Compact Lie Group Actions on Closed Manifolds of Non-positive Curvature

A. Borel proved that, if a finite group F acts effectively and continuously on a closed aspherical manifold M with centerless fundamental group π1(M), then a natural homomorphism ψ from F to the outer automorphism group Outπ1(M) of π1(M), called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel’s theorem holds for a compact Lie group G acting effectively and smoothly on a particular orientable aspherical manifold N admitting a Riemannian metric g0 of non-positive curvature in case that π1(N) has a non-trivial center. It turns out that if G attains the maximal dimension equal to the rank of Center π1(N) and the metric g0 is real analytic, then any element of G defining a diffemorphism homotopic to the identity of N must be contained in the identity component G0 of G. Moreover, if the inner automorphism group of π1(N) is torsion free, then the associated abstract kernel ψ : G/G0 → Outπ1(N) is a monomorphism. The same result holds for the nonorientable N’s under certain techical assumptions. Our result is an application of a theorem by Schoen-Yau (Topology, 18 (1979), 361-380) on harmonic mappings. Mathematics Subject Classifications (2000): Primary: 57S25, 53C43; Secondary: 20F34.