Property of defect diminishing and stability

Abstract

Let $\Gamma$ be a group and $\mathscr{C}$ a class of groups endowed with bi-invariant metrics. We say that $\Gamma$ is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $(G,d) \in \mathscr{C}$, is $\delta_\varepsilon$-close to a homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon < C \varepsilon$ for some $C$ we say that $\Gamma$ is $ \mathscr{C} $-stable with a linear rate. We say that $\Gamma$ 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.