Manifolds of negative curvature

1.1. For a Riemannian manifold V we denote by c(V) and c~(V) respectively the upper and the lower bounds of the sectional curvature, by vol(F) the volume, and by d(V) the diameter. 1.2. Let V be an ^-dimensional closed Riemannian manifold of negative curvature and c~(V) > 1 . If n > 8, then vol (V) > C(l + d{V)\ where the constant C > 0 depends only on n. Remark. This inequality is exact: For each n there exists an infinite sequence Vi with d(Vi) —> oo, / —> oo, and with uniformly bounded ratio vol {Vi)jd{Vi). Proof. Take a manifold V of constant negative curvature with infinite group Hλ(V) (see [8]) and a sequence of its finite cyclic coverings. For n = 4, 5, 6, 7 we shall prove here the following weaker result: vol (V) > C(l + d(V)). Notice that arguments from § 4 show that for n > 4 an ndimensional manifold V with — ε > c(V) > c~(V) > — 1, ε > 0, satisfies: vol (V) > C(l + d{V)) where C depends on n and ε. 1.3. Theorem 1.2 sharpens the Margulis-Heintze theorem (see [6], [4]) stating the inequality vol (V) > C = Cn. In this paper we prove the following generalization. 1.3A. Let X be a complete simply connected manifold of negative curvature with c~(X) > — 1. Let Γ be a discrete group (possibly with torsion) of isometries of V. Then vol(X/Γ) > C, where C > 0 depends only on dim(X). This fact is still true for manifolds of nonpositive curvature with c~(X) > — 1 and negative Ricci curvature (see [5]). In the homogeneous case this is the Kazhdan-Margulis theorem (see [9]).