Property of defect diminishing and stability

Year 2022, Volume: 31 Issue: 31, 49 - 54, 17.01.2022

Marco Antonio Garcia Morales Lev Glebsky

https://doi.org/10.24330/ieja.1058399

Cited By: 2

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.

Keywords

Group theoretic stability, $~$, $~~$

References

• G. Arzhantseva and L. Paunescu, Almost commuting permutations are near commuting permutations, J. Funct. Anal., 269(3) (2015), 745-757.
• O. Becker and J. Mosheiff, Abelian groups are polynomially stable, Int. Math. Res. Not. IMRN, 20 (2021), 15574-15632.
• M. Burger, N. Ozawa and A. Thom On Ulam stability, Israel J. Math., 193(1) (2013), 109-129.
• V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, Lecture Notes in Mathematics, vol. 2136, Springer, Cham, 2015.
• M. De Chiffre, L. Glebsky, A. Lubotzky and A. Thom, Stability, cohomology vanishing, and nonapproximable groups, Forum Math. Sigma, 8 (2020), Paper No. e18, 37 pp.
• A. Thom, Finitary approximations of groups and their applications, In Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018, vol. III. Invited Lectures, World Sci. Publ., Hackensack, NJ, (2018), 1779-1799