The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

Abstract

Abstract. For an isometric action of a finitely generated group on the ultralimit 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. Furthermore, 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.

Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors

“…Proof of Theorem 1.2. Since F α,r is continuous and convex, inf v∈B |∇ − F α,r |(v) = 0 by Lemma 5.4 in [12]. Hence, if condition (ii) holds, there exists x 0 ∈ N with F α,r (x 0 ) = 0.…”