Property of defect diminishing and stability

Morales, Marcela; Glebsky, Lev

doi:10.24330/ieja.1058399

Cited by 3 publications

References 6 publications

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Testability in group theory

Becker,

Lubotzky,

Mosheiff

2023

Isr. J. Math.

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This paper is a journal counterpart to [5], in which we initiate the study of property testing problems concerning a finite system of relations E between permutations, generalizing the study of stability in permutations. To every such system E, a group Γ = ΓE is associated and the testability of E depends only on Γ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group). This leads to the notion of testable groups, and, more generally, Benjamini–Schramm rigid groups. The paper presents an ensemble of tools to check if a given group Γ is testable/BS-rigid or not.

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Testability in group theory

Becker,

Lubotzky,

Mosheiff

2023

Isr. J. Math.

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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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Garland’s method with Banach coefficients

Oppenheim

2023

Analysis & PDE

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We prove a Banach version of Garland's method of proving vanishing of cohomology for groups acting on simplicial complexes. The novelty of this new version is that our new condition applies to every reflexive Banach space.This new version of Garland's method allows us to deduce several criteria for vanishing of group cohomology with coefficients in several classes of Banach spaces (uniformly curved spaces, Hilbertian spaces and L p spaces).Using these new criteria, we improve recent results regarding Banach fixed-point theorems for random groups in the triangular model and give a sharp lower bound for the conformal dimension of the boundary of such groups. Also, we derive new criteria for group stability with respect to p-Schatten norms. Lécureux et al. 2020] for results regarding vanishing of the first cohomology, and see [Oppenheim 2017;Lubotzky and Oppenheim 2020] for results regarding vanishing of higher cohomologies. However, much less is known when one considers the less-structured setting of a group acting on a simplicial complex without assuming the extra structure of an affine building.Recently, the results for vanishing of the first cohomology of random groups with coefficients in Banach spaces were improved: First, Drut , u and Mackay [2019] proved vanishing of the first cohomologyThe author was partially supported by ISF grant no. 293/18 . MSC2020: primary 20J06; secondary 20F67, 20P05, 46B20, 46M20.

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Property of defect diminishing and stability

Cited by 3 publications

References 6 publications

Testability in group theory

Testability in group theory

Testability of relations between permutations

Garland’s method with Banach coefficients

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