Almost equivalence for Anosov flows: from genus 2 to genus 1

From a topological perspective, it is natural to classify Anosov flows up to orbit equivalence. While the number of such equivalence classes is infinite, it remains countable. A more robust relation is provided by almost equivalence, obtained by quotienting out by Goodman-Fried surgery. Fried, Christie, and Ghys investigated the strength of this relation, specifically questioning whether all flows with orientable invariant foliations are equivalent—which would provide an analog of Lickorish-Wallace theorem for Anosov flows. We will present several results pointing toward a positive answer, and in particular a recent result claiming that if an Anosov flow admits a genus two Birkhoff section whose first-return has only only singularity, then it admits a genus one Birkhoff section.