Coarse equivalences of Euclidean buildings. (With an appendix by Jeroen Schillewaert and Koen Struyve.)
The main results of this paper can be summarized as follows: Coarse equivalences of products of (possibly non-discrete) Euclidean buildings induce combinatorial isomorphisms between the spherical buildings at infinity. If one supposes in addition that none of the factors is a tree, i.e. all irreducible factors have dimension at least two, then coarsely equivalent Euclidean buildings are isometric (up to scaling factors). Suppose further that none of the irreducible factors is a Euclidean cone over its boundary, then the isometry is unique and has finite distance from the coarse equivalence. This paper offers an alternative approach to (the weaker) rigidity results which were earlier proved by Kleiner and Leeb as Theorem 1.1.3 in [\textit{B. Kleiner} and \textit{B. Leeb}, Publ. Math., Inst. Hautes Étud. Sci. 86, 115--197 (1997; Zbl 0910.53035)] and by Leeb as Theorem 1.3 in [\textit{B. Leeb}, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry. Bonn: Univ. Bonn, Mathematisch-Naturwissenschaftliche Fakultät (2000; Zbl 1005.53031)] under the additional hypothesis that either the Euclidean buildings are locally finite and simplicial or their buildings at infinity satisfy the Moufang condition. One of the main tools is Morse's Lemma and a higher dimensional version of it which essentially says that for a coarse equivalence f between buildings X and Y and every apartment A in X the image of A under f is at uniformly bounded Hausdorff distance from an apartment in Y. Further the authors make use of Tits' rigidity result about tree-preserving (ecological) boundary isomorphisms of Euclidean buildings. Compare Theorem 2 in [\textit{J. Tits}, Lect. Notes Math. 1181, 159--190 (1986; Zbl 0611.20026)]. Finally, in higher rank the fact that ultralimits of Euclidean buildings are metrically complete Euclidean buildings is used in the proofs. In an appendix, Jeroen Schillewaert and Koen Struyve prove that the assumption that X1 and X2 are metrically complete, which is made in the main text in several places, is in fact not necessary.
scientific article