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 1-skeleton of the pants complex, each isomorphic to the product of two
Farey graphs. We deduce the existence of many convex planes in the 1-skeleton
of the pants complex.
KEYWORDS: pants complex; Weil-Petersson metric
2000 MSC: 57M50 (primary); 05C12 (secondary)
Let be a compact, connected and orientable surface, possibly with non-empty
boundary, of genus g() and || boundary components, and refer to as the
mapping class group Map() the group of all self-homeomorphisms of up to
After Hatcher-Thurston [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 path-metric d on P() by first assigning length 1
to each edge and then regarding the result as a length space.