論文

The purpose of this article is to present sufficient conditions for an isometric action of a finitely generated group to have a fixed point in terms of an energy of equivariant maps (Proposition 2.1 and Theorem 3.1), which turn out to be useful in proving that groups with fixed-point property form a large family in the set of finitely presented groups. Most of the results are included in [7] and [8].

In this paper, using the generalized version of the theory of combinatorial harmonic maps, we give a criterion for a finitely generated group Gamma 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 Gamma-space to the limit space of a sequence of Hadamard spaces with the isometric actions of a finitely generated group Gamma. 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.