Summary: Totally geodesic subgraphs of the pants complex.
Javier Aramayona, Hugo Parlier, Kenneth J. Shackleton
[First draft: August 2006; Revised: July 2007]
ABSTRACT: Our main theorem asserts that every Farey graph embedded in
the 1-skeleton of the pants complex of any finite type surface is totally geodesic.
KEYWORDS: pants complex; Weil-Petersson metric; Farey graph
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.
The pants graph, with its own geometry, is a fundamental object to study.
Brock [B] revealed deep connections with hyperbolic 3-manifolds and proved the