Approximations of groups, characterizations of sofic groups, and equations over groups

Glebsky, Lev

doi:10.1016/j.jalgebra.2016.12.012

Cited by 13 publications

(20 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This latter statement was observed by Glebsky in[5], although with a more complicated proof 5. L ω 1 ,ω is the extension of first-order logic that allows countable conjunctions and disjunctions provided only finitely many free variables appear in all of the formulae involved in the conjunction or disjunction.…”

mentioning

confidence: 78%

See 1 more Smart Citation

On the nonexistence of Følner sets

Goldbring¹

2019

Preprint

View full text Add to dashboard Cite

A. We show that there is n ∈ N, a finite system Σ( x, y) of equations and inequations having a solution in some group, where x has length n, and ǫ > 0 such that: for any group G and any a ∈ G n , if the system Σ( a, y) has a solution in G, then there is no ( a, ǫ)-Følner set in G. The proof uses ideas from modeltheoretic forcing together with the observation that no amenable group can be existentially closed. Along the way, we also observe that no existentially closed group can be exact, have the Haagerup property, or have property (T). Finally, we show that, for n large enough and for ǫ small enough, the existence of (F, ǫ)-Følner sets, where F has size at most n, cannot be expressed in a first-order way uniformly in all groups.

show abstract

mentioning

confidence: 78%

“…The sentences σ n,ǫ are prototypical examples of ∀ ∃-sentences, which are special kind of L ω 1 ,ω -sentences 5 . It is known that if P is a ∀ ∃-property and there is a locally universal object with property P, then there is an e.c.…”

Section: Preliminaries On Existentially Closed and Locally Universal ...mentioning

confidence: 99%

On the nonexistence of Følner sets

Goldbring¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The use of metric ultraproducts can be replaced by the following notion of approximation, cf. [30] and [17, Definition 3]. In this definition and below we always assume that metric groups are considered with respect to invariant metrics.…”

Section: Decidability Of Theories Of Pseudo Finite Dimensional Hilbert Spacesmentioning

confidence: 99%

“…It is known that when the metrics of

scriptK

are bounded by some fixed number r , a group G is

scriptK

‐approximable if and only if it embeds into a metric ultraproduct of groups from

scriptK

[17, 30]. Moreover in the case of sofic and hyperlinear groups the function α can be taken constant on

G ∖ {1}

with the value equal to any real number strictly between 0 and 1 (between 0 and

r = \sqrt{2}

in the hyperlinear case).…”

Section: Decidability Of Theories Of Pseudo Finite Dimensional Hilbert Spacesmentioning

confidence: 99%

Continuous theory of operator expansions of finite dimensional Hilbert spaces and decidability

Ivanov

2021

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

“…Note that the above definition differs slightly from Definition 1.6 from the one in [27] as we impose no restrictions on the invariant length functions. However, it is equivalent to Definition 6 from [9].…”

Section: 2mentioning

confidence: 99%

Some remarks on finitarily approximable groups

Nikolov¹,

Schneider²,

Thom³

2018

Journal De L’École Polytechnique — Mathématiques

View full text Add to dashboard Cite

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group.Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. On a related note, we prove that any non-trivial group which can be approximated by finite groups has a non-trivial quotient that can be approximated by finite special linear groups.Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that a the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well.All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal. In Section 4 we also use results of Liebeck and Shalev on bounded generation in finite simple groups.

show abstract

Approximations of groups, characterizations of sofic groups, and equations over groups

Cited by 13 publications

References 20 publications

On the nonexistence of Følner sets

On the nonexistence of Følner sets

Continuous theory of operator expansions of finite dimensional Hilbert spaces and decidability

Some remarks on finitarily approximable groups

Contact Info

Product

Resources

About