Counting Locally Symmetric Manifolds

We give quantitive estimates for the number of locally symmetric spaces of a given type with bounded volume. Explicitly, let S be a symmetric space of non-compact type without Euclidean de Rham factors. Then, after rescaling appropriately the Riemannian metric, the following hold. Theorem A If rank(S) = 1 and S ≇ H2,H3, then there are at most V V Riemannian manifolds, locally isometric to S, with total volume ≤ V . Theorem B If rank(S) > 1 and S ≇ H2 × H2, then there are at most V V regular Riemannian manifolds, locally isometric to S, with volume ≤ V . Theorem C If S = (H2)a × (H3)b where (a, b) 6= (1, 0), (0, 1), (2, 0), then V V bounds the number of all irreducible S-manifolds with volume ≤ V .