Summary: CONVEXITY OF STRATA IN DIAGONAL PANTS GRAPHS OF SURFACES
J. ARAMAYONA, C. LECUIRE, H. PARLIER & K. J. SHACKLETON
Abstract. We prove a number of convexity results for strata of the diagonal pants graph of a sur-
face, 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
Let S be a connected orientable surface, with empty boundary and negative Euler characteristic.
The pants graph P(S) is the graph whose vertices correspond to homotopy classes of pants decom-
positions of S, and where two vertices are adjacent if they are related by an elementary move; see
Section 2 for an expanded definition. The graph P(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 P(S) stems from the result of Brock  which
asserts that P(S) is quasi-isometric to T (S), the Teichm¨uller space of S equipped with the Weil-
Petersson metric. As such, P(S) (or any of its relatives also discussed in this article) is a combina-
torial model for Teichm¨uller space.
By results of Wolpert  and Chu , the space T (S) is not complete. Masur  proved that its
completion ^T (S) is homeomorphic to the augmented Teichm¨uller space of S, obtained from T (S)
by extending Fenchel-Nielsen coordinates to admit zero lengths. The completion ^T (S) admits a
natural stratified structure: each stratum TC(S) ^T (S) corresponds to a multicurve C S, and