is called a subgraph of Γ,Γ\Gamma,roman_Γ , denoted H⊂Γ,ݐ»ΓH\subset\Gamma,italic_H ⊂ roman_Γ , if V′⊂Vsuperscriptݑ‰′ݑ‰V^\prime\subset Vitalic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ italic_V and E′⊂E.superscriptݐ¸′ݐ¸E^\prime\subset E.italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ italic_E . In particular, for any subset S⊂V(Γ),ݑ†ݑ‰ΓS\subset V(\Gamma),italic_S ⊂ italic_V ( roman_Γ ) , the complete subgraph of Sݑ†Sitalic_S in Γnormal-Γ\Gammaroman_Γ, denoted Γ[S]Γdelimited-[]ݑ†\Gamma[S]roman_Γ [ italic_S ], is the subgraph of ΓΓ\Gammaroman_Γ with vertex set Sݑ†Sitalic_S and edges between any pair of vertices x,y∈Sݑ¥ݑ¦ݑ†x,y\in Sitalic_x , italic_y ∈ italic_S if and only if there is an edge e∈E(Γ)ݑ’ݐ¸Γe\in E(\Gamma)italic_e ∈ italic_E ( roman_Γ ) connecting the vertices xݑ¥xitalic_x and y.ݑ¦y.italic_y . By definition Γ[S]⊂ΓΓdelimited-[]ݑ†Γ\Gamma[S]\subset\Gammaroman_Γ [ italic_S ] ⊂ roman_Γ. As usual, we can make any graph ΓΓ\Gammaroman_Γ into a metric space by endowing the graph with the usual graph metric. Specifically, we assign each edge to have length one, and then define the distance between any two vertices to be the length of the shortest path in the graph connecting the two vertices if the vertices are in the same connected component of Γ,Γ\Gamma,roman_Γ , or infinity otherwise. The diameter of a graph, denoted diam(Γ),ݑ‘ݑ–ݑŽݑšΓdiam(\Gamma),italic_d italic_i italic_a italic_m ( roman_Γ ) , is the maximum of the distance function over all pairs of vertices in Γ×Γ.ΓΓ\Gamma\times\Gamma.roman_Γ × roman_Γ . This diameter function can be restricted to subgraphs in the obvious manner.

Given a graph Γ,Γ\Gamma,roman_Γ , a walk is a sequence of alternating vertices and edges, beginning and ending with a vertex, where each vertex is incident to both the edge that precedes it and the edge that follows it in the sequence. The length of a walk is the number of vertices in the walk. A cycle is a closed walk in which all edges and all vertices other than first and last are distinct. A loop is a cycle of length one. A graph ΓΓ\Gammaroman_Γ is acyclic if it contains no cycles, i.e. its connected components are trees. The girth of a graph ΓΓ\Gammaroman_Γ is defined to be the length of a shortest cycle in Γ,Γ\Gamma,roman_Γ , unless ΓΓ\Gammaroman_Γ is acyclic, in which case the girth is defined to be infinity.

A graph ΓΓ\Gammaroman_Γ is connected if there is a walk between any two vertices of the graph. Otherwise, it is said to be disconnected. If a subset of vertices, C⊂V,ݐ¶ݑ‰C\subset V,italic_C ⊂ italic_V , has the property that the deletion subgraph, Γ[V∖C]Γdelimited-[]ݑ‰ݐ¶\Gamma[V\setminus C]roman_Γ [ italic_V ∖ italic_C ], is disconnected, then Cݐ¶Citalic_C is called a cut-set of a graph. If the deletion subgraph Γ[V∖C]Γdelimited-[]ݑ‰ݐ¶\Gamma[V\setminus C]roman_Γ [ italic_V ∖ italic_C ], is disconnected and moreover it has at least two connected components each consisting of at least two vertices or a single vertex with a loop, Cݐ¶Citalic_C is said to be a non-trivial cut-set. A (nontrivial) cut-set Cݐ¶Citalic_C is called a minimal sized (non-trivial) cut-set if |C|ݐ¶|C|| italic_C | is minimal over all (non-trivial) cut-sets of Γ.Γ\Gamma.roman_Γ . On the other hand, a cut-set Cݐ¶Citalic_C is said to be a minimal (non-trivial) connected cut-set if |C|ݐ¶|C|| italic_C | is minimal over all (non-trivial) cut-sets Cݐ¶Citalic_C of ΓΓ\Gammaroman_Γ such that Γ[C]Γdelimited-[]ݐ¶\Gamma[C]roman_Γ [ italic_C ] is connected.

In this paper we are interested in a family of graphs that are robust with regard to non-trivial disconnection by the removal of connected cut-sets. More formally, we define an infinite family of graphs, Γi(Vi,Ei)subscriptΓݑ–subscriptݑ‰ݑ–subscriptݐ¸ݑ–\Gamma_i(V_i,E_i)roman_Γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), with increasing vertex size to be log length connected if they have the property that the size of minimal non-trivial connected cut-sets of the graphs, asymptotically grows logarithmically in the vertex size of the graphs. Specifically, if we set the function f(i)ݑ“ݑ–f(i)italic_f ( italic_i ) to be equal to the cardinality of a minimal non-trivial connected cut-set of the graph Γi,subscriptΓݑ–\Gamma_i,roman_Γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , then f(i)=Θ(log(|Vi|)).ݑ“ݑ–Θsubscriptݑ‰ݑ–f(i)=\Theta(\log(|V_i|)).italic_f ( italic_i ) = roman_Θ ( roman_log ( | italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) ) .

((3,g)3ݑ”(3,g)( 3 , italic_g )-cages) In the literature on graph theory, a family of graphs called (3,g)3ݑ”(3,g)( 3 , italic_g )-cages are a well studied, although not very well understood family of graphs. By definition a (k,g)ݑ˜ݑ”(k,g)( italic_k , italic_g )-cage is a graph of minimum vertex size among all kݑ˜kitalic_k-regular graphs with girth g.ݑ”g.italic_g . Note that (k,g)ݑ˜ݑ”(k,g)( italic_k , italic_g )-cages need not be unique, and generally are not. In [ES] it is shown that for k≥2,g≥3,formulae-sequenceݑ˜2ݑ”3k\geq 2,g\geq 3,italic_k ≥ 2 , italic_g ≥ 3 , there exist (k,g)ݑ˜ݑ”(k,g)( italic_k , italic_g )-cages. Moreover if we let μ(g)ݜ‡ݑ”\mu(g)italic_μ ( italic_g ) represent the number of vertices in a (3,g)3ݑ”(3,g)( 3 , italic_g )-cage, then it is well known that 2g/2≤μ(g)≤23g/4,superscript2ݑ”2ݜ‡ݑ”superscript23ݑ”42^g/2\leq\mu(g)\leq 2^3g/4,2 start_POSTSUPERSCRIPT italic_g / 2 end_POSTSUPERSCRIPT ≤ italic_μ ( italic_g ) ≤ 2 start_POSTSUPERSCRIPT 3 italic_g / 4 end_POSTSUPERSCRIPT , see [Big]. Furthermore, a theorem of Jiang and Mubayi, guarantees that the cardinality of a minimal non-trivial connected cut-set of a (3,g)3ݑ”(3,g)( 3 , italic_g )-cage is at least ⌊g2⌋.ݑ”2\lfloor\fracg2\rfloor.⌊ divide start_ARG italic_g end_ARG start_ARG 2 end_ARG ⌋ . Combining the two previous sentences it follows that the family of (3,g)3ݑ”(3,g)( 3 , italic_g )-cages are log length connected.

2.2. Curve and Pants Complex

Given any surface of finite type, S=Sg,n,ݑ†subscriptݑ†ݑ”ݑ›S=S_g,n,italic_S = italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT , that is a genus gݑ”gitalic_g surface with nݑ›nitalic_n boundary components (or punctures), the complexity of S,ݑ†S,italic_S , denoted ξ(S)∈ℤ,ݜ‰ݑ†ℤ\xi(S)\in\mathbbZ,italic_ξ ( italic_S ) ∈ blackboard_Z , is a topological invariant defined to be 3g-3+n.3ݑ”3ݑ›3g-3+n.3 italic_g - 3 + italic_n . To be sure, while in terms of the ℳCGℳݐ¶ݐº\mathcalMCGcaligraphic_M italic_C italic_G there is a distinction between boundary components of a surface and punctures on a surface, as elements of the ℳCGℳݐ¶ݐº\mathcalMCGcaligraphic_M italic_C italic_G must fix the former, yet can permute the latter, for the purposes of this paper such a distinction is not relevant. Accordingly, throughout this paper while we will always refer to surfaces with boundary components, the same results hold mutatis mutandis for surfaces with punctures.

A simple closed curve in Sݑ†Sitalic_S is peripheral if it bounds a disk containing at most one boundary component; a non-peripheral curve is essential. For Sݑ†Sitalic_S any surface with positive complexity, the curve complex of S,ݑ†S,italic_S , denoted ݒž(S),ݒžݑ†\mathcalC(S),caligraphic_C ( italic_S ) , is the simplicial complex obtained by associating to each isotopy class of an essential simple closed curve a 0-cell, and more generally a k-cell to each unordered tuple γ0,…,γksubscriptݛ¾0…subscriptݛ¾ݑ˜\\gamma_0,...,\gamma_k\ italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of k+1ݑ˜1k+1italic_k + 1 isotopy classes of disjoint essential simple closed curves, or multicurves. This simplicial complex first defined by Harvey [Har] has many natural applications to the study of the ℳCGℳݐ¶ݐº\mathcalMCGcaligraphic_M italic_C italic_G and is a well studied complex in geometric group theory.

Among simple closed curves on a surface of finite type we differentiate between two types of curves. Specifically, a simple closed curve γ⊂Sݛ¾ݑ†\gamma\subset Sitalic_γ ⊂ italic_S is called a non-trivially separating curve, or simply a separating curve, if S∖γݑ†ݛ¾S\setminus\gammaitalic_S ∖ italic_γ consists of two connected components Y1subscriptݑŒ1Y_1italic_Y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and Y2subscriptݑŒ2Y_2italic_Y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT such that ξ(Yi)≥1.ݜ‰subscriptݑŒݑ–1\xi(Y_i)\geq 1.italic_ξ ( italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ 1 . Any other simple closed curve is non-separating. It should be stressed that, perhaps counterintuitively, a trivially separating curve, that is a simple closed curve that cuts off two boundary components of the surface, under our definition, is considered a non-separating curve. In light of the dichotomy between separating curves and non-separating curves, there is an important natural subcomplex of the curve complex called the complex of separating curves, denoted ݒžsep(S),subscriptݒžݑ ݑ’ݑݑ†\mathcalC_sep(S),caligraphic_C start_POSTSUBSCRIPT italic_s italic_e italic_p end_POSTSUBSCRIPT ( italic_S ) , which is the restriction of the curve complex to the set of separating curves.

For Sݑ†Sitalic_S a surface of positive complexity, a pair of pants decomposition, or simply a pants decomposition, PݑƒPitalic_P is a multicurve of maximal cardinality. Equivalently, a pants decomposition PݑƒPitalic_P is a set of disjoint homotopically distinct curves such that the complement S∖Pݑ†ݑƒS\setminus Pitalic_S ∖ italic_P consists of a disjoint union of topological pairs of pants, or spheres with three boundary components.

Related to the curve complex, ݒž(S),ݒžݑ†\mathcalC(S),caligraphic_C ( italic_S ) , there is another natural complex associated to any surface of finite type with positive complexity: the pants complex. In particular, the 1-skeleton of the pants complex, the pants graph, denoted ݒ«(S),ݒ«ݑ†\mathcalP(S),caligraphic_P ( italic_S ) , is a graph with vertices corresponding to different pants decompositions of the surface, and edges between two vertices when the two corresponding pants decompositions differ by a so called elementary pants move. Specifically, two pants decompositions of a surface differ by an elementary pants move, if the two decompositions differ in exactly one curve and those differing curves intersect minimally inside the unique complexity one component of the surface, topologically either an S0,4subscriptݑ†04S_0,4italic_S start_POSTSUBSCRIPT 0 , 4 end_POSTSUBSCRIPT or an S1,1,subscriptݑ†11S_1,1,italic_S start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT , in the complement of all the other agreeing curves in the pants decompositions. By a theorem of Hatcher and Thurston, [HT], the pants graph is connected, and hence we have a notion of distance between different vertices, or pants decompositions P1,P2∈ݒ«(S),subscriptݑƒ1subscriptݑƒ2ݒ«ݑ†P_1,P_2\in\mathcalP(S),italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_P ( italic_S ) , obtained by endowing ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) with the graph metric. We denote this distance by dݒ«(P1,P2).subscriptݑ‘ݒ«subscriptݑƒ1subscriptݑƒ2d_\mathcalP(P_1,P_2).italic_d start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) .

Just as with the curve complex, there is an important subcomplex of the pants complex called the pants complex of separating curves, denoted ݒ«sep(S),subscriptݒ«ݑ ݑ’ݑݑ†\mathcalP_sep(S),caligraphic_P start_POSTSUBSCRIPT italic_s italic_e italic_p end_POSTSUBSCRIPT ( italic_S ) , which is the restriction of the pants graph to the set of those pants decompositions that contain a separating curve. This paper analyzes the net of the pants complex of separating curves in the entire pants complex, for all surfaces of finite type.

3. Pants Decomposition Graph

By elementary topological considerations, it follows that for any pants decomposition P∈P(Sg,n),ݑƒݑƒsubscriptݑ†ݑ”ݑ›P\in P(S_g,n),italic_P ∈ italic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ) , the number of curves in the pants decomposition PݑƒPitalic_P, is equal to ξ(S)=3g-3+n,ݜ‰ݑ†3ݑ”3ݑ›\xi(S)=3g-3+n,italic_ξ ( italic_S ) = 3 italic_g - 3 + italic_n , while the number of pairs of pants into which the pants decomposition decomposes the surface is equal to 2(g-1)+n.2ݑ”1ݑ›2(g-1)+n.2 ( italic_g - 1 ) + italic_n . Corresponding to any pants decomposition PݑƒPitalic_P we define its pants decomposition graph, Γ(P),Γݑƒ\Gamma(P),roman_Γ ( italic_P ) , as follows: For P∈ݒ«(S),ݑƒݒ«ݑ†P\in\mathcalP(S),italic_P ∈ caligraphic_P ( italic_S ) , Γ(P)Γݑƒ\Gamma(P)roman_Γ ( italic_P ) is a graph with vertices corresponding the connected components of S∖P,ݑ†ݑƒS\setminus P,italic_S ∖ italic_P , and edges between vertices corresponding to connected components that share a common boundary curve. See Figure 1 for an example of a pants decomposition graph. Pants decomposition graphs classify pants decompositions up to topological type. Specifically, two pants decompositions have the same pants decomposition graph if and only if they divide the surface in the same topological manner, or equivalently the two pants decompositions differ by an element of the mapping class group.

The notion of pants decomposition graphs is considered in [Bus] as well as in [Par]. Moreover, replacing the vertices in a pants decomposition graph with edges and vice versa yields the adjacency graph of Behrstock and Margalit [BeMa] developed in the course of proving that the mapping class group is co-Hopfian with regard to finite index subgroups.

The following elementary lemma, whose proof follows immediately, organizes elementary properties of Γ(P)Γݑƒ\Gamma(P)roman_Γ ( italic_P ) and gives a one to one correspondence between certain graphs and pants decomposition graphs:

For P∈ݒ«(Sg,n),ݑƒݒ«subscriptݑ†ݑ”ݑ›P\in\mathcalP(S_g,n),italic_P ∈ caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ) , and Γ(P)normal-Γݑƒ\Gamma(P)roman_Γ ( italic_P ) its pants decomposition graph:

(1)
Γ(P)Γݑƒ\Gamma(P)roman_Γ ( italic_P ) is a connected graph with 2(g-1)+n2ݑ”1ݑ›2(g-1)+n2 ( italic_g - 1 ) + italic_n vertices and 3(g-1)+n3ݑ”1ݑ›3(g-1)+n3 ( italic_g - 1 ) + italic_n edges

(2)
Γ(P)Γݑƒ\Gamma(P)roman_Γ ( italic_P ) is at most cubic

Moreover, for all q,p∈ℕݑžݑℕq,p\in\mathbbNitalic_q , italic_p ∈ blackboard_N, given any connected, at most cubic graph Γ=Γ(V,E)normal-Γnormal-Γݑ‰ݐ¸\Gamma=\Gamma(V,E)roman_Γ = roman_Γ ( italic_V , italic_E ) with |V|=2(p-1)+qݑ‰2ݑ1ݑž|V|=2(p-1)+q| italic_V | = 2 ( italic_p - 1 ) + italic_q and |E|=3(p-1)+q,ݐ¸3ݑ1ݑž|E|=3(p-1)+q,| italic_E | = 3 ( italic_p - 1 ) + italic_q , there exists a pants decomposition P∈ݒ«(Sp,q)ݑƒݒ«subscriptݑ†ݑݑžP\in\mathcalP(S_p,q)italic_P ∈ caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ) with pants decomposition graph Γ(P)≅Γ.normal-Γݑƒnormal-Γ\Gamma(P)\cong\Gamma.roman_Γ ( italic_P ) ≅ roman_Γ .

Euler characteristic considerations imply the following corollary of Lemma 3.2:

For P∈ݒ«(Sg,n),ݑƒݒ«subscriptݑ†ݑ”ݑ›P\in\mathcalP(S_g,n),italic_P ∈ caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ) , π1(Γ(P))subscriptݜ‹1normal-Γݑƒ\pi_1(\Gamma(P))italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( roman_Γ ( italic_P ) ) is the free group of rank g.ݑ”g.italic_g .

Another relevant elementary lemma is the following:

Let P∈ݒ«(Sg,n),ݑƒݒ«subscriptݑ†ݑ”ݑ›P\in\mathcalP(S_g,n),italic_P ∈ caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ) , and let πݒž:ݒž(Sg,n)↠ݒž(Sg,n-1)∪∅normal-:subscriptݜ‹ݒžnormal-↠ݒžsubscriptݑ†ݑ”ݑ›ݒžsubscriptݑ†ݑ”ݑ›1\pi_\mathcalC\colon\mathcalC(S_g,n)\twoheadrightarrow\mathcalC(S_g,% n-1)\cup\emptysetitalic_π start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT : caligraphic_C ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ) ↠caligraphic_C ( italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT ) ∪ ∅ be a projection map which fills in a boundary component. Then the map πݜ‹\piitalic_π extends to a surjection

πݒ«:ݒ«(Sg,n)↠ݒ«(Sg,n-1).:subscriptݜ‹ݒ«↠ݒ«subscriptݑ†ݑ”ݑ›ݒ«subscriptݑ†ݑ”ݑ›1\pi_\mathcalP\colon\mathcalP(S_g,n)\twoheadrightarrow\mathcalP(S_g,% n-1).italic_π start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT : caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ) ↠caligraphic_P ( italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT ) .

Note that the map πݒžsubscriptݜ‹ݒž\pi_\mathcalCitalic_π start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT has range ݒž(Sg,n-1)∪∅ݒžsubscriptݑ†ݑ”ݑ›1\mathcalC(S_g,n-1)\cup\emptysetcaligraphic_C ( italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT ) ∪ ∅ as an essential curve that cuts off a pair of boundary components can become peripheral in the event that one of the cut off boundary components is filled in.

Under the map πݒ«,subscriptݜ‹ݒ«\pi_\mathcalP,italic_π start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT , all but one of the pairs of pants in a pants decomposition of Sg,nsubscriptݑ†ݑ”ݑ›S_g,nitalic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT are left unaffected. The one affected pair of pants, which contains the boundary component being filled, becomes an annulus in Sg,n-1.subscriptݑ†ݑ”ݑ›1S_g,n-1.italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT . After identifying the two isotopic boundary curves of the annulus in Sg,n-1,subscriptݑ†ݑ”ݑ›1S_g,n-1,italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT , we have a pants decomposition of Sg,n-1.subscriptݑ†ݑ”ݑ›1S_g,n-1.italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT . The fact that the projection πݒ«subscriptݜ‹ݒ«\pi_\mathcalPitalic_π start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT is surjective follows the observation that given any pants decomposition of Sg,n-1,subscriptݑ†ݑ”ݑ›1S_g,n-1,italic_S start_POSTSUBSCRIPT italic_g , italic_n - 1 end_POSTSUBSCRIPT , one can easily construct a lift under πݒ«subscriptݜ‹ݒ«\pi_\mathcalPitalic_π start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT of the pants decomposition in Sg,n.subscriptݑ†ݑ”ݑ›S_g,n.italic_S start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT . ∎

In the next three subsections we explore certain aspects of pants decomposition graphs. 3.1. Calculus of elementary pants moves. Their action on pants decomposition graphs. 1. Calculus of elementary pants moves. Their action on pants decomposition graphs. Calculus of elementary pants moves. Their action on pants decomposition graphs.

Recall that there are two types of elementary pants moves depending on the type of complexity one piece in which the move takes place:

E1:

Inside a S1,1subscriptݑ†11S_1,1italic_S start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT component of the surface in the complement of all of the pants curves except α,ݛ¼\alpha,italic_α , the curve αݛ¼\alphaitalic_α is replaced with βݛ½\betaitalic_β where αݛ¼\alphaitalic_α and βݛ½\betaitalic_β intersect once.

E2:

Inside a S0,4subscriptݑ†04S_0,4italic_S start_POSTSUBSCRIPT 0 , 4 end_POSTSUBSCRIPT component of the surface in the complement of all of the pants curves except α,ݛ¼\alpha,italic_α , the curve αݛ¼\alphaitalic_α is replaced with βݛ½\betaitalic_β where αݛ¼\alphaitalic_α and βݛ½\betaitalic_β interse

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