The Vanishing Theorem for `1-cohomology in Negative Curvature

If G is the fundamental group of a closed Riemannian manifold of negative curvature, then H n (1) (G; V) = 0 for all n 2 and for all Banach spaces V. This implies that the linear isoperimetric inequality is satissed for llings of higher dimensional real cycles. x1. Introduction. Hyperbolic groups in the sense of Gromov Gr] are known to be characterized as nitely presented groups G which satisfy the linear isoperimetric inequality for llings of real 1-cycles on the universal cover X of a K(G; 1)-complex X 0 with a nite 2-skeleton Ge1]. Since it is known that such an X 0 can be chosen for a hyperbolic group G with nite n-skeleton for all n, it is natural to ask whether the linear isoperimetric inequality is satissed for llings of n-cycles for all n 2. That this is true for n = 2 follows from results of A-B], but it is an open question for all n 3. It is shown in Ge2] Theorem 6.1 that for hyperbolic G the linear isoperimetric inequality for llings of n-cycles for some n 2 has a cohomological interpretation; namely, it is equivalent to the vanishing of the cohomology group H n (1) (G; ` 1). In this note we shall show that if G is the fundamental group of a closed Rie-mannian manifold of strictly negative curvature, then H n (1) (G; V) = 0 for all Ba-nach spaces V and all n 2. This is on the face of it stronger than the linear isoperimetric inequality for higher dimensional real cycles, by the last result quoted in the preceding paragraph. Since it is known from Ge1] that hyperbolicity of the nitely presented group G is equivalent to the vanishing of H 2 (1) (G; ` 1), it suggests that there may possibly be a diierence between \negatively curved groups", which act properly discontinuously and cocompactly by isometries on spaces of negative curvature, and hyperbolic groups, and that the groups H 2 (1) (G; V) are obstructions to hyperbolic groups being \negatively curved". It would be a remarkable fact of considerable depth if all these obstructions vanished. There are various ways to express the condition H n (1) (G; V) = 0 for all Banach spaces V and some integer n. One such is that the canonical inclusion B n?1 (X; R) C n?1 (X; R) have a bounded linear …