Metric building exercises (Q1605101)

The author proves some results concerning the fixed point sets and the translation distances of isometries of Euclidean buildings. He proves inequalities of the following sort: Let \(I\) be a Euclidean building and let \(\sigma\) be an automorphism of \(I\) with non empty fixed-point-set \(I^\sigma\). Then, there exists a constant \(k\) such that \(d(x,\sigma x)/d(x,I^\sigma)>k>0\) for every \(x\in I\setminus I^\sigma\).