Metric Properties of Euclidean Buildings

Abstract

This is a survey on nondiscrete euclidean buildings, with a focus on metric properties of these spaces.Euclidean buildings are higher dimensional generalizations of trees. Indeed, the euclidean product X of two (leafless) metric trees T 1 , T 2 is already a good "toy example" of a 2-dimensional euclidean building. The space X contains lots of copies of the euclidean plane E 2 and has at the same time a complicated local branching.Euclidean building were invented by Jacques Tits in the seventies. Similarly as in the case of spherical buildings, their definition was motivated by group theoretic questions. While spherical buildings are by now a standard tool in the structure theory of reductive algebraic groups over arbitrary fields, euclidean buildings are important for the advanced structure theory of reductive groups over fields with valuations. In particular, they are very much linked to number theory and arithmetic geometry.In the last 25 years, however, euclidean buildings have also become important in geometry. This is due to the fact that euclidean buildings are spaces of nonpositive curvature. But more is true. Together with the Riemannian symmetric spaces of nonpositive curvature, euclidean buildings could be called the islands of high symmetry in the world of CAT(0) spaces. This claim will be made more precise below. Almost inevitably, questions about symmetry, rigidity, or higher rank for CAT(0) spaces lead to these geometries.I thank Alexander Lytchak, Koen Struyve and Richard Weiss for helpful comments.

Fixed points for group actions on 2-dimensional affine buildings

“…This was extended by Morgan and Shalen to R-trees [14,Proposition II.2.15]. We prove a similar result for 2-dimensional affine buildings of crystallographic type, meaning that the associated Weyl group is crystallographic; as discussed in [10,Section 9] this assumption holds for all Bruhat-Tits buildings and all discrete buildings, but there exist non-crystallographic R-buildings. We also assume that the building is not of type G2 , since our method fails in this case.…”