Sofic groups: graph products and graphs of groups. (Q471287): Difference between revisions

For a finite set \(A\), let \(S(A)\) be the group of all its permutations. For any positive \(\varepsilon\), two elements \(f_1,f_2\) of \(S(A)\) are \(\varepsilon\)-similar, if they are coincidend \(f_1(a)=f_2(a)\) at most at \(\varepsilon|A|\) points. If \(\varepsilon\geq 1\) the condition is trivial. A group \(G\) is called sofic, if for each \(\varepsilon\in(0,1)\) and any finite set \(F\subseteq G\), \(G\) admits a special \((F,\varepsilon)\)-quasi-action (Definition 1.3), i.e. there is a function \(\varphi\colon G\to S(F)\) with properties: [a] \(\varphi(1)=1\); [b] \(\varphi(g^{-1})=\varphi(g)^{-1}\), \(\forall g\in G\); [c] for any \(g\neq 1\), \(\varphi(g)\) has no fixed points; [d] for any \(g_1,g_2\in F\) the elements \(\varphi(g_1g_2)\) and \(\varphi(g_1)\varphi(g_2)\) are \((F,\varepsilon)\)-similar. In the paper under review, the authors prove that: 1. The graph product of sofic groups is sofic (Theorem 1.1 and \S2); 2. The fundamental group (Definition 3.1) of a graph of groups is sofic if each vertex group is sofic and each edge group is amenable (Theorem 1.2 and \S3).