  We prove two generalizations of results proved by 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, but
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.

Metric Geometry