Abstract.
We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

Let Sݑ†Sitalic_S be a connected orientable surface, with empty boundary and negative Euler characteristic. The pants graph ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) is the graph whose vertices correspond to homotopy classes of pants decompositions of Sݑ†Sitalic_S, and where two vertices are adjacent if they are related by an elementary move; see Section 2 for an expanded definition. The graph ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) is connected, and becomes a geodesic metric space by declaring each edge to have length 1.

A large part of the motivation for the study of ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) stems from the result of Brock [4] which asserts that P(S)ݑƒݑ†P(S)italic_P ( italic_S ) is quasi-isometric to ݒ¯(S)ݒ¯ݑ†\mathcalT(S)caligraphic_T ( italic_S ), the Teichmüller space of Sݑ†Sitalic_S equipped with the Weil-Petersson metric. As such, P(S)ݑƒݑ†P(S)italic_P ( italic_S ) (or any of its relatives also discussed in this article) is a combinatorial model for Teichmüller space.

By results of Wolpert [12] and Chu [7], the space ݒ¯(S)ݒ¯ݑ†\mathcalT(S)caligraphic_T ( italic_S ) is not complete. Masur [8] proved that its completion ݒ¯^(S)^ݒ¯ݑ†\hat\mathcalT(S)over^ start_ARG caligraphic_T end_ARG ( italic_S ) is homeomorphic to the augmented Teichmüller space of Sݑ†Sitalic_S, obtained from ݒ¯(S)ݒ¯ݑ†\mathcalT(S)caligraphic_T ( italic_S ) by extending Fenchel-Nielsen coordinates to admit zero lengths. The completion ݒ¯^(S)^ݒ¯ݑ†\hat\mathcalT(S)over^ start_ARG caligraphic_T end_ARG ( italic_S ) admits a natural stratified structure: each stratum ݒ¯C(S)⊂ݒ¯^(S)subscriptݒ¯ݐ¶ݑ†^ݒ¯ݑ†\mathcalT_C(S)\subset\hat\mathcalT(S)caligraphic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) ⊂ over^ start_ARG caligraphic_T end_ARG ( italic_S ) corresponds to a multicurve C⊂Sݐ¶ݑ†C\subset Sitalic_C ⊂ italic_S, and parametrizes surfaces with nodes exactly at the elements of Cݐ¶Citalic_C. By Wolpert’s result [11] on the convexity of length functions, ݒ¯C(S)subscriptݒ¯ݐ¶ݑ†\mathcalT_C(S)caligraphic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) is convex in ݒ¯^(S)^ݒ¯ݑ†\hat\mathcalT(S)over^ start_ARG caligraphic_T end_ARG ( italic_S ) for all multicurves C⊂Sݐ¶ݑ†C\subset Sitalic_C ⊂ italic_S.

The pants graph admits an analogous stratification, where the stratum ݒ«C(S)subscriptݒ«ݐ¶ݑ†\mathcalP_C(S)caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) corresponding to the multicurve Cݐ¶Citalic_C is the subgraph of ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) spanned by those pants decompositions that contain Cݐ¶Citalic_C. Moreover, Brock’s quasi-isometry between ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) and ݒ¯(S)ݒ¯ݑ†\mathcalT(S)caligraphic_T ( italic_S ) may be chosen so that the image of ݒ«C(S)subscriptݒ«ݐ¶ݑ†\mathcalP_C(S)caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) is contained in ݒ¯C(S)subscriptݒ¯ݐ¶ݑ†\mathcalT_C(S)caligraphic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ).

In light of this discussion, it is natural to study which strata of ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) are convex. This problem was addressed in [1, 2], where certain families of strata in ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) were proven to be totally geodesic; moreover, it is conjectured that this is the case for all strata of ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ), see Conjecture 5 of [1]. As was observed by Lecuire, the validity of this conjecture is equivalent to the existence of only finitely many geodesics between any pair of vertices of ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ); we will give a proof of the equivalence of these two problems in the Appendix.

The main purpose of this note is to study the extrinsic geometry of strata in certain graphs of pants decompositions closely related to ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ), page_seo_titlely the diagonal pants graph ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) and the cubical pants graph. Concisely, ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) (resp. ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S )) is obtained from ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) by adding an edge of length 1 (resp. length kݑ˜\sqrtksquare-root start_ARG italic_k end_ARG) between any two pants decompositions that differ by kݑ˜kitalic_k disjoint elementary moves. Note that Brock’s result [4] implies that both ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) and ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S ) are quasi-isometric to ݒ¯(S)ݒ¯ݑ†\mathcalT(S)caligraphic_T ( italic_S ), since they are quasi-isometric to ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ). These graphs have recently arisen in the study of metric properties of moduli spaces; indeed, Rafi-Tao [10] use ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) to estimate the Teichmüller diameter of the thick part of moduli space, while ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S ) has been used by Cavendish-Parlier to give bounds on the Weil-Petersson diameter of moduli space.

As above, given a multicurve C⊂Sݐ¶ݑ†C\subset Sitalic_C ⊂ italic_S, denote by ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) (resp. ݒžݒ«C(S))fragmentsCsubscriptݒ«ݐ¶fragments(S))\mathcalCP_C(S))caligraphic_C caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) ) the subgraph of ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) (resp. ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S )) spanned by those pants decompositions which contain Cݐ¶Citalic_C. Our first result is:

Let Sݑ†Sitalic_S be a sphere with punctures and C⊂Sݐ¶ݑ†C\subset Sitalic_C ⊂ italic_S a multicurve. Then ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) is convex in ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ).

We remark that, in general, strata of ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) are not totally geodesic; see Remark 4.10 below.

The proof of Theorem 1.1 will follow directly from properties of the forgetful maps between graphs of pants decompositions. In fact, the same techniques will allow us to prove the following more general result:

Let YݑŒYitalic_Y be an essential subsurface of a surface Sݑ†Sitalic_S such that YݑŒYitalic_Y has the same genus as Sݑ†Sitalic_S. Let Cݐ¶Citalic_C be the union of a pants decomposition of S∖Yݑ†ݑŒS\setminus Yitalic_S ∖ italic_Y with all the boundary components of YݑŒYitalic_Y. Then:

(2)
If YݑŒYitalic_Y is connected, then ݒ«C(S)subscriptݒ«ݐ¶ݑ†\mathcalP_C(S)caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) and ݒžݒ«C(S)ݒžsubscriptݒ«ݐ¶ݑ†\mathcalCP_C(S)caligraphic_C caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) are totally geodesic inside ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) and ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S ), respectively.

Next, we will use an enhanced version of the techniques in [1] to show an analog of Theorem 1.1 for general surfaces, provided the multicurve Cݐ¶Citalic_C has deficiency 1; that is, it has one curve less than a pants decomposition. page_seo_titlely, we will prove:

Let C⊂Sݐ¶ݑ†C\subset Sitalic_C ⊂ italic_S be a multicurve of deficiency 1111. Then:

(1)
ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) is convex in ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ).

(2)
ݒžݒ«C(S)ݒžsubscriptݒ«ݐ¶ݑ†\mathcalCP_C(S)caligraphic_C caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) is totally geodesic in ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S ).

We observe that part (2) of Theorem 1.3 implies the main results in [1].

We now turn to discuss some applications of our main results. The first one concerns the existence of convex flat subspaces of ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) of every possible rank. Since ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) is quasi-isometric to ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ), the results of Behrstock-Minsky [3], Brock-Farb [5] and Masur-Minsky [9] together yield that ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) admits a quasi-isometrically embedded copy of ℤrsuperscriptℤݑŸ\mathbbZ^rblackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT if and only if r≤[3g+p-22]ݑŸdelimited-[]3ݑ”ݑ22r\leq[\frac3g+p-22]italic_r ≤ [ divide start_ARG 3 italic_g + italic_p - 2 end_ARG start_ARG 2 end_ARG ]. If one considers T^(S)^ݑ‡ݑ†\hatT(S)over^ start_ARG italic_T end_ARG ( italic_S ) instead of ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ), one obtains isometrically embedded copies of ℤrsuperscriptℤݑŸ\mathbbZ^rblackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT for the exact same values of rݑŸritalic_r. In [2], convex copies of ℤ2superscriptℤ2\mathbbZ^2blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT were exhibited in ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ), and it is unknown whether higher rank convex flats appear. The corresponding question for the diagonal pants graph is whether there exist convex copies of ℤrsuperscriptℤݑŸ\mathbbZ^rblackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT with a modified metric which takes into account the edges added to obtain ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ); we will denote this metric space ݒŸℤrݒŸsuperscriptℤݑŸ\mathcalD\mathbbZ^rcaligraphic_D blackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT. Using Theorems 1.1 and 1.2, we will obtain a complete answer to this question, page_seo_titlely:

There exists an isometric embedding ݒŸℤr→ݒŸݒ«(S)normal-→ݒŸsuperscriptℤݑŸݒŸݒ«ݑ†\mathcalD\mathbbZ^r\to\mathcalDP(S)caligraphic_D blackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT → caligraphic_D caligraphic_P ( italic_S ) if and only if r≤[3g+p-22]ݑŸdelimited-[]3ݑ”ݑ22r\leq[\frac3g+p-22]italic_r ≤ [ divide start_ARG 3 italic_g + italic_p - 2 end_ARG start_ARG 2 end_ARG ].

Our second application concerns the finite geodesicity of different graphs of pants decompositions. Combining Theorem 1.1 with the main results of [1, 2], we will obtain the following:

Let Sݑ†Sitalic_S be the six-times punctured sphere, and let P,Qݑƒݑ„P,Qitalic_P , italic_Q be two vertices of ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ). Then there are only finitely many geodesics in ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) between PݑƒPitalic_P and Qݑ„Qitalic_Q.

The analogue of Corollary 1.5 for ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) is not true. Indeed, we will observe that if Sݑ†Sitalic_S is a sphere with at least six punctures, one can find pairs of vertices of ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) such that there are infinitely many geodesics in ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) between them. However, we will prove:

Given any surface Sݑ†Sitalic_S, and any kݑ˜kitalic_k, there exist points in ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) at distance kݑ˜kitalic_k with only a finite number of geodesics between them. Acknowledgements. The authors would like to thank Jeff Brock. Saul Schleimer for conversations. The authors would like to thank Jeff Brock. Saul Schleimer for conversations.

2. S be a connected orientable surface with empty boundary. Negative Euler characteristic. The complexity of Sݑ†Sitalic_S is the number κ(S)=3g-3+pݜ…ݑ†3ݑ”3ݑ\kappa(S)=3g-3+pitalic_κ ( italic_S ) = 3 italic_g - 3 + italic_p, where gݑ”gitalic_g and pݑpitalic_p denote, respectively, the genus and number of punctures of Sݑ†Sitalic_S.

2.1. Curves and multicurves

By a curve on Sݑ†Sitalic_S we will mean a homotopy class of simple closed curves on Sݑ†Sitalic_S; we will often blur the distinction between a curve and any of its representatives. We say that a curve α⊂Sݛ¼ݑ†\alpha\subset Sitalic_α ⊂ italic_S is essential if no representative of αݛ¼\alphaitalic_α bounds a disk with at most one puncture. The geometric intersection number between two curves αݛ¼\alphaitalic_α and βݛ½\betaitalic_β is defined as

i(α,β)=mina∩b.ݑ–ݛ¼ݛ½:ݑŽݑformulae-sequenceݑŽݛ¼ݑݛ½i(\alpha,\beta)=\min\.italic_i ( italic_α , italic_β ) = roman_min : italic_a ∈ italic_α , italic_b ∈ italic_β .
If α≠βݛ¼ݛ½\alpha\neq\betaitalic_α ≠italic_β and i(α,β)=0ݑ–ݛ¼ݛ½0i(\alpha,\beta)=0italic_i ( italic_α , italic_β ) = 0 we say that αݛ¼\alphaitalic_α and βݛ½\betaitalic_β are disjoint; otherwise, we say they intersect. We say that the curves αݛ¼\alphaitalic_α and βݛ½\betaitalic_β fill Sݑ†Sitalic_S if i(α,γ)+i(β,γ)>0ݑ–ݛ¼ݛ¾ݑ–ݛ½ݛ¾0i(\alpha,\gamma)+i(\beta,\gamma)>0italic_i ( italic_α , italic_γ ) + italic_i ( italic_β , italic_γ ) >0 for all curves γ⊂Sݛ¾ݑ†\gamma\subset Sitalic_γ ⊂ italic_S. A multicurve is a collection of pairwise distinct. Pairwise disjoint essential curves. The deficiency of a multicurve Cݐ¶Citalic_C is defined as κ(S)-|C|ݜ…ݑ†ݐ¶\kappa(S)-|C|italic_κ ( italic_S ) - | italic_C |, where |C|ݐ¶|C|| italic_C | denotes the number of elements of Cݐ¶Citalic_C.

2.2. Graphs of pants decompositions

A pants decomposition PݑƒPitalic_P of Sݑ†Sitalic_S is a multicurve that is maximal with respect to inclusion. As such, PݑƒPitalic_P consists of exactly κ(S)ݜ…ݑ†\kappa(S)italic_κ ( italic_S ) curves, and has a representative P′superscriptݑƒ′P^\primeitalic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT such that every connected component of S∖P′ݑ†superscriptݑƒ′S\setminus P^\primeitalic_S ∖ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is homeomorphic to a sphere with three punctures, or pair of pants.

We say that two pants decompositions of Sݑ†Sitalic_S are related by an elementary move if they have exactly κ(S)-1ݜ…ݑ†1\kappa(S)-1italic_κ ( italic_S ) - 1 curves in common, and the remaining two curves either fill a one-holed torus and intersect exactly once, or else they fill a four-holed sphere and intersect exactly twice. Observe that every elementary move determines a unique subsurface of Sݑ†Sitalic_S of complexity 1; we will say that two elementary moves are disjoint if the subsurfaces they determine are disjoint.

As mentioned in the introduction, the pants graph ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) of Sݑ†Sitalic_S is the simplicial graph whose vertex set is the set of pants decompositions of Sݑ†Sitalic_S, considered up to homotopy, and where two pants decompositions are adjacent in ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) if and only if they are related by an elementary move. We turn ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) into a geodesic metric space by declaring the length of each edge to be 1.

The diagonal pants graph ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) is the simplicial graph obtained from ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) by adding an edge of unit length between any two vertices that differ by k≥2ݑ˜2k\geq 2italic_k ≥ 2 disjoint elementary moves.

The cubical pants graph ݒžݒ«(S)ݒžݒ«ݑ†\mathcalCP(S)caligraphic_C caligraphic_P ( italic_S ) is obtained from ݒ«(S)ݒ«ݑ†\mathcalP(S)caligraphic_P ( italic_S ) by adding an edge of length kݑ˜\sqrtksquare-root start_ARG italic_k end_ARG between any two edges that differ by k≥2ݑ˜2k\geq 2italic_k ≥ 2 disjoint elementary moves.

3. The forgetful maps: proofs of Theorems 1.1 and 1.2

The idea of the proof of Theorems 1.1 and 1.2 is to use the so-called forgetful maps to define a distance non-increasing projection from the diagonal pants graph to each of its strata.

3.1. Forgetful maps

Let S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be connected orientable surfaces with empty boundary. Suppose that S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT have equal genus, and that S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT has at most as many punctures as S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Choosing an identification between the set of punctures of S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and a subset of the set of punctures of S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT yields a map ψ~:S1→S~1:~ݜ“→subscriptݑ†1subscript~ݑ†1\tilde\psi:S_1\to\tildeS_1over~ start_ARG italic_ψ end_ARG : italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, where S~1subscript~ݑ†1\tildeS_1over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is obtained by forgetting all punctures of S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT that do not correspond to a puncture of S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Now, S~1subscript~ݑ†1\tildeS_1over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are homeomorphic; by choosing a homeomorphism φ:S~1→S2:ݜ‘→subscript~ݑ†1subscriptݑ†2\varphi:\tildeS_1\to S_2italic_φ : over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, we obtain a map ψ=φ∘ψ~:S1→S2:ݜ“ݜ‘~ݜ“→subscriptݑ†1subscriptݑ†2\psi=\varphi\circ\tilde\psi:S_1\to S_2italic_ψ = italic_φ ∘ over~ start_ARG italic_ψ end_ARG : italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. We refer to all such maps ψݜ“\psiitalic_ψ as forgetful maps.

Observe that if α⊂S1ݛ¼subscriptݑ†1\alpha\subset S_1italic_α ⊂ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is a curve then ψ(α)ݜ“ݛ¼\psi(\alpha)italic_ψ ( italic_α ) is a (possibly homotopically trivial) curve on S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Also, i(ψ(α),ψ(β))≤i(α,β)ݑ–ݜ“ݛ¼ݜ“ݛ½ݑ–ݛ¼ݛ½i(\psi(\alpha),\psi(\beta))\leq i(\alpha,\beta)italic_i ( italic_ψ ( italic_α ) , italic_ψ ( italic_β ) ) ≤ italic_i ( italic_α , italic_β ) for al curves α,β⊂Sݛ¼ݛ½ݑ†\alpha,\beta\subset Sitalic_α , italic_β ⊂ italic_S. Lastly, observe that if PݑƒPitalic_P and Qݑ„Qitalic_Q are pants decompositions of S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT that differ by at most kݑ˜kitalic_k pairwise disjoint elementary moves, then ψ(P)ݜ“ݑƒ\psi(P)italic_ψ ( italic_P ) and ψ(Q)ݜ“ݑ„\psi(Q)italic_ψ ( italic_Q ) differ by at most kݑ˜kitalic_k pairwise disjoint elementary moves. We summarize these observations as a lemma.

Let ψ:S1→S2normal-:ݜ“normal-→subscriptݑ†1subscriptݑ†2\psi:S_1\to S_2italic_ψ : italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be a forgetful map. Then:

(1)
If PݑƒPitalic_P is a pants decomposition of S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then ψ(P)ݜ“ݑƒ\psi(P)italic_ψ ( italic_P ) is a pants decomposition of S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

(2)
If PݑƒPitalic_P and Qݑ„Qitalic_Q are related by kݑ˜kitalic_k disjoint elementary moves in S1subscriptݑ†1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then ψ(S1)ݜ“subscriptݑ†1\psi(S_1)italic_ψ ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and ψ(S2)ݜ“subscriptݑ†2\psi(S_2)italic_ψ ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) are related by at most kݑ˜kitalic_k disjoint elementary moves in S2subscriptݑ†2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. □□\square□

In light of these properties, we obtain that forgetful maps induce distance non-increasing maps

ݒ«(S1)→ݒ«(S2)→ݒ«subscriptݑ†1ݒ«subscriptݑ†2\mathcalP(S_1)\to\mathcalP(S_2)caligraphic_P ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → caligraphic_P ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )
and, similarly,

ݒŸݒ«(S1)→ݒŸݒ«(S2) and ݒžݒ«(S1)→ݒžݒ«(S2).formulae-sequence→ݒŸݒ«subscriptݑ†1ݒŸݒ«subscriptݑ†2 and →ݒžݒ«subscriptݑ†1ݒžݒ«subscriptݑ†2\mathcalDP(S_1)\to\mathcalDP(S_2)\hskip 14.22636pt\text and % \hskip 14.22636pt\mathcalCP(S_1)\to\mathcalCP(S_2).caligraphic_D caligraphic_P ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → caligraphic_D caligraphic_P ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) and caligraphic_C caligraphic_P ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → caligraphic_C caligraphic_P ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) .

3.2. Projecting to strata

Let Sݑ†Sitalic_S be a sphere with punctures, Cݐ¶Citalic_C a multicurve on Sݑ†Sitalic_S, and consider S∖C=X1⊔…⊔Xnݑ†ݐ¶square-unionsubscriptݑ‹1…subscriptݑ‹ݑ›S\setminus C=X_1\sqcup\ldots\sqcup X_nitalic_S ∖ italic_C = italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊔ … ⊔ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. For each iݑ–iitalic_i, we proceed as follows. On each connected component of S∖Xiݑ†subscriptݑ‹ݑ–S\setminus X_iitalic_S ∖ italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT we choose a puncture of Sݑ†Sitalic_S. Let X¯isubscript¯ݑ‹ݑ–\barX_iover¯ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT be the surface obtained from Sݑ†Sitalic_S by forgetting all punctures of Sݑ†Sitalic_S in S∖Xiݑ†subscriptݑ‹ݑ–S\setminus X_iitalic_S ∖ italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT except for these chosen punctures. Noting that X¯isubscript¯ݑ‹ݑ–\barX_iover¯ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is naturally homeomorphic to Xisubscriptݑ‹ݑ–X_iitalic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, as above we obtain a map

ϕi:ݒ«(S)→ݒ«(Xi),:subscriptitalic-ϕݑ–→ݒ«ݑ†ݒ«subscriptݑ‹ݑ–\phi_i:\mathcalP(S)\to\mathcalP(X_i),italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : caligraphic_P ( italic_S ) → caligraphic_P ( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ,
for all i=1,…,nݑ–1…ݑ›i=1,\ldots,nitalic_i = 1 , … , italic_n. We now define a map

ϕC:ݒ«(S)→ݒ«C(S):subscriptitalic-ϕݐ¶→ݒ«ݑ†subscriptݒ«ݐ¶ݑ†\phi_C:\mathcalP(S)\to\mathcalP_C(S)italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT : caligraphic_P ( italic_S ) → caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S )
by setting

ϕC(P)=ϕ1(P)∪…∪ϕn(P)∪C.subscriptitalic-ϕݐ¶ݑƒsubscriptitalic-ϕ1ݑƒ…subscriptitalic-ϕݑ›ݑƒݐ¶\phi_C(P)=\phi_1(P)\cup\ldots\cup\phi_n(P)\cup C.italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_P ) = italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_P ) ∪ … ∪ italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_P ) ∪ italic_C .
Abusing notation, observe that we also obtain maps

ϕC:ݒŸݒ«(S)→ݒŸݒ«C(S) and ϕC:ݒžݒ«(S)→ݒžݒ«C(S).:subscriptitalic-ϕݐ¶→ݒŸݒ«ݑ†ݒŸsubscriptݒ«ݐ¶ݑ† and subscriptitalic-ϕݐ¶:→ݒžݒ«ݑ†ݒžsubscriptݒ«ݐ¶ݑ†\phi_C:\mathcalDP(S)\to\mathcalDP_C(S)\hskip 14.22636pt\text and % \hskip 14.22636pt\phi_C:\mathcalCP(S)\to\mathcalCP_C(S).italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT : caligraphic_D caligraphic_P ( italic_S ) → caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) and italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT : caligraphic_C caligraphic_P ( italic_S ) → caligraphic_C caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) .
The following is an immediate consequence of the definitions and Lemma 3.1:

Let C⊂Sݐ¶ݑ†C\subset Sitalic_C ⊂ italic_S be a multicurve. Then

(1)
If PݑƒPitalic_P is a pants decomposition of Sݑ†Sitalic_S that contains Cݐ¶Citalic_C, then ϕC(P)=Psubscriptitalic-ϕݐ¶ݑƒݑƒ\phi_C(P)=Pitalic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_P ) = italic_P.

(2)
For all P,Q∈ݒŸݒ«(S)ݑƒݑ„ݒŸݒ«ݑ†P,Q\in\mathcalDP(S)italic_P , italic_Q ∈ caligraphic_D caligraphic_P ( italic_S ), dݒŸݒ«(S)(ϕC(P),ϕC(Q))≤dݒŸݒ«(S)(P,Q).subscriptݑ‘ݒŸݒ«ݑ†subscriptitalic-ϕݐ¶ݑƒsubscriptitalic-ϕݐ¶ݑ„subscriptݑ‘ݒŸݒ«ݑ†ݑƒݑ„d_\mathcalDP(S)(\phi_C(P),\phi_C(Q))\leq d_\mathcalDP(S)(P,Q).italic_d start_POSTSUBSCRIPT caligraphic_D caligraphic_P ( italic_S ) end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_P ) , italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_Q ) ) ≤ italic_d start_POSTSUBSCRIPT caligraphic_D caligraphic_P ( italic_S ) end_POSTSUBSCRIPT ( italic_P , italic_Q ) . □□\square□

3.3. Proof of Theorem 1.1

After the above discussion, we are ready to give a proof of our first result:

Proof of Theorem 1.1.

Let PݑƒPitalic_P and Qݑ„Qitalic_Q be vertices of ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ), and let ωݜ”\omegaitalic_ω be a path in ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) between them. By Lemma 3.2, ϕC(ω)subscriptitalic-ϕݐ¶ݜ”\phi_C(\omega)italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ω ) is a path in ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) between ϕC(P)=Psubscriptitalic-ϕݐ¶ݑƒݑƒ\phi_C(P)=Pitalic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_P ) = italic_P and ϕC(Q)=Qsubscriptitalic-ϕݐ¶ݑ„ݑ„\phi_C(Q)=Qitalic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_Q ) = italic_Q, of length at most that of ωݜ”\omegaitalic_ω. Hence ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) is convex. ∎

Remark 3.3 (Strata that are not totally geodesic).

If Sݑ†Sitalic_S has at least 6 punctures, there exist multicurves Cݐ¶Citalic_C in Sݑ†Sitalic_S for which ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) fails to be totally geodesic. For instance, let C=α∪βݐ¶ݛ¼ݛ½C=\alpha\cup\betaitalic_C = italic_α ∪ italic_β, where αݛ¼\alphaitalic_α cuts off a four-holed sphere Z⊂Sݑݑ†Z\subset Sitalic_Z ⊂ italic_S and β⊂Zݛ½ݑ\beta\subset Zitalic_β ⊂ italic_Z. Let P,Q∈ݒŸݒ«C(S)ݑƒݑ„ݒŸsubscriptݒ«ݐ¶ݑ†P,Q\in\mathcalDP_C(S)italic_P , italic_Q ∈ caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) at distance at least 3, and choose a geodesic path

P=P1,P2,…,Pn-1,Pn=Qformulae-sequenceݑƒsubscriptݑƒ1subscriptݑƒ2…subscriptݑƒݑ›1subscriptݑƒݑ›ݑ„P=P_1,P_2,\ldots,P_n-1,P_n=Qitalic_P = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_Q
in ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) between them. For each i=2,…,n-1ݑ–2…ݑ›1i=2,\ldots,n-1italic_i = 2 , … , italic_n - 1, let Pi′superscriptsubscriptݑƒݑ–′P_i^\primeitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be a pants decomposition obtained from Pisubscriptݑƒݑ–P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT by replacing the curve βݛ½\betaitalic_β with a curve β′⊂Zsuperscriptݛ½′ݑ\beta^\prime\subset Zitalic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ italic_Z such that Pisubscriptݑƒݑ–P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and Pi′subscriptsuperscriptݑƒ′ݑ–P^\prime_iitalic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are related by an elementary move for all iݑ–iitalic_i. Observing that PݑƒPitalic_P and P1′subscriptsuperscriptݑƒ′1P^\prime_1italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (resp. Pn-1′subscriptsuperscriptݑƒ′ݑ›1P^\prime_n-1italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT and Qݑ„Qitalic_Q) are related by two disjoint elementary moves, and therefore are adjacent in ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ), we obtain a geodesic path

P=P1,P2′,…,Pn-1′,Pn=Qformulae-sequenceݑƒsubscriptݑƒ1subscriptsuperscriptݑƒ′2…subscriptsuperscriptݑƒ′ݑ›1subscriptݑƒݑ›ݑ„P=P_1,P^\prime_2,\ldots,P^\prime_n-1,P_n=Qitalic_P = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_Q
between PݑƒPitalic_P and Qݑ„Qitalic_Q that lies entirely outside ݒŸݒ«C(S)ݒŸsubscriptݒ«ݐ¶ݑ†\mathcalDP_C(S)caligraphic_D caligraphic_P start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_S ) except at the endpoints. In particular, ݒŸݒ«(S)ݒŸݒ«ݑ†\mathcalDP(S)caligraphic_D caligraphic_P ( italic_S ) is not totally geodesic.

3.4. Proof of Theorem 1.2

Let YݑŒYitalic_Y be a subsurface of Sݑ†Sitalic_S such that YݑŒYitalic_Y has the same genus as Sݑ†Sitalic_S. Each component of ∂YݑŒ\partial Y∂ italic_Y bounds in Sݑ†Sitalic_S a punctured disc. As in Section 3.2, we may define a projection

ϕY:ݒ«(S)→ݒ«(Y),:subscriptitalic-ϕݑŒ→ݒ«ݑ†ݒ«ݑŒ\phi_Y:\mathcalP(S)\to\mathcalP(Y),italic_ϕ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT : caligraphic_P ( italic_S ) → caligraphic_P ( italic_Y ) ,
this time by choosing one puncture of Sݑ†Sitalic_S in each punctured disc bounded by a component of ∂YݑŒ\partial Y∂ italic_Y. Let Cݐ¶Citalic_C be the union of a pants de

Costway US $143.00 Steel Frame Outdoor Loveseat Patio Canopy Swing with Cushion-Blue Costway US $278.00 2 Pieces Patio Loveseat Bench Table Furniture Set with Cushioned Chair-Blue Costway US $484.00 4 Pieces Patio Acacia Wood Thick Cushion Loveseat Sofa Set-White Costway US $431.00 4 Pieces Patio Rattan Conversation Set with Loveseat Sofas and Coffee Table Costway US $348.00 Modern Fabric Loveseat Sofa for with Metal Legs and Armrest Pillows-Army Green Costway US $311.00 Modern Loveseat Sofa Couch with Side Storage Pocket-Light Gray Costway US $184.00 Patio Rocking Bench Double Rocker Chair with Ergonomic Seat 2-Person Loveseat Costway US $348.00 Modern Fabric Loveseat Sofa for with Metal Legs and Armrest Pillows-Peacock Blue Costway US $225.00 2-Person Rocking Loveseat with Mesh Fabric and Tempered Glass Table Costway US $311.00 Modern Loveseat Sofa Couch with Side Storage Pocket and Sponged Padded Seat Cushions-Orange