On surjectively universal Polish groups

Ding, Lei

doi:10.1016/j.aim.2012.06.029

Cited by 7 publications

(10 citation statements)

References 9 publications

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“…It has been known for about 30 years that there exist injectively universal Polish groups [13,14], that is, Polish groups G such that every Polish group H is isomorphic (as a topological group) to a (necessarily closed) subgroup of G. Those examples in particular answered a Scottish Book question by Schreier-Ulam (question 103 in [8]); for a recent new example, see [3]. A few years ago L. Ding proved [6], answering a long-standing question of A. Kechris (see [7,Problem 2.10], [2,Problem 1.4.2]), that there also exists a projectively universal, or couniversal, Polish group, that is, such a Polish group G that every Polish group H is isomorphic (as a topological group) to the quotient group G/N for some closed invariant subgroup N ⊳ G.…”

Section: Introductionmentioning

confidence: 87%

Projectively universal countable metrizable groups

Pestov

Uspenskij

2016

Proc. Amer. Math. Soc.

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Abstract. We prove that there exists a countable metrizable topological group G such that every countable metrizable group is isomorphic to a quotient of G. The completion H of G is a Polish group such that every Polish group is isomorphic to a quotient of H. IntroductionA topological group is Polish if it is homeomorphic to a complete separable metric space. It has been known for about 30 years that there exist injectively universal Polish groups [13,14], that is, Polish groups G such that every Polish group H is isomorphic (as a topological group) to a (necessarily closed) subgroup of G. , that there also exists a projectively universal, or couniversal, Polish group, that is, such a Polish group G that every Polish group H is isomorphic (as a topological group) to the quotient group G/N for some closed invariant subgroup N ⊳ G.The aim of this note is to provide a shorter proof of a stronger theorem: there exists a projectively universal countable metrizable group. The completion of such a group is a projectively universal Polish group (Theorem 2.1), so our result indeed implies that of Ding. We give two constructions in sections 3 and 4, due to the first and the second author, respectively.We mention that a projectively universal Abelian Polish group was constructed in [11], and an injectively universal Abelian Polish group was constructed in [12]. The question remains open, due to Kechris, Date: October 26, 2015.

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Section: Introductionmentioning

confidence: 87%

Projectively universal countable metrizable groups

Pestov

Uspenskij

2016

Proc. Amer. Math. Soc.

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“…It says that we may realize f in F * Z by the generator of the new copy of the integers so that the extended metric on F * Z is still rational and finitely generated, i.e. F * Z ∈ G. However, then by Fact 1.8 (2) we may suppose that this extension of F to F * Z actually exists in G 1 and thus the generator of this new copy of the integers, denoted by x f , belongs to G 1 and we are done by Fact 1.17.…”

Section: 2mentioning

confidence: 99%

“…It remains to check that D G is a G δ subset of D. Again by Theorem 1.19 (2), it suffices to check that the subset of all those metrics ρ from D satisfying ∀ε > ε ′ > 0 ∀0 ≤ m < n ∀(F n , p) and any monomorphism ι :…”

Section: 5mentioning

confidence: 99%

“…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. projectively universal for the whole class of Polish groups) and then further constructions of projectively universal Polish groups by Pestov and Uspenskij in [25].…”

Section: Introductionmentioning

confidence: 99%

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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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“…An important question raised in [Kec94] and further advertised in [BK96] is whether there is a universal Polish group. Motivated by this question, L. Ding and S. Gao [DG07b] constructed generalized Graev metrics, and based on this construction Ding [Din12] answered the question of Kechris in the affirmative. In a recent paper Gao [Gao13] addressed the question of the existence of surjectively universal Polish ultrametric groups and gave yet another modification of Graev's original definition.…”

Section: Introductionmentioning

confidence: 99%

Graev ultrametrics and free products of Polish groups

Slutsky

2015

Forum Mathematicum

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We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups G and H can be embedded into a Polish group T in such a way that the subgroup of T generated by G and H is isomorphic to the free product G * H.Theorem (Folklore, see Theorem 2.11 in [Kec94]). The group F (N N ) is surjectively universal in the class of Polish groups that admit compatible two-sided invariant metrics. An important question raised in [Kec94] and further advertised in [BK96] is whether there is a universal Polish group. Motivated by this question, L. Ding and S. Gao [DG07b] constructed generalized Graev metrics, and based on this construction Ding [Din12] answered the question of Kechris in the affirmative. In a recent paper Gao [Gao13] addressed the question of the existence of surjectively universal Polish ultrametric groups and gave yet another modification of Graev's original definition. The latter paper of Gao motivates our study of the Graev ultrametrics on free products of ultrametric groups.1.1. Main results. The main results of this work are twofold. In Section 2 we give the constructions of Graev ultrametrics for free products and HNN extensions of groups with two-sided invariant ultrametrics. In particular we prove Theorem (see Theorem 2.13). Let (G, d G ) and (H, d H ) be groups with two-sided invariant ultrametrics, and let A = G ∩ H be a common closed subgroup. There exists a two-sided invariant ultrametric on the free product with amalgamation G * A H that extends ultrametrics d G and d H .Theorem (see Theorem 2.21). Let (G, d ) be a group with a two-sided invariant ultrametric d, A and B be closed subgroups of G and φ : A → B be an isometric isomorphism. If diam(A) ≤ K, then there exists a twosided invariant ultrametric δ on the HNN extension H of (G, φ) which extends d and such that δ(t, e) = K, where t is the stable letter of H.While we follow closely the methods of [Slu12], the formalism for trivial words used in this paper is different. We introduce a new notion of a maximal evaluation forest and argue that it provides a more unified tool for studying Graev metrics on free products than the notion of an evaluation tree.

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On surjectively universal Polish groups

Cited by 7 publications

References 9 publications

Projectively universal countable metrizable groups

Projectively universal countable metrizable groups

Metrical universality for groups

Graev ultrametrics and free products of Polish groups

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