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In this paper, using the generalized version of the theory of combinatorial harmonic maps, we give a criterion for a finitely generated group to have the fixed-point property for a certain class of Hadamard spaces, and prove a fixed-point theorem for randomgroup actions on the same class of Hadamard spaces. We also study the existence of an equivariant energy-minimizing map from a -space to the limit space of a sequence of Hadamard spaces with the isometric actions of a finitely generated group . 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.

Abstract. We prove that a random group of the graph model associated with a sequence of expanders has fixed-point property for a certain class of CAT (0) spaces. We use Gromov's criterion for fixed-point property in terms of the growth of n-step energy of equivariant maps from a finitely generated group into a CAT(0) space, to which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL(m, Q r ), and deduce from the general result above that the same random group has fixed-point property for all of these Euclidean buildings with m bounded from above.

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