Combinatorial harmonic maps and discrete-group actions on Hadamard spaces.

*(English)*Zbl 1108.58014The 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)].

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)].

Reviewer: Andreas Gastel (Erlangen)

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\textit{H. Izeki} and \textit{S. Nayatani}, Geom. Dedicata 114, 147--188 (2005; Zbl 1108.58014)

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##### References:

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[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 |

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