Nonstandard hulls of locally uniform groups

TL;DR

This paper introduces a nonstandard hull construction for locally uniform groups, providing new insights into their structure and relationships with existing concepts like Banach-Lie groups and small subgroups.

Contribution

It develops a local group nonstandard hull construction, analyzes its dependence on pseudometrics, and connects it to Enflo's and Pestov's frameworks for infinite-dimensional Lie groups.

Findings

Nonstandard hull is a local group, not global.

The construction varies with the choice of pseudometrics.

The hull is locally isomorphic to Pestov's for Banach-Lie groups.

Abstract

We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxembourg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used to construct the hull. We use the nonstandard hull construction to give a nonstandard characterization of Enflo's notion of groups that are uniformly free from small subgroups. We prove that our nonstandard hull is locally isomorphic to Pestov's nonstandard hull for Banach-Lie groups. We also give some examples of infinite-dimensional Lie groups that are locally uniform.