Fixed-point property of random groups

In this paper, using the generalized version of the theory of combinatorial harmonic maps, we give a criterion for a finitely generated group Γ to have the fixed-point property for a certain class of Hadamard spaces, and prove a fixed-point theorem for random-group actions on the same class of Hadamard spaces. We also study the existence of an equivariant energy-minimizing map from a Γ-space to the limit space of a sequence of Hadamard spaces with the isometric actions of a finitely generated group Γ. As an application, we present the existence of a constant which may be regarded as a Kazhdan constant for isometric discrete-group actions on a family of Hadamard spaces.