THE ENERGY OF EQUIVARIANT MAPS AND A FIXED-POINT PROPERTY FOR BUSEMANN NONPOSITIVE CURVATURE SPACES

摘要

For an isometric action of a finitely generated group on the ultra-limit of a sequence of global Busemann nonpositive curvature spaces, we state a sufficient condition for the existence of a fixed point of the action in terms of the energy of equivariant maps from the group into the space. Further-more, we show that this energy condition holds for every isometric action of a finitely generated group on any global Busemann nonpositive curvature space in a family which is stable under ultralimit, whenever each of these actions has a fixed point. We also discuss the existence of a fixed point of affine isometric actions of a finitely generated group on a uniformly convex, uniformly smooth Banach space in terms of the energy of equivariant maps.