Separable ultrametric spaces and their isometry groups (Q470230)

The author studies the isometry groups of Polish metric spaces. He describes explicitly the isometry groups of such spaces which are non locally rigid and satisfy the condition that distances between orbits are realized by points. The groups that appear are a variant of the generalized wreath product. An essential ingredient of the proof is the construction of \(d\)-transversals. A subset \(Y\) of a metric space \((X, d)\) is called a \(d\)-transversal if \(Y\) intersects every orbit and distances between points of \(Y\) are equal to the distance of their orbits. As an application of his results the author characterizes those of the spaces considered whose isometry groups have uncountable strong cofinality.