We prove two generalizations of results of Bruhat and Tits involving metrical completeness and -buildings. Firstly, we give a generalization of the Bruhat–Tits fixed point theorem also valid for non-complete -buildings with the added condition that the group is finitely generated. Secondly, we generalize a criterion which reduces the problem of completeness to the wall trees of the -building. This criterion was proved by Bruhat and Tits for -buildings arising from root group data with valuation.
Cite this article
Koen Struyve, (Non)-completeness of ℝ-buildings and fixed point theorems. Groups Geom. Dyn. 5 (2011), no. 1, pp. 177–188DOI 10.4171/GGD/121