The pants complex has only one end

Summary

Definitions and statement of the main theorem

The purpose of this note is to prove the following theorem:

Theorem 4.1.Let S be a closed, connected, orientable surface with genus g(S) ≥ 3. Then the pants complex of S has only one end. In fact, there are constants K = K(S) and M = M(S) so that: if R > M, and P and Q are pants decompositions at distance greater than KR from a basepoint, then P and Q may be connected by a path which remains at least distance R from the basepoint.

A pants decomposition of S consists of 3g(S)—3 disjoint essential non-parallel simple closed curves on S. Each component of the complement of the curves is a three-holed sphere; a pants. Then the pants complex P(S) is the metric graph whose vertices are pants decompositions of S, up to isotopy. Two vertices P, P′ are connected by an edge of length one if P, P′ differ by an elementary move. In an elementary move all curves of the pants are fixed except for one curve α. Remove α and let V be the component of the complement of the remaining curves which is not a pants. Then V contains α and is either a once-holed torus or a four-holed sphere. Now α is replaced by any curve β contained in V that intersects α minimally; in the torus case once, and in the sphere case twice.