Graev metrics on free products and HNN extensions

Slutsky, Konstantin

doi:10.1090/s0002-9947-2014-06010-8

Cited by 3 publications

(12 citation statements)

References 7 publications

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“…Our arguments here are very similar to those in [Slu12], and we shall outline the proofs and give references for more details. Essentially the proofs are repetitions of the proofs fore the metric case when the summation operation is substituted with the operation of taking maximum, e.g., like in Proposition 1.1.…”

Section: Graev Ultrametrics On Free Productsmentioning

confidence: 64%

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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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Section: Graev Ultrametrics On Free Productsmentioning

confidence: 64%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Graev ultrametrics and free products of Polish groups

Slutsky

2015

Forum Mathematicum

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“…Then we have: Theorem 1.7 (Slutsky, Theorem 5.10 [30]). p is a bi-invariant metric on G 3 extending d 1 on G 1 and d 2 on G 2 .…”

Section: 2mentioning

confidence: 98%

“…We need to prove the other inequality. To do that, we use a result of Slutsky from [30]. To state his result, we introduce some notation.…”

Section: 2mentioning

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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“…We also refer the reader to [13] where a variant of Graev metric on free products of groups having a common closed subgroup was defined.…”

Section: Introductionmentioning

confidence: 99%

Non-abelian group structure on the Urysohn universal space

Doucha

2015

Fund. Math.

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Following the continuing interest in the Urysohn space and, more specifically, the recent problem area of finding and comparing group structures on the Urysohn space we prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, we introduce a variant of the Graev metric that enables us to construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We provide several open questions and problems related to this subject.2010 Mathematics Subject Classification. Primary 22A05; Secondary 54E50, 03C98.

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Graev metrics on free products and HNN extensions

Cited by 3 publications

References 7 publications

Graev ultrametrics and free products of Polish groups

Graev ultrametrics and free products of Polish groups

Metrical universality for groups

Non-abelian group structure on the Urysohn universal space

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