Combinatorial Harmonic Maps and Discrete-group Actions on Hadamard Spaces

A harmonic map between two Riemannian manifolds is defined to be a critical point of the energy functional; the energy of a smooth map is by definition the integration of the squared norm of its differential over the source manifold. By solving the heat equation associated with the energy functional, Eells and Sampson [7] proved that if two manifolds are compact and the target has nonpositive sectional curvature, any smooth map can be deformed to a harmonic map. This existence result was later extended to the equivariant setting [4, 6, 20, 24]. Application of the harmonic map theory to the superrigidity of lattices started with the work of Corlette [5], in which it was proved that superrigidity over archimedian fields holds for lattices in Sp(n, 1) (n ≥ 2) and F 4 . By developing the relevant harmonic map theory, Gromov and Schoen [12] carried the preceding result over to the p-adic case, and thereby established the arithmeticity of such lattices. These works were followed by those of Mok-Siu-Yeung [28] and Jost-Yau [21], which established a result including Corlette’s one mentioned above as well as the cocompact case of the superrigidity theorem of Margulis [26] concerning lattices in real semisimple Lie groups of rank ≥ 2. Wang [31, 32] has taken an important step in developing an analogy of the story described hitherto when the source and target spaces are respectively a simplicial complex and a Hadamard space, that is, a complete CAT(0) space; this is the setting necessary for the geometric-variational approach to the Margulis superrigidity for lattices in semisimple algebraic groups over p-adic fields. Wang formulated the notion of energy of an equivariant map between the spaces of the above sort, and established the existence of an energy-minimizing equivariant map assuming that the target space is locally compact and the action is reductive. He then gave an application to the isometric action of a discrete group, arising as the covering transformation group of a finite simplicial complex, on a Hadamard space. Under the same assumptions as above, he formulated a general criterion for the existence of a fixed point in terms of a Poincaré-type constant for maps from the link of a vertex of the source to a tangent cone of the target. In this paper we shall refer to this constant as Wang’s invariant. He also proved a more concrete fixed-point theorem when the target space was a Hadamard manifold. It should be mentioned that there are also various forms of harmonic map theories with singular source spaces [16, 17, 23, 25]. The purpose of the present paper is to push forward with Wang’s work, especially when the target Hadamard space is non-locally compact and/or singular. We shall