Coarse equivalences of Euclidean buildings. (With an appendix by Jeroen Schillewaert and Koen Struyve.)
type: Item
instance of: scholarly article
title: Coarse equivalences of Euclidean buildings. (With an appendix by Jeroen Schillewaert and Koen Struyve.)
zbMATH Open document ID: 1327.51015
published in: Advances in Mathematics
publication date: 2014-04-15T00:00:00Z
review text: 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 assumpt…
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6285041
zbMATH Keywords: metric geometry
zbMATH Keywords: quai-isometric rigidity
zbMATH Keywords: coarse equivalence
zbMATH Keywords: Euclidean buildings
zbMATH Keywords: Bruhat-Tits buildings
zbMATH Keywords: trees
zbMATH Keywords: ultraproducts
author: Linus Kramer
author: Richard M. Weiss
reviewed by: Petra Schwer
MaRDI profile type: Publication
full work available at URL: j.aim.2013.10.031
OpenAlex ID: W2962914400
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DOI: 10.1016/J.AIM.2013.10.031
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