 
Summary: Constructing convex planes in the pants complex.
Javier Aramayona, Hugo Parlier, Kenneth J. Shackleton
[First draft: February 2007]
ABSTRACT: Our main theorem identifies a class of totally geodesic subgraphs
of the 1skeleton of the pants complex, each isomorphic to the product of two
Farey graphs. We deduce the existence of many convex planes in the 1skeleton
of the pants complex.
KEYWORDS: pants complex; WeilPetersson metric
2000 MSC: 57M50 (primary); 05C12 (secondary)
§1. Introduction.
Let be a compact, connected and orientable surface, possibly with nonempty
boundary, of genus g() and  boundary components, and refer to as the
mapping class group Map() the group of all selfhomeomorphisms of up to
homotopy.
After HatcherThurston [HT], to the surface one may associate a simpli
cial graph P(), the pants graph, whose vertices are all the pants decompositions
of and any two vertices are connected by an edge if and only if they differ
by an elementary move; see §2.2 for an expanded definition. This graph is con
nected, and one may define a pathmetric d on P() by first assigning length 1
to each edge and then regarding the result as a length space.
