COMPACT LIE GROUP ACTIONS ON CLOSED MANIFOLDS OF NON-POSITIVE CURVATURE

Abstract

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 Out π1(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 → Out π1(N) is a monomorphism. The same result holds for the non-orientable N's under certain technical assumptions. Our result is an application of a theorem by Schoen–Yau [12] on harmonic mappings.