A Local-to-Global Result for Topological Spherical Buildings

Abstract

Suppose that \Delta, \Delta' are two buildings each arising from a semisimpe algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of \Delta, generated by a nonempty open set of chambers, and whose codomain is \Delta', is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups.