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.