Groups of Lipschitz functions and metric duality

In this talk we will consider the metric group of all pointed, Lipschitz functions defined on a pointed metric space and with values on the compact group . We will present and explore the duality between this metric group and the free Abelian group on the space , equipped with the Graev extension of the original metric. The analogous construction for Banach spaces has remained an important area of study for several decades. As expected, the group counterpart resembles the original theory but is also both challenging and revealing in its own ways.
This is an ongoing project with M. J. Chasco and M. Tkachenko. The talk will report on some recent developments concerning separability of the group and the question whether the natural embedding of the group into the metric dual group of is an isometry onto its image.