Vector Bundles and Arithmetical Groups I. The higher Bruhat-Tits tree

Abstract

We define and study a simplicial complex which is a homogeneous space for the group $PGL(2, K)$ over a two-dimensional local field $K$. The complex is a generalization of the tree studied by F. Bruhat, J. Tits, J.-P. Serre and P. Cartier in the 60's and early 70's. Such complex can be canonically attached to the triples $x \in C \subset X$ where $X$ is an algebraic surface, $C$ is an irreducible curve and $x$ is a smooth point on $C$ and $X$. This construction can be used for a description of the isomorphism set of vector bundles on $X$.