# Buildings and Curvature

### Ernst Heintze

Universität Augsburg, Germany### Linus Kramer

Universität Münster, Germany### Bernhard Mühlherr

Universität Gießen, Giessen, Germany### Bertrand Rémy

Université Claude Bernard Lyon 1, Villeurbanne, France

## Abstract

It was the aim of the meeting to bring together international experts from the theory of buildings, differential geometry and geometric group theory. Buildings are combinatorial structures (simplicial complexes) which can be seen as simultaneously generalizing projective spaces and trees. Already from these examples it is clear that there will be interesting groups acting on buildings. Conversely, groups can be studied using their actions on given buildings. Groups coming up in this context are in particular groups having a $BN$-pair. Examples of such groups include the classical groups, simple Lie groups and algebraic groups (also over local fields), Kac-Moody groups and loop groups. This already indicates that these groups play an important role in many different areas of mathematics such as algebra, geometry, number theory, physics and analysis. Kac-Moody groups correspond to so-called twin buildings, a particularly active area in the theory of buildings. Geometric group theory is concerned with the investigation of group actions on metric spaces using the interplay of group theoretic properties and metric properties like curvature in the sense of Alexandrov, or CAT$(0)$-spaces. The geometric realization of a building is a metric space with interesting curvature properties on which the above mentioned groups as well as their subgroups like uniform lattices or arithmetic groups act in a natural way by isometries. In this respect there are a number of canonical connections between the theory of buildings and geometric group theory. One of the current problems concerns the characterization of buildings as metric spaces. In differential geometry these aspects also play an important role, e.g. in connection with Hadamard manifolds, (simply connected Riemannian manifolds of nonpositive curvature). A special role is played by the Riemannian symmetric spaces and their quotients of finite volume which one wants to characterize geometrically. By considering the fundamental groups, one obtains discrete group actions also studied in geometric group theory. Buildings come up in differential geometry as the compactifications of Riemannian symmetric spaces yielding examples of topological buildings. Asymptotic cones (and ultrapowers) of symmetric spaces present non-discrete affine buildings and create new and interesting relations to model theory. These constructions are important in new proofs of differential geometric rigidity theorems, like Mostow Rigidity and the Margulis Conjecture. This shows that there are close connections between the areas, and this meeting was the first in a number of years in Oberwolfach having these connections as its topic. Geometric group theory has recently introduced interesting aspects into the theory of buildings, in particular the hyperbolic buildings. Conversely, new developments in the theory of buildings, e.g. the twin buildings have interesting group theoretic applications, for example in the theory of $S$-arithmetic groups or in the theory of Kac-Moody groups. All these aspects played an important part in this meeting and the interaction between the participants from different areas was very lively.

## Cite this article

Ernst Heintze, Linus Kramer, Bernhard Mühlherr, Bertrand Rémy, Buildings and Curvature. Oberwolfach Rep. 1 (2004), no. 2, pp. 1233–1284

DOI 10.4171/OWR/2004/23