Fundamental groups of manifolds of nonpositive curvature (Q914168)

Let M be a manifold of finite volume with sectional curvature nonpositive and bounded below. The authors study the relation between the geometric structure of M and the algebraic structure of its fundamental group \(\Gamma\). This relation is exemplified by the following theorem: Either M is flat or \(\Gamma\) contains a nonabelian free subgroup. With respect to the notion of the rank of M defined in [the authors and \textit{M. Brin}, Ann. Math., II. Ser. 122, 171-203 (1985; Zbl 0589.53047)], the authors prove that the rank of M equals the rank of \(\Gamma\) first studied by \textit{G. Prasad} and \textit{M. S. Raghunathan} [ibid. 96, 296-317 (1972; Zbl 0245.22013)]. The authors characterize the irreducible locally symmetric spaces of noncompact type and rank at least 2 in terms of algebraic data in \(\Gamma\). As a consequence, if \(\Gamma\) is isomorphic to the fundamental group of such a locally symmetric space \(M^*\), then M and \(M^*\) are isometric up to scaling.