Remark on ordered braid groups

Abstract

We recover the Dehornoy order on the braid group [Formula: see text] from the tracial state on a cluster [Formula: see text]-algebra [Formula: see text] associated to the surface [Formula: see text] of genus [Formula: see text] with [Formula: see text] boundary components. It is proved that the space of left-ordering of the fundamental group [Formula: see text] is a totally disconnected dense subspace of the projective Teichmüller space [Formula: see text]. In particular, each left-ordering of [Formula: see text] defines the orbit of a Riemann surface [Formula: see text] under the geodesic flow on the space [Formula: see text].