An example of a non-commutative uniform Banach group

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.

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