Generating Infinite Symmetric Groups

Bergman, George M.

doi:10.1112/s0024609305018308

Cited by 81 publications

(140 citation statements)

References 21 publications

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“…And even if groups that we consider here are abstract (no topology), they naturally appear as subgroups of certain "infinite-dimensional" groups, objects some of the deep recent insights in whose structure we owe, again, to logicians, see, e.g. [10,45,51,61,70].…”

Section: Introductionmentioning

confidence: 99%

Hyperlinear and Sofic Groups: A Brief Guide

2008

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Abstract. This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U (n) and symmetric groups S n , n ∈ N. Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular it is still unknown if every group is hyperlinear and/or sofic.

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Section: Introductionmentioning

confidence: 99%

Hyperlinear and Sofic Groups: A Brief Guide

2008

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“…Note that this was already a consequence of the strong boundedness of symmetric groups [Ber04], but provides a better cardinality: if |G| = κ, we obtain a group of cardinality κ ℵ0 rather that 2 κ .…”

Section: ω 1 -Existentially Closed Groupsmentioning

confidence: 97%

“…The combination of these two properties, sometimes referred as "uncountable strong cofinality" 1 , has been introduced and is extensively studied in Bergman's preprint [Ber04]; see also [DG05]. Note that an uncountable group with cofinality = ω is not necessarily Cayley bounded: the free product of two uncountable groups of cofinality = ω, or the direct product of an uncountable group of cofinality = ω with Z, are obvious counterexamples.…”

Section: Strongly Bounded Groupsmentioning

confidence: 99%

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Strongly Bounded Groups and Infinite Powers of Finite Groups

Cornulier¹

2006

Communications in Algebra

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Abstract. We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently introduced by Bergman.Our main result is that G I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that ω 1 -existentially closed groups are strongly bounded.

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“…For more general topological groups the two properties have been distinguished: as observed e.g. in [13], every (necessarily uncountable) discrete group having Bergman's property [8] has property (F H), while it is well-known and easily proved that a discrete group with property (T ) is finitely generated (cf. [6], Th.…”

Section: Theorem 11 a Locally Compact Group G With Property (T ) Ismentioning

confidence: 99%

Amenability versus property (𝑇) for non-locally compact topological groups

Pestov

2018

Trans. Amer. Math. Soc.

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Abstract. For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugationinvariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T ) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T ) admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property (T ), and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property (T ) and property (F H) in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property (T ), having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property (T ) is closed under arbitrary infinite products with the usual product topology.

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Generating Infinite Symmetric Groups

Abstract: We prove that for any infinite κ, the full symmetric group Sym(κ) is the product of at most 14 abelian subgroups. This is a strengthening of a recent result of M. Abért.

Cited by 81 publications

References 21 publications

Hyperlinear and Sofic Groups: A Brief Guide

Hyperlinear and Sofic Groups: A Brief Guide

Strongly Bounded Groups and Infinite Powers of Finite Groups

Amenability versus property (𝑇) for non-locally compact topological groups

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