Abelian Groups Are Polynomially Stable

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group Γ with respect to a sequence of groups {G n } ∞ n=1 , equipped with bi-invariant metrics {d n } ∞ n=1 . We consider the case G n = U (n) (resp. G n = Sym (n)), equipped with the normalized Hilbert-Schmidt metric d HS n (resp. the normalized Hamming metric d Hamming n ). Our main result is that if Γ is infinite, hyperlinear (resp. sofic) and has Property (T), then it is not stable with respect to U (n) , d HS n (resp. Sym (n) , d Hamming n ). This answers a question of Hadwin and Shulman regarding the stability of SL 3 (Z). We also deduce that the mapping class group MCG (g), g ≥ 3, and Aut (F n ), n ≥ 3, are not stable with respect to Sym (n) , d Hamming n .Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on U (n) and the (unnormalized) p-Schatten metrics, since many groups with Property (T) are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim.We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to U (n) , d HS n and Sym (n) , d Hamming n . The proof of Theorem 1.4Before we begin, we record a simple observation regarding P-stability. Fix formal elements {v i } ∞ i=1 , and for every n ∈ N, let B n = (v 1 , . . . , v n ) serve as an ordered basis for a complex vector space H n . A permutation σ ∈ Sym (n) ≅ Sym (B n ) extends uniquely to an element of O.B.,

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“…The proof uses the following proposition, which is a special case of Proposition A.3 of [5]. Proposition 3.2.…”

Section: The Non-p-stability Of Autmentioning

confidence: 99%

Group stability and Property (T)

Becker

Lubotzky

2020

Journal of Functional Analysis

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“…That is, different systems of equations may correspond to isomorphic groups. The starting point of the grouptheoretic approach to stability is the following observation from [6] (see also [11,Proposition 1.11]).…”

Section: Testability and Stability Of Systems Of Equationsmentioning

confidence: 99%

“…To express the more specific fact that XY = YX admits a family of testers coming from the Sample and Substitute algorithm, we say that XY = YX is stable (see Definition 1.5). A later work [11] improved upon the result of [6] by showing that we may take k(ε) = 1 ε O (1) , where the implied constant is absolute, and by providing a constructive proof.…”

Section: Introductionmentioning

confidence: 98%

Testability of relations between permutations

Becker¹,

Lubotzky

Mosheiff

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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We initiate the study of property testing problems concerning equations in permutations. In such problems, the input consists of permutations σ 1 , . . . , σ d ∈ Sym(n), and one wishes to determine whether they satisfy a certain system of equations E, or are far from doing so. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d = 2 and E consists of the single equation XY = YX, this corresponds to testing whetherWe formulate the well-studied group-theoretic notion of stability in permutations as a testability concept, and interpret all works on stability as testability results. Furthermore, we establish a close connection between testability and group theory, and harness the power of group-theoretic notions such as amenability and property (T) to produce a large family of testable equations, beyond those afforded by the study of stability, and a large family of non-testable equations.Finally, we provide a survey of results on stability from a computational perspective and describe many directions for future research.

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“…The rate of stability is, roughly speaking, the dependence of ϵ and δ in Definition 2.2. See [2] for details. To make this precise we define the function D (S ,R) : R + → R + as follows:…”

Section: [6]mentioning

confidence: 99%

“…Mosheiff [2] showed that the rate of stability of Z d , d ≥ 2 is polynomial but not linear (with respect to symmetric groups with normalized Hamming distance) 1 .…”

Section: Introductionmentioning

confidence: 99%