The Hirzebruch-Mumford volume for the orthogonal group and applications

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an application of Hirzebruch-Mumford proportionality allows us to determine the leading term of the growth of the dimension of the spaces S_k(\Gamma) of cusp forms of weight k, as k goes to infinity. We compute this in a number of examples, which are important for geometric applications.