Stability and invariant random subgroups
By Oren Becker, Alexander Lubotzky, Andreas Thom. Duke mathematical journal, 2018
66 citations recorded by the source corpus.
Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for every $g,h\in\Gamma$, $\lim_{k\rightarrow\infty}d_{n_k}( \rho_{n_k}(gh),\rho_{n_k}(g)\rho_{n_k}(h))=0$) is close to an actual homomorphism $\varphi_{n_k} \colon\Gamma\rightarrow\operatorname{Sym}(n_k)$. Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and P\u{a}unescu showed the same for abelian groups and raised many questions, especially about P-stability of amenable groups. We develop P-stability in general, and in particular for amenable groups. Our main tool is the theory of invariant random subgroups (IRS), which enables us to give a characterization of P-stability among amenable groups, and to deduce stability and instability of various families of amenable groups.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/89d6bb3b946ceadba6f76a24d0d662e0898ffccc
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Citations in the corpus
- STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS (2017, methodology): “This method has failed so far for two reasons: (1) it is very difficult to prove stability directly and (2) even in the case where stability was proven (see, for example, [4] and the references therein as well as [8]), it was proven in a way that completely classifies asymptotic homomorphism and it ”