PROPERTY OF DEFECT DIMINISHING AND STABILITY

Let Γ be a group and C a class of groups endowed with bi-invariant metrics. We say that Γ is C-stable if every ε-homomorphism Γ → G, (G, d) ∈ C, is δε-close to a homomorphism, δε → 0 when ε → 0. If δε < Cε for some C we say that Γ is C-stable with a linear rate. We say that Γ has the property of defect diminishing if any asymptotic homomorphism can be changed a little to make errors essentially better. We show that the defect diminishing is equivalent to the stability with a linear rate. © 2022, Hacettepe University. All rights reserved.

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