(Non)-completeness of ℝ-buildings and fixed point theorems

We prove two generalizations of results of Bruhat and Tits involving metrical completeness and R-buildings. Firstly, we give a generalization of the Bruhat-Tits fixed point theorem also valid for non-complete R-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 R-building. This criterion was proved by Bruhat and Tits for R-buildings arising from root group data with valuation