Higher genus meanders and Masur–Veech volumes

A meander is a pair consisting of a straight line in the plane and of a smooth closed curve transversally intersecting the line, considered up to an isotopy preserving the straight line. The number of meanders with 2 N intersections grows exponentially with N , but asymptotics still remains conjectural. A meander defines a pair of transversally intersecting simple closed curves on a 2 -sphere. In this paper we consider such pairs on a closed oriented surface of arbitrary genus. The number of these higher genus meanders still admits exponential upper and lower bounds as N grows. Fixing the number n of bigons in the complement to the union of the two curves, we compute the precise asymptotics of genus  g meanders with n bigons and with at most 2 N intersections and show that it grows polynomially with N . We obtain a similar result in the case of oriented curves.