Remark on approximability of groups
Let G be a group generated by \(a_ 1,a_ 2,...,a_ S\); \(G_ N\) be the set of elements in G presented as words of lengths \(\leq N\). A group G is N-isomorphic to a group H generated by \(b_ 1,b_ 2,...,b_ S\) if there exists a one-to-one correspondence \(\phi\) between \(G_ N\) and \(H_ N\) such that \(\phi (g_ 1g_ 2)=\phi (g_ 1)\phi (g_ 2)\) whenever \(g_ 1,g_ 2,g_ 1g_ 2\in G_ N\). A finitely generated group G is called locally-approximable by finite groups, if for every N the group G is N-isomorphic to a finite group. Proposition. (a) a finitely generated residually finite group is locally- approximable by finite groups, (b) a finitely presented group, that is locally-approximable by finite groups, is residually finite. Theorem. A finitely generated group G has a free-approximable action if and only if G is locally-approximable by finite groups. The action of the group G on a space with measure (X,\(\mu)\) is called approximable, if for every \(g_ 1,g_ 2,...,g_ n\in G\) and \(\epsilon >0\) there exist elements \(k_ 1,k_ 2,...,k_ n\) in the completion [G] generated by a finite subgroup of transformations such that \(\mu\) \(\{\) \(x: g_ ix=k_ ix,i=1,2,...,n\}>1-\epsilon\). We have a free-approximable action of G if it is free and the elements \(k_ 1,k_ 2,...,k_ n\) may be taken such that \(gr(k_ 1,k_ 2,...,k_ n)\) is free.
finitely generated residually finite group
scientific article; zbMATH DE number 3901019