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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.

05C30 Enumeration in graph theory
20E08 Groups acting on trees
58E20 Harmonic maps, etc.
Full Text: DOI Numdam EuDML
[1] H. BASS, Covering theory for graphs of groups, J. Pure Appl. Alg., 89 (1993), 3-47. · Zbl 0805.57001
[2] M. BRIDSON, A. HAEFLIGER, Metric spaces of nonpositive curvature, livre en préparation. · Zbl 0988.53001
[3] M. BURGER, S. MOZES, CAT(-1) spaces, divergence groups and their commensurators, J. Am. Math. Soc., vol. 9, 1 (1996). · Zbl 0847.22004
[4] H. BASS, R. KULKARNI, Uniform tree lattices, J. Amer. Math. Soc., 3 (1990), 843-902. · Zbl 0734.05052
[5] K. CORLETTE, Archimedian superrigidity and hyperbolic geometry, Ann. of Math., 135 (1992), 165-182. · Zbl 0768.53025
[6] J. EELLS, J. SAMPSON, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 85 (1964), 109-160. · Zbl 0122.40102
[7] Y. GAO, Superrigidity for homomorphisms into isometry groups of non-proper CAT(-1) spaces, prépublication. · Zbl 0888.22008
[8] É. GHYS, P. DE LA HARPE, Sur LES groupes hyperboliques d’après mikhael Gromov, Birkhäuser, Boston, 1990. · Zbl 0731.20025
[9] M. GROMOV, P. PANSU, Rigidity of lattices : an introduction, in geometric topology : recent developments, Montecatini Terme, 1990, Lecture Note 1504, 39-137. · Zbl 0786.22015
[10] M. GROMOV, R. SCHOEN, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math. IHES, 76 (1992), 165-246. · Zbl 0896.58024
[11] P. HARTMAN, On homotopic harmonic maps, Can. J. Math., 19 (1967), 673-687. · Zbl 0148.42404
[12] J. JOST, Riemmannian geometry and geometric analysis, Universitext, Springer-Verlag, 1995.
[13] N.J. KOREVAAR, R. SCHOEN, Sobolev spaces and harmonic maps for metric space targets, Comm. in Anal. and Geom., vol. 1, 4 (1993), 561-659. · Zbl 0862.58004
[14] A. LUBOTZKY, S. MOZES, R.J. ZIMMER, Superrigidity of the commensurability groups of tree lattices, Comm. Math. Helv., 69 (1994), 523-548. · Zbl 0839.22011
[15] Y. LIU, Density of the commensurability groups of uniform tree lattices, J. of Alg., 165 (1994), 346-359. · Zbl 0813.20024
[16] J.E. LOS, A variational calculus for automorphisms of free groups, prépublication Univ. Nice, janvier 1993.
[17] G.A. MARGULIS, Superrigidity for commensurability subgroups and generalized harmonic maps, prépublication.
[18] P. PANSU, Sous-groupes discrets des groupes de Lie : rigidité, arithméticité, Sém. Bourbaki, 46-ème année, 1993-1994, n° 778, Soc. Math. France, Astérisque, 227 (1995), 69-105. · Zbl 0835.22011
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