A structure theorem for euclidean buildings

TL;DR

This paper establishes a structure theorem for Euclidean buildings, showing how they can be decomposed into products involving their spherical boundaries and providing a converse embedding result.

Contribution

It introduces an affine analog of Scharlau's reduction theorem, linking Euclidean buildings with their spherical boundaries and extending embeddings.

Findings

Decomposition of Euclidean buildings into products with Euclidean space

Existence of a Euclidean building with a given spherical boundary

Extension of embeddings from spherical to Euclidean buildings

Abstract

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 $\partial X$ at infinity. Then there exists a euclidean building $\bar X$ such that $X$ splits as a product of $\bar X$ with some euclidean $k$-space such that $\partial \bar X$ is the thick reduction of $\partial X$ 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.