On the isometry groups of certain CAT(0) spaces and trees (Q1591081)

Let \(X\) be a locally finite tree considered as a locally finite 1-dimensional simply connected simplicial complex. The group of automorphisms \(\Aut(X)\) consists of simplicial homeomorphisms of \(X\). Equipped with the compact open topology, this group is a locally compact totally disconnected Hausdorff topological group. There is a natural metric on \(X\) which makes it a CAT(0) space. The space \(X\) is a CAT(0) space if for any geodesic triangle \(\triangle\) in \(X\) the mapping \(\triangle\rightarrow{\mathbb R^2}\), which is an isometry on the sides of \(\triangle\), does not decrease distances. It is proved that \(\Aut(X)\) is either discrete, or pro-finite, or not the inverse limit of an inverse system of discrete groups. For a subgroup \(G\) of \(\Aut(X)\), the situations when these cases occur are described precisely. By a result of Tits, any fixed-point free automorphism acts on some line in \(X\) by translation. This line is called translation axis. If \(G\) contains some fixed-point free automorphism then \(G\) is either discrete or not the inverse limit of discrete groups; the first possibility occurs if there exists an edge in some translation axis which has the trivial stabilizer, the second if the stabilizer is nontrivial for some such edge. If \(G\) does not contain a fixed-point free automorphism then \(G\) is either pro-finite or not the inverse limit of discrete groups if it has a bounded orbit or not, respectively.