Searching for the power rangers

Abstract

We develop a method for studying centralizers of generic elements in some large Polish groups. We identify a condition which enables us to show that the range of each power map is comeager; a strengthening of this condition allows us to prove that a generic element embeds in a flow (a continuous, one-parameter subgroup). This approach enables us to answer the question that motivated this work : we prove that a generic isometry of the Urysohn space (or the Urysohn sphere) embeds in a flow. We also obtain information on the centralizer of a generic isometry; we show that it contains an infinite-dimensional torus and that torsion elements are dense in it. To illustrate the flexibility of our method, we apply it to two other Polish groups. The first is the group of measure-preserving automorphisms of a standard probability space, and we recover results of King and of de la Rue and de Sam Lazaro. Finally we apply our technique to the group of nonsingular automorphisms of a standard probability space, and we deduce that a generic nonsingular automorphism embeds in a flow, answering a question of Danilenko and Silva.