Counting curves in hyperbolic surfaces (Q311137)

Let \(\Sigma\) be a hyperbolic surface of genus \(g\) with \(r\) punctures; an \textit{immersed multicurve} or just \textit{multicurve} in \(\Sigma\) is an immersed finite collection of closed, non-peripheral curves in \(\Sigma\) representing primitive elements in the fundamental group. Two multicurves are of the same \textit{type} if they belong to the same mapping class orbit (i.e., are isotopic as immersed 1-submanifolds after applying a diffeomorphism of the surface to one of them). It is known that the number of multicurves of a fixed type and of length at most \(L\) grows coarsely like a polynomial of degree \(6g-6 +2r\) (see [\textit{M. Rees}, Ergodic Theory Dyn. Syst. 1, 461--488 (1981; Zbl 0539.58018)], for the case of simple multicurves, and [\textit{J. Sapir}, Geom. Dedicata 184, 1--25 (2016; Zbl 1353.53051)], or Corollary 3.6 of the present paper for the general case). Now ``the perhaps most grappling question'' is whether the limit exists when dividing the number of multicurves of a given type by \(L^{6g-6 +2r}\) and letting \(L\) go to infinity. It is the main result of the present paper that this limit exists if \(\Sigma\) is a once-puntured torus; in other words, the cardinality of the set of isotopy classes of multicurves of a fixed type and of length at most \(L\) is asymptotic to \(L^2\) times a constant. For the case of simple multicurves, \textit{M. Mirzakhani} proved that the limit exists for all \(g\) and \(r\) [Ann. Math. (2) 168, No. 1, 97--125 (2008; Zbl 1177.37036)] and, as the authors note, in a recent preprint Mirzakhani established the existence of the limit also in complete generality (independently and by different methods).