## Prof. Dr. Petra Schwer

### Project leader

Otto von Guericke Universität Magdeburg

E-mail: petra.schwer(at)ovgu.de

Homepage: http://www.math.kit.edu/iag2/~schwer/

## Project

**20**Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings

## Publications within SPP2026

In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

**Related project(s):****20**Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings

We prove an affine analog of Scharlau's reduction theorem for spherical buildings. To be a bit more precise let *X* be a euclidean building with spherical building ∂*X* at infinity. Then there exists a euclidean building *X*¯ such that *X* splits as a product of *X*¯ with some euclidean *k*-space such that ∂*X*¯ is the thick reduction of ∂*X* in the sense of Scharlau. \newline In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.

**Related project(s):****20**Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings