Metrical universality for groups

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 κ with bi-invariant metric for any uncountable cardinal κ. We moreover deduce that under GCH there is a universal SIN group of weight κ for any infinite cardinal κ.2010 Mathematics Subject Classification. 22A05,54E50,03C98.

Metrical universality for groups

“…This assumption is what was taken for granted in the original proof of Proposition 1.15 in [1]. Let us show how to conclude the proof provided that Claim 12 is proved.…”

“…In case that w j ∈ J ρ , we may analogously assume that w j is the left-most letter of V(j). (1) In case that w j ∈ J ρ , we can move the trivial subword w i+1 . .…”

“…We note that in the original statement there was an equality p(f i , f i ) = δ. While it is possible to prove the equality (under the assumption stated just before the statement of Proposition 2.15 in [1] that δ ≤ p 1 (f i , 1) + p 2 (f i , 1)), in order to keep the size of the corrigendum appropriate we note that the inequality p(f i , f i ) ≤ δ is sufficient for Embedding Construction 2.3 in [1]. Indeed, the reader can readily verify that the construction employs a sequence (ε i ) i with ∑ i ε i < ∞, and sequences of group generators (f ) ≤ ε n , the sequence is Cauchy too.…”

“…From now on, we assume that G is a finitely generated free group over some finite set of generators and λ is a CI-norm on G, generated by values on some finite symmetric subset A ⊆ G, which we assume that, without loss of generality, contains the free generators of G (as was also done in the appropriate section in [1] -see [1,Definition 2.13]). We identify the elements of G with the reduced words over the alphabet consisting of these generators and their inverses.…”

Universal and homogeneous structures on the Urysohn and Gurarij spaces

“…the p-Gurarij space, were enriched by a universal and homogeneous linear operator. We also mention the author's constructions of universal metric groups in [2] and [3] which can also be viewed as an enriching the Urysohn space by group structures.…”