 
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 WeilPetersson completion.
As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants
graph.
1. Introduction
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 [4] which
asserts that P(S) is quasiisometric 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 [12] and Chu [7], the space T (S) is not complete. Masur [8] proved that its
completion ^T (S) is homeomorphic to the augmented Teichm¨uller space of S, obtained from T (S)
by extending FenchelNielsen 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
