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

Abstract

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