In this talk, we study the isometry group \(\text{Iso}(\mathbb{U}_\Delta)\) of the \(\Delta\)-metric Urysohn space \(\mathbb{U}_\Delta\) equipped with the pointwise convergence topology for a countable distance set \(\Delta\) with \(\inf\Delta=0\). We showed that \(\text{Iso}(\mathbb{U}_\Delta)\) is a Lévy group, so it is extremely amenable. Moreover, we can choose the Lévy family such that its increasing union is isomorphic to Hall's group. This generalizes the results that \(\text{Iso}(\mathbb{U})\) is Lévy by Pestov and \(\text{Iso}(\mathbb{U}_\Delta)\) contains a dense subgroup which is isomorphic to Hall's group by Etedadialiabadi, Gao, Le Maître and Melleray. Then we will discuss its analogy for the continuous logic case.
It is an ongoing project with Su Gao and Víctor Hugo Yañez.