An example of a non non-archimedean Polish group with ample generics

TL;DR

This paper constructs a non-archimedean Polish group with ample generics by analyzing permutation groups associated with analytic P-ideals, expanding the class of known groups with this property.

Contribution

It demonstrates the existence of a non non-archimedean Polish group with ample generics using permutation groups derived from analytic P-ideals.

Findings

If Fin is properly contained in I, then S_I has ample generics.

Existence of a non non-archimedean Polish group with ample generics.

Provides new examples of groups with ample generics.

Abstract

For an analytic $P$-ideal $I$, $S_I$ is the Polish group of all the permutations of $\mathbb{N}$ whose support is in $I$, with Polish topology given by the corresponding submeasure on $I$. We show that if $\mbox{Fin} \subsetneq I$, then $S_I$ has ample generics. This implies that there exists a non non-archimedean Polish group with ample generics.