A structure theorem for 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.

13 pages, 1 figure. Final version to appear in Journal of Geometry

References in corpus (2)

• Generalized affine buildings
• Notes on completely reducible subcomplexes of spherical buildings