A note on invariants of flows induced by Abelian differentials on Riemann surfaces

The real and imaginary part of any Abelian differential on a compact Riemann surface define two flows on the underlying compact orientable $C^{\mathrm{\infty }}$ surface. Furthermore, these flows induce an interval exchange transformation on every transversal simple closed curve, via Poincar'e recurrence. This note shows that the ordered $K_0$ groups of several $C^{\ast }$ algebras naturally associated to one of the flows resp. interval exchange transformations are isomorphic, mainly using methods of I. Putnam.