Metrical universality for groups

TL;DR

This paper constructs a universal separable group with a bi-invariant metric bounded by a constant, analogous to the Gurarij space, and explores its properties and limitations in the context of metric groups.

Contribution

It introduces a new universal separable group with bi-invariant metric, exhibiting almost-universal disposition and rich automorphism group, extending the concept of universality in metric group theory.

Findings

Constructed a universal separable group with bi-invariant metric bounded by K.

Showed the group contains isometric copies of all separable groups with bounded bi-invariant metrics.

Proved non-existence of a universal bounded metric group without diameter restrictions.

Abstract

We prove that for any constant $K>0$ there exists a separable group equipped with a complete bi-invariant metric bounded by $K$, isometric to the Urysohn sphere of diameter $K$, that is 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 bi-invariant metric bounded by $K$. As a consequence, we get that it is a universal Polish group admitting compatible bi-invariant metric, resp. 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 group with bi-invariant metric (bounded by $K$). On the other hand, we prove there is no metrically universal separable group with bi-invariant metric when there is no restriction on diameter. The same is true for separable locally compact groups with bi-invariant metric. Assuming the generalized continuum hypothesis, we prove that there exists a metrically universal (unbounded) group of density $\kappa$ with bi-invariant metric for any uncountable cardinal $\kappa$. We moreover deduce that under GCH there is a universal SIN group of weight $\kappa$ for any infinite cardinal $\kappa$.