Orderability and the Weinstein conjecture

In this article we prove that the Weinstein conjecture holds for contact manifolds (sigma;, ε) for which Cont₀ (sigma;, ε) is non-orderable in the sense of Eliashberg and Polterovich [Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal. 10 (2000), 1448-1476]. More precisely, we establish a link between orderable and hypertight contact manifolds. In addition, we prove for certain contact manifolds a conjecture by Sandon [A Morse estimate for translated points of contactomorphisms of spheres and projective spaces, Geom. Dedicata 165 (2013), 95-110] on the existence of translated points in the non-degenerate case.