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

The project is concerned with rigidity, compactifications and local-to-global principles in CAT(0) geometry.

One aim is to give a uniform construction of compactifications of euclidean buildings, using Gromov's embedding into spaces of continuous functions. The ultimate goal is to study the dynamics of discrete group actions on the building, using the compactification.

LG-rigidity of a metric space $$X$$ means that there is some $$r>0$$ such that if $$Y$$ is a metric space in which every $$r$$-ball is isometric to some $$r$$-ball in $$X$$, then there is a covering map $$X\to Y$$ which is a local isometry on all $$r$$-balls. The project intends to investigate LG-rigidity and non-rigidity for the 1-skeletons and chamber graphs of general Bruhat-Tits buildings.

## Publications

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

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## Team Members

Prof. Dr. Petra Schwer