Group actions on Polish spaces

TL;DR

This paper studies actions of Polish groups on uncountable Polish spaces, exploring measurability and orbit size, and constructing examples with nonmeasurable sets and subgroups with special properties.

Contribution

It introduces new results on nonmeasurable subgroups and sets under Polish group actions, linking these to set theory and providing explicit examples.

Findings

Existence of nonmeasurable subgroups with respect to certain $\sigma$-ideals.

Construction of nonmeasurable sets from group actions on Polish spaces.

Examples involving isometry groups of the Cantor space and homeomorphism groups.

Abstract

In this paper we investigate the action of Polish groups (not necessary abelian) on an uncountable Polish spaces. We consider two main situations. First, when the orbits given by group action are small and the second when the family of orbits are at most countable. We have found some subgroups which are not measurable with respect to a given $\sigma$-ideals on the group and the action on some subsets gives a completely nonmeasurable sets with respect to some $\sigma$-ideals with a Borel base on the Polish space. In most cases the general results are consistent with ZFC theory and are strictly connected with cardinal coefficients. We give some suitable examples, namely the subgroup of isometries of the Cantor space where the orbits are sufficiently small. In the opposite case we give an example of the group of the homeomorphisms of a Polish space in which there is a large orbit and we have found the subgroup without Baire property and a subset of the mentioned space such that the action of this subgroup on this set is completely nonmeasurable set with respect to the $\sigma$-ideal of the subsets of first category.