Equivariant Euler characteristics of M_{g, n}

Let M_{g, n} be the moduli space of n-pointed stable genus g curves, and let M_{g, n} be the moduli space of n-pointed smooth curves of genus g. In this paper, we obtain an asymptotic expansion for the characteristic of the free modular operad MV generated by a stable S-module V, allowing to effectively compute S_n-equivariant Euler characteristics of M_{g, n} in terms of S_n'-equivariant Euler characteristics of M_{g', n'} with 0 ≤ g' ≤ g and max[0; 3-2g'] ≤ n' ≤ 2(g-g') + n. This answers a question posed by Getzler and Kapranov by making their integral representation of the characteristic of the modular operad MV effective. To illustrate how the asymptotic expansion is used, we give formulas expressing the generating series of the S_n-equivariant Euler characteristics of M_{g, n}, for g = 0; 1 and 2, in terms of the corresponding generating series associated with M_{g, n}.