Rigidity at infinity of trees and Euclidean buildings (Q265488)

\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)] and \textit{L. Kramer} and \textit{R. M. Weiss} [Adv. Math. 253, 1--49 (2014; Zbl 1327.51015)] obtained rigidity results for Euclidean buildings and related spaces. The paper under review is based on and extends their work. The author's main result is that a locally finite leafless \(\mathbb{R}\)-tree \(T\) with at least three ends and which admits a group \(G\) of isometries acting 2-transitively on the ends of \(T\) can be reconstructed, up to a scaling factor, from the boundary at infinity of \(T\) and the group action. The main step of the proof is a study of the set of branch points of \(T\) and the recognition of the bounded subgroups by the action on \(\partial_\infty T\). From then on, the author is able to follow the same arguments used by Kramer and Weiss [loc. cit.]. As a corollary it is then shown that an isometry \(f :\partial_\infty X_1\to \partial_\infty X_2\) between the buildings at infinity of two locally-finite irreducible Euclidean buildings \(X_1\) and \(X_2\) of dimension at least two extends (after rescaling the metric on \(X_2\)) to an isometry \(\tilde f :X_1\to X_2\). Furthermore, in case that \(X_1\) is not a Euclidean cone this isometry is unique. This result is achieved by considering the projectivity group associated to a panel \(a\) of \(\partial_\infty X_1\) and its action on the ends of the panel tree \(X_1(a)\).