Infinitesimal generators and quasi non-archimedean topological groups

TL;DR

This paper explores the properties of infinitesimal generation in connected locally compact groups and introduces quasi non-archimedean groups, revealing new structural insights and examples in Polish groups.

Contribution

It generalizes existing theorems on infinitesimal generation and introduces the concept of quasi non-archimedean groups, expanding understanding of topological group structures.

Findings

Connected separable locally compact groups are infinitesimally finitely generated.

Separable connected compact groups are infinitesimally 2-generated.

A locally compact group is quasi non-archimedean iff it is totally disconnected.

Abstract

We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer $n$ such that every neighborhood of the identity contains $n$ elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally $2$-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. We prove that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, and provide various examples which show that the picture is much richer for Polish groups. In particular, we get an example of a Polish group which is infinitesimally $1$-generated but totally disconnected, strengthening Stevens' negative answer to Problem 160 from the Scottish book.