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Applications harmoniques entre graphes finis et un théorème de superrigidité. (Harmonic maps between finite graphs and a theorem of superrigidity.). (French) Zbl 0861.05032
Summary: We define the energy of a map between two finite metric graphs, and study the problem of minimizing the energy in a homotopy class. In this context, we prove analogues of Eells-Sampson’s and Hartman’s theorems concerning existence and uniqueness of harmonic maps into nonpositively curved manifolds. We also show that energy minimizing maps behave well under finite covering of the source. As an application of these results, we give a new (elementary) proof of a theorem of superrigidity for commensurability groups of tree lattices.

MSC:
05C30 Enumeration in graph theory
20E08 Groups acting on trees
58E20 Harmonic maps, etc.
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