On Polish groups admitting a compatible complete left-invariant metric

Malicki, Maciej

doi:10.2178/jsl/1305810757

Cited by 11 publications

(8 citation statements)

References 5 publications

(11 reference statements)

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“…The following statement gives a partial answer. It somehow corresponds to the result of M.Malicki [28] that the set of all Polish groups admitting compatible complete left-invariant metrics is coanalytic non-Borel as a subset of a standard Borel space of Polish groups.…”

Section: Complexity Of Sets Of Approximately Ultrahomogeneous Structuressupporting

confidence: 60%

Polish G-spaces and continuous logic

Ivanov

Majcher-Iwanow

2017

Annals of Pure and Applied Logic

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Section: Complexity Of Sets Of Approximately Ultrahomogeneous Structuressupporting

confidence: 60%

Polish G-spaces and continuous logic

Ivanov

Majcher-Iwanow

2017

Annals of Pure and Applied Logic

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“…object all Polish groups yes [32], [34], [1] yes [2], [25] abelian Polish groups yes [28] yes [27] compact Polish groups yes, [26] yes, [26] loc. compact P.groups no, [26] no, [26] CLI groups no [18] no [18] SIN P. groups yes, this paper yes, [27] Parallely, there has been a research on universal groups for subclasses of non-Archimedean Polish groups. We refer to the recent article [10] of Gao and Xuan, where they in particular prove that in case of non-Archimedean Polish groups there is no universal object admitting bi-invariant compatible metric.…”

Section: Class Of Groupsmentioning

confidence: 99%

“…They are usually called CLI (complete left-invariant) groups. Malicki in [18] proved that there is neither a universal CLI group nor a projectively universal CLI group. The last major contribution to this area were in [2] the Ding's construction of a projectively universal Polish group (i.e.…”

Section: Introductionmentioning

confidence: 99%

Metrical universality for groups

Doucha

2016

Forum Mathematicum

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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.

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“…A Polish group G is weakly universal if every Polish group is isomorphic to a closed subgroup of a topological quotient group of G. For example, ℓ 1 under addition is a weakly universal for abelian Polish groups. On the negative side, most recently, Malicki [9] proved that there is no weakly universal CLI group. So neither universal CLI group, nor surjectively universal CLI group exists.…”

Section: Question 11 Is There a Surjectively Universal Polish Group?mentioning

confidence: 99%

On surjectively universal Polish groups

Ding

2012

Advances in Mathematics

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A Polish group is surjectively universal if it can be continuously homomorphically mapped onto every Polish group. Making use of a type of new metrics on free groups [2], we prove the existence of surjectively universal Polish groups, answering in the positive a question of Kechris. In fact, we give several examples of surjectively universal Polish groups.We find a sufficient condition to guarantee that the new metrics on free groups can be computed directly. We also compare this condition with CLI groups.

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On Polish groups admitting a compatible complete left-invariant metric

Cited by 11 publications

References 5 publications

Polish G-spaces and continuous logic

Polish G-spaces and continuous logic

Metrical universality for groups

On surjectively universal Polish groups

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