Separability properties of certain tree products of groups
The authors study double coset separability and conjugacy separability properties of an amalgamated free product with amalgamating subgroup which is central in one of the factors. A group \(G\) said to be conjugacy separable if any two non-conjugate elements of \(G\) are non-conjugate in some finite quotient of \(G\). A subgroup \(H\) of a group \(G\) is said to be separable if for every \(g\not\in H\) there is a finite quotient of \(G\), where the image of \(H\) does not contain the image of \(g\); equivalently \(H\) is separable if \(H\) is closed in the profinite topology of \(G\). The authors call a group \(G\) central subgroup separable if every finitely generated central subgroup is separable in \(G\). Among others the following results are proved in the paper. \(\bullet\) Let \(A\), \(B\) be conjugacy separable and central subgroup separable groups and \(H\) a finitely generated common central subgroup of \(A\) and \(B\). Then the free amalgamated product \(G=A*_HB\) is conjugacy separable. \(\bullet\) Let \(G\) be a tree product of finitely many central subgroup separable and conjugacy separable groups amalgamating finitely generated central edge groups. Then \(G\) is conjugacy separable. -- As immediate consequence of this the authors deduce that a tree product of polycyclic groups amalgamating finitely generated central edge groups is conjugacy separable.
scientific article; zbMATH DE number 1797475