From veering triangulations to link spaces and back again

This paper is the third in a sequence establishing a dictionary between the combinatorics of veering triangulations equipped with appropriate filling slopes, and the dynamics of pseudo-Anosov flows (without perfect fits) on closed three-manifolds. Our motivation comes from the work of Agol and Gu\'eritaud. Agol introduced veering triangulations of mapping tori as a tool for understanding the surgery parents of pseudo-Anosov mapping tori. Gu\'eritaud gave a new construction of veering triangulations of mapping tori using the orbit spaces of their suspension flows. Generalising this, Agol and Gu\'eritaud announced a method that, given a closed manifold with a pseudo-Anosov flow (without perfect fits), produces a veering triangulation equipped with filling slopes. In this paper we build, from a veering triangulation, a canonical circular order on the cusps of the universal cover. Using this we build the veering circle and the link space. These are the first entries in the promised dictionary. The link space and the circle are, respectively, analogous to the orbit space of a flow and to Fenley's boundary at infinity of the orbit space. In the other direction, and using our previous work, we prove that the veering triangulation is recovered (up to canonical isomorphism) from the dynamics of the fundamental group acting on the link space. This is the first step in proving that our dictionary gives a bijection between the two theories.