Bounded normal generation is not equivalent to topological bounded normal generation

TL;DR

The paper demonstrates that topological bounded normal generation is a strictly weaker property than bounded normal generation by providing examples of non-simple Polish groups with the former but not the latter.

Contribution

It introduces examples of Polish groups that have topological bounded normal generation without having bounded normal generation, clarifying their distinction.

Findings

Existence of non-simple Polish groups with topological bounded normal generation

Topological bounded normal generation is weaker than bounded normal generation

Examples differentiate the two properties in group theory

Abstract

We show that some derived $\mathrm{L}^1$ full groups provide examples of non simple Polish groups with the topological bounded normal generation property. In particular, it follows that there are Polish groups with the topological bounded normal generation property but not the bounded normal generation property.