On invariant means and pre-syndetic subgroups
Vladimir G. Pestov, Friedrich Martin Schneider
TL;DR
The paper surveys amenability notions beyond local compactness, focusing on skew-amenability and Malliavin--Malliavin amenability (M--M) and their behavior for nontraditional subgroups. It proves that pre-syndetic subgroups of skew-amenable groups are skew-amenable and deduces that co-compact subgroups of amenable SIN groups are amenable, using right-invariant means and ultrafilter limits. It introduces M--M amenability, showing it is stronger than ordinary amenability and providing Reiter- and Følner-type criteria that characterize M--M amenability via invariant means on bounded Borel functions and partition- or matching-based formulations. The work also collects open problems about the relationships among these notions, the amenability of various infinite-dimensional groups, and the existence of asymptotically invariant nets of measures for M--M amenable Polish groups, inviting further exploration of invariant dynamics in non-locally compact settings.
Abstract
Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.