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IntroductionLet M be a smooth manifold of dimension n endowed with a conformal structure C , by which we mean a conformal class of Riemannian metrics on M . Throughout this paper, we assume that the dimension n is not less than three unless otherwise stated. We say that C is flat if, for each point p ∈ M , we can find local coordinates defined in a neighborhood U of p so that some (hence any) metric g ∈ C has an expression of the form λ(x ) n i =1 (dx i ) 2 on U for some positive function λ. In this case, M is called a conformally flat manifold, and each metric in C is called a conformally flat metric. In dimensions not less than four, such a metric is characterized as one with vanishing Weyl curvature. A conformally flat manifold typically arises as the quotient of a domain Ω of the sphere S n by a Kleinian group Γ . Such a conformally flat manifold is called a Kleinian manifold. Schoen-Yau [16] showed that an extensive class of conformally flat manifolds are uniformized, that is, realized as Kleinian manifolds. This class contains any conformally flat manifold which admits a compatible complete metric with nonnegative scalar curvature.In this paper, we construct a canonical conformally flat metric g on a given Kleinian manifold M = Ω/Γ which is compatible with the conformal structure. This is carried out in Sect. 2 and the properties of the constructed metric are studied in the later sections. A major ingredient in our construction is the measure supported on the limit set of Γ which was first introduced by Patterson [13] for Fuchsian groups and subsequently generalized by Sullivan [17] to Kleinian groups in any dimension. It should be mentioned that the metric we present here is different in general from the C 1,1 metric constructed by Apanasov [3].In Sect. 3 we compute the curvature of the metric g. It turns out that the curvature well reflects the size (= critical exponent) of Γ , which coincides with Partly supported by the Grant-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan. 116S. Nayatani the Hausdorff dimension of the limit set for geometrically finite Γ . In particular, supposing that the limit set consists of more than one point, it follows that if δ(Γ ) < (resp. =, >) (n − 2)/2, the scalar curvature of g is positive (resp. zero, negative) everywhere, where δ(Γ ) is the critical exponent of Γ . Using this, we can slightly refine a result of Schoen-Yau [16]. It also follows that if δ(Γ ) > n −2, then the Ricci curvature of g is negative.In Sect. 4 we study the symmetries of the metric g. Under certain assumptions, we prove that the isometry group of g coincides with the conformal transformation group of M . These assumptions are satisfied if M is compact and Γ is geometrically finite.In Sect. 5 we give a topological application. We compute the curvature term of the Weitzenböck formula for the Hodge-de Rham Laplacian defined with respect to our metric. It follows by the classical Bochner-Weitzenböck argument that if M = Ω/Γ is a compact Kleinian manif...

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.