Almost equivalence for transitive Anosov flows

The talk is based on a joint preprint [DS] with M. Shannon.
Two flows are almost-equivalent if one can go from one to the other by a finite number of Dehn surgeries on periodic orbits. Examples of almost equivalence go back to Fried who showed that any transitive Anosov flow is almost-equivalent to the suspension of a pseudo-Anosov homeomorphism [F], and even to Birkhoff whose construction [B] was popularized by Fried and implies that every geodesic flow on a hyperbolic surface is almost equivalent to some suspension of an Anosov map of the 2-torus.
An open question of Ghys asks whether all transitive Anosov flows in dimension 3 are pairwise almost-equivalent. Using so-called Birkhoff sections and a result of Minakawa, we show that the answer is positive for suspension of automorphisms of the torus and for geodesic flows on hyperbolic orbifolds.
[B] G. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), 199–300.
[DS] P. Dehornoy, M. Shannon, Almost equivalence for algebraic Anosov flows, arXiv 1910.08457
[F] D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology 22 (1983), 299–303.