An example of a non-commutative uniform Banach group

TL;DR

This paper constructs the first known example of a non-commutative uniform Banach group, featuring a dense free group of countably many generators and a bi-invariant metric, answering longstanding open questions.

Contribution

It provides the first explicit example of a non-commutative uniform Banach group, advancing understanding in geometric nonlinear functional analysis.

Findings

Constructed a non-commutative uniform Banach group.

The group contains a dense free group of countably many generators.

The metric induced by the norm is bi-invariant.

Abstract

Benyamini and Lindenstrauss mention in their monograph \emph{Geometric nonlinear functional analysis Vol. 1., American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000} that there is no known example of a non-commutative uniform Banach group. Prassidis and Weston also asked whether there is a non-commutative example. We answer this problem affirmatively. We construct a non-commutative uniform Banach group which has the free group of countably many generators as a dense subgroup. Moreover, we show that our example is a free one-generated uniform Banach group whose metric induced by the norm is bi-invariant.