Convexity of strata in diagonal pants graphs of surfaces
  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. 
Javier Aramayona, Cyril Lecuire, Hugo Parlier, Kenneth J. Shackleton
math.GT