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Stability of approximate group actions: uniform and probabilistic

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We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky’s original question by showing that the group \mathbb{Z} is not uniformly strictly stable. Furthermore, we show that \operatorname{SL}_{r}(\mathbb{Z}) , r\geq3 , is uniformly flexibly stable, but the free group F_{r} , r\geq2 , is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.

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... There are many (non-equivalent) ways to formulate Ulam's problem (cf. [9,13,20,26,37,40,52]). In this section we study a finite group presentations variant of it. ...
... So, the linearity test is Algorithm 1 under the restriction that the range of f is Sym(2) and not any Sym(N ). In [9], it is proved that the multiplication table presentation of any finite group is ρ-homomorphism stable with linear rate ρ(ε) = 3000ε. This generalizes [17], though with worse parameters. ...
... This notion is also called ρ-robustness and ρ-soundness in the literature.9 Our notion of homomorphism stability is usually referred to in the literature as pointwise flexible stability in permutations (see[10]). ...
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... Similarly to stability in permutations [4] and in the normalized Hilbert-Schmidt norm [6], we consider also a notion of flexible stability, where the homomorphism ψ : Γ → GL N (F ) is allowed to take values in matrices of larger dimension N ≥ n, and ϕ (Γ) is embedded in M N (F ) as a corner, see Section 2.1. We will usually refer to stability in the normal sense (where an increase in dimension is not allowed) as strict stability. ...
... Finally, we provide counterexamples for flexible stablity of groups, by a construction in the spirit of Rolli [15] who proved that free groups are not stable in the operator norm, in addition to the proof in [4] that free groups are not flexibly stable in permutations. Theorem 1.3. ...
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... Our proof will involve a bounded generation argument for stability that was pioneered in [10]. We will only use a simple version thereof, closer to the one from [5]. Recall that is said to be boundedly generated by the collection of subgroups H if there exists k ≥ 1 such that the sets {H 1 · · · H k : H i ∈ H} cover . ...
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... The corollary immediately implies that F, F and T are not uniformly Sapproximable, and that F and T are uniformly S-stable. Since F has infinite abelianization, it follows from [5,Theorem 1.4] that it is not uniformly S-stable. However the corollary together with [5,Theorem 1.2] implies that it is flexibly uniformly S-stable; that is, every uniform asymptotic homomorphism is uniformly close to a sequence of homomorphisms taking values in a symmetric group of slightly larger degree. ...
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... Though it is not clear that d H satisfies the triangle inequality, it does (see[BC22]). 3 Note that in [CL23a; CL23b] the L 1 -defect and L 1 -distance between functions is used. In the context of this paper it makes the calculations nicer to use the L ∞ -setup, so we chose it.4 ...
... In the context of this paper it makes the calculations nicer to use the L ∞ -setup, so we chose it.4 This is commonly called pointwise flexible stability in permutations (see the discussions in[BC22]). ...
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A long standing problem asks whether every group is sofic, i.e., can be separated by almost-homomorphisms to the symmetric group Sym(n). Similar problems have been asked with respect to almost-homomorphisms to the unitary group U(n), equipped with various norms. One of these problems has been solved for the first time in [De Chiffre, Gelbsky, Lubotzky, Thom, 2020]: some central extensions Γ~\widetilde{\Gamma} of arithmetic lattices Γ\Gamma of Sp(2g,Qp)Sp(2g,\mathbb{Q}_p) were shown to be non-Frobenius approximated by almost homomorphisms to U(n). Right after, it was shown that similar results hold with respect to the p-Schatten norms in [Lubotzky, Oppenheim, 2020]. It is natural, and has already been suggested in [Chapman, Lubotzky, 2024] and [Gohla, Thom, 2024], to check whether the Γ~\widetilde{\Gamma} are also non-sofic. In order to show that they are (also) non-sofic, it suffices: (a) To prove that the permutation Cheeger constant of the simplicial complex underlying Γ\Gamma is positive, generalizing [Evra, Kaufman, 2016]. This would imply that Γ\Gamma is stable. (b) To prove that the (flexible) stability of Γ\Gamma implies the non-soficity of Γ~\widetilde{\Gamma}. Clause (b) was proved by Gohla and Thom. Here we offer a more algebraic/combinatorial treatment to their theorem.
... Our proof will involve a bounded generation argument for stability that was pioneered in [BOT13]. We will only use it a simple version thereof, closer to the one from [BC20]. Recall that Γ is said to be boundedly generated by the collection of subgroups H if there exists k ≥ 1 such that the sets {H 1 · · · H k : H i ∈ H} cover Γ. ...
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We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms. These include lamplighters ΓΛ\Gamma \wr \Lambda where Λ\Lambda is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups F,F,TF, F', T and V. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.
... It is of some interest in geometry as well as algebra to consider maps which are almost homomorphisms ϕ: G → H in the sense that ∂(ϕ(gh), ϕ(g)ϕ(h)) < ε uniformly for all g, h ∈ G. This has been studied in various situations, see for example [1,3,6,7,10]) and the references therein. Note that ...
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... It essentially measures how much one Schreier graph needs to be changed to get to the other graph. For more on that, see Section 4 of [CL23] and the Γ-graph notion in [BC22]. ...
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... So, the linearity test is Algorithm 1 under the restriction that the range of f is Sym(2) and not any Sym(N ). In [BC22], it is proved that the multiplication table presentation of any finite group is ρ-homomorphism stable with linear rate ρ(ε) = 3000ε. This generalizes [BLR93], though with worse parameters. ...
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... It is a new form of a statement asserting that almost commuting unitaries are close to unitaries, that does not seem to be comparable with existing results, even when ϕ is a trace. Comparing with the probabilistic results in [2] for permutation actions (that Michael Chapman kindly pointed out to me) raises the question whether there is a form of Corollary 5 that is valid for arbitrary amenable groups and not just Z/nZ × Z/mZ. ...
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... Given a graph G ∈ G S (n), Local Statistics Matcher computes an approximation of the distribution N G,r (for a large enough r, independent of n), which we denote by N Empirical and N G ′ ,r are at most δ-far from each 7 That is, the proof in [10] provides an explicit algorithm that, given a graph G ∈ G S (n) that is accepted by SAS XY=YX k(ε) with high probability, produces a graph G ′ ∈ GSol E (n) close to G. 8 Readers familiar with Benjamini-Schramm convergence will recognize a close connection between Local Statistics Matcher and the Benjamini-Schramm metric. 9 This ball is the graph whose vertex set V consists of all vertices wx of G, where w is a word of length at most r over S ± , and whose edge set consists of all the edges of G that start and end in V . ...
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We introduce a simple two-player test which certifies that the players apply tensor products of Pauli σX and σZ observables on the tensor product of n EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive of the optimal must be poly(ε)-close, in the appropriate distance measure, to the honest n-qubit strategy. The test involves 2n-bit questions and 2-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld. As applications of our result we give (i) the first robust self-test for n EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completeness-soundness gap independent of system size; (iii) a robust protocol for verifiable delegated quantum computation with a constant number of quantum polynomial-time provers sharing entanglement.
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We prove an inverse theorem for the Gowers U2U^2-norm for maps GMG\to\mathcal M from an countable, discrete, amenable group G into a von Neumann algebra M\mathcal M equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm \Vert\cdot\Vert. We use this result to prove a stability result for unitary-valued ε\varepsilon-representations GU(M)G\to\mathcal U(\mathcal M) with respect to \Vert\cdot \Vert.
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We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations. Our result extends to k-tuples of almost commuting permutations, for any given k, and allows restrictions, for instance, to even permutations.
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this paper, we will show that non-elementary word-hyperbolic groups have infinite dimensional second bounded cohomology. In order to state our main result, we recall that `
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