On the commensurating full group
Antoine Derimay
TL;DR
The paper introduces the commensurating full group $[T]_{\rm com}$ for ergodic measure-class preserving transformations, establishing it as a Polish-group analogue of the $L^1$-full group and proving it is a complete invariant of flip conjugacy. It shows the index map $I:[T]_{\rm com}\to\mathbf Z$ is a quasi-isometry, making $\overline{D([T]_{\rm com})}$ bounded and a robust, bounded invariant; combined with Fremlin-type rigidity, this yields that abstract or topological isomorphisms of $[T]_{\rm com}$ (or its derived subgroup) imply flip conjugacy of the base transformations. The work highlights distinct behavior from $[T]_1$, notably the bounded geometry of the derived commensurating full group and the resulting quasi-isometric relation to $\mathbf Z$, while proving topological simplicity of $\overline{D([T]_{\rm com})}$ and transitivity properties of the associated boolean actions. Overall, the results provide a sharp, structure-preserving invariant theory for flip conjugacy in ergodic dynamics via the commensurating full group, and extend Le Maître's ideas to the measure-class setting.
Abstract
We introduce a new Polish group, called the commensurating full group, associated to an ergodic measure-class preserving transformation of a standard atomless probability space. It is an analogue of the $\rm L^1$ full group defined by Le Maître, which does not need the transformation to preserve a measure to be defined. We prove, among others, that it is a complete invariant of flip conjugacy, and that it is quasi-isometric to the line in the sense of Rosendal.