Metrical universality for groups

We prove that for any constant K > 0, there exists a separable group equipped with a complete bi-invariant metric bounded by K, which is isometric to the Urysohn sphere of diameter K and of `almost-universal disposition'. It is thus an object in the category of separable groups with bi-invariant metric, analogous in its properties to the Gurarij space from the category of separable Banach spaces. We show that this group contains an isometric copy of any separable group equipped with a bi-invariant metric bounded by K. As a consequence, we get that it is a universal Polish group admitting a compatible bi-invariant metric or a universal second countable SIN group. Moreover, the almost-universal disposition shows that the automorphism group of this group is rich and it characterizes the group uniquely up to isometric isomorphism. We also show that this group is in a certain sense generic in the class of separable groups with bi-invariant metric (bounded by K). On the other hand, we prove that there is no metrically universal separable group with bi-invariant metric when there is no restriction on the diameter. The same is true for separable locally compact groups with bi-invariant metric. Assuming the generalized continuum hypothesis (GCH), we prove that there exists a metrically universal (unbounded) group of density kappa with bi-invariant metric for any uncountable cardinal kappa. Moreover, under GCH, we deduce that there exists a universal SIN group of weight kappa for any infinite cardinal kappa.