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Metrical universality for groups

  • Michal Doucha EMAIL logo
Veröffentlicht/Copyright: 19. August 2016
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 29 Heft 4

Abstract

We prove that for any constant K>0, there exists a separable group equipped with a complete bi-invariant metric bounded by K, which is isometric to the Urysohn sphere of diameter K and 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 a bi-invariant metric bounded by K. As a consequence, we get that it is a universal Polish group admitting a compatible bi-invariant metric or a 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 groups with bi-invariant metric (bounded by K).

On the other hand, we prove that there is no metrically universal separable group with bi-invariant metric when there is no restriction on the diameter. The same is true for separable locally compact groups with bi-invariant metric.

Assuming the generalized continuum hypothesis (GCH), we prove that there exists a metrically universal (unbounded) group of density κ with bi-invariant metric for any uncountable cardinal κ. Moreover, under GCH, we deduce that there exists a universal SIN group of weight κ for any infinite cardinal κ.

Keywords: Metrically universal group; free groups; Fraïssé theory; Urysohn space; Graev metric; SIN group
MSC 2010: 22A05; 54E50; 03C98

Communicated by Manfred Droste

Funding statement: The author was supported by IMPAN’s international fellowship program partially sponsored by PCOFUND-GA-2012-600415.

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Received: 2015-9-13
Revised: 2016-6-22
Published Online: 2016-8-19
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
https://doi.org/10.1515/forum-2015-0181
Schlagwörter für diesen Artikel
Metrically universal group; free groups; Fraïssé theory; Urysohn space; Graev metric; SIN group

Artikel in diesem Heft

  1. Frontmatter
  2. Hurewicz fibrations, almost submetries and critical points of smooth maps
  3. Lower bounds for regular genus and gem-complexity of PL 4-manifolds
  4. Diffusion semigroup on manifolds with time-dependent metrics
  5. Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
  6. A computational approach to Milnor fiber cohomology
  7. Metrical universality for groups
  8. The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
  9. Slender domains and compact domains
  10. Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
  11. A note on local Hardy spaces
  12. On the value group of a model of Peano Arithmetic
  13. Extremal values of the (fractional) Weinstein functional on the hyperbolic space
  14. Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
  15. Unitals in shift planes of odd order
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