A Local-to-Global Result for Topological Spherical Buildings

TL;DR

This paper establishes conditions under which a partially defined chamber map between topological spherical buildings extends uniquely, linking local properties to global structure and relating to the Borel-Tits theorem.

Contribution

It provides a new local-to-global extension result for chamber maps in topological spherical buildings, connecting local conditions to global isomorphisms.

Findings

Partial chamber maps extend uniquely under certain conditions

Conditions relate local open sets to global structure

Connects to the Borel-Tits theorem on algebraic groups

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.