A Local-to-Global Result for Topological Spherical Buildings

Suppose that Δ, Δ' 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 Δ, generated by a nonempty open set of chambers, and whose codomain is Δ', 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.

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