×

zbMATH — the first resource for mathematics

Combinatorial harmonic maps and discrete-group actions on Hadamard spaces. (English) Zbl 1108.58014
The paper continues studies by M.-T. Wang [J. Differ. Geom. 50, 249–267 (1998; Zbl 0951.58020), Commun. Anal. Geom. 8, 545–563 (2000; Zbl 0977.58018)] on applications of harmonic maps (from a simplicial complex to a Hadamard space) to superrigidity.
A main subject of the paper is to study isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex, and acting also isometrically on a Hadamard space, the authors find criteria for the latter action to have a global fixed point. In the criterion, Wang’s constant plays a role, which is derived from Poincaré-type constants for maps from the link of a vertex in the domain to a tangent cone of the target.
Firstly, one of Wang’s results is generalized to the effect that both nonlocally compact and/or singular target Hadamard spaces may be considered. Then, by estimating Wang’s constant in special cases, more concrete criteria for the existence of fixed points are obtained, involving estimates on the first nonzero eigenvalues of Laplacians. Crucial ingredients of the proof are suitable definitions of equivariant harmonic maps and criteria under which they are constant, thus providing a deformation of any equivariant map to a constant one by the heat flow.
The authors inform us that closely related results can also be found in [M. Gromov, Geom. Funct. Anal. 13, 73–146 (2003; Zbl 1122.20021)].

MSC:
58E20 Harmonic maps, etc.
22E40 Discrete subgroups of Lie groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[13] de la Harpe, P. and Valette, A.: La propriété (T) de Kazhdan pour les groupes localement compactes, Astérisque 175 (1989). · Zbl 0759.22001
[14] Izeki, H. and Nayatani, S.: Combinatorial harmonic maps and superrigidity (Japanese), In: Hyperbolic Spaces and Discrete Groups II (Kyoto, 2001), RIMS Kokyuroku 1270 (2002), 182–194.
[19] Jost, J.: Nonlinear Dirichlet forms, In New Directions in Dirichlet Forms, Stud. Adv. Math. Amer. Math. Soc., Providence, 1998, pp. 1–48. · Zbl 0914.31006
[30] Sarnak, P.: Some Applications of Modular Forms, Cambridge Tracts in Math. 99, Cambridge Univ. Press, 1990. · Zbl 0721.11015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.