Finite volume and fundamental group on manifolds of negative curvature (Q801541)

It is proved that the finiteness of volume can be read off from the fundamental group for a complete Riemannian manifold of bounded negative curvature and dimension \(\geq 3\). Theorem: Let V be a complete Riemannian manifold with dimension \(n\geq 3\) and curvature \(-b^ 2\leq K\leq -a^ 2<0\). Then the volume of V is finite if and only if (1): \(\pi\) (V) contains only finitely many conjugation classes of maximal almost nilpotent subgroups of rank n-1, and (2): if \(\Delta\) is the amalgamated product of \(\pi_ 1(V)\) with itself on these subgroups, then \(H_ n(\Delta,{\mathbb{Z}}_ 2)={\mathbb{Z}}_ 2\).