Distances between fixed-point sets in metrically complete \(2\)-dimensional Euclidean buildings are realised (Q6060082)

Let \((X,d)\) be a metric space and \(A,B \subseteq X\), the distance between \(A\) and \(B\) is defined as \(d(A,B)=\inf \{d(a,b) \mid a\in A, b \in B \}\). In this interesting paper, the authors prove the following result (Main theorem): Let \(\Delta\) be a metrically complete 2-dimensional Euclidean building. Let \(G_{A}\) and \(G_{B}\) be finitely generated groups acting by isometries on \(\Delta\) with nonempty fixed-point sets \(A\) and \(B\), respectively. Then there exist points \(a \in A\) and \( b\in B\) such that \(d(a,b) = d(A,B)\). The proof uses the geometry of Euclidean buildings which the authors view as \(\mathsf{CAT}(0)\) spaces and properties of ultrapowers of Euclidean buildings.