Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

We study the problem of finite approximability with respect to conjugacy of amalgamated free products by a normal subgroup and prove the following assertions. A) IfG is the amalgamated free productG=G _1*H G ₂ of polycyclic groupsG ₁ andG ₂ by a normal subgroupH, whereH is an almost free Abelian group of rank 2, thenG is finitely approximate with respect to conjugacy. B) (i) IfG ₁ =G ₂ =L is a polycyclic group andG=G _1*H G ₂ is the amalgamated product of two copies of the groupL by a normal subgroupH, thenG is finitely approximable with respect to conjugacy. (ii) IfG is an amalgamated free productG=G _1*H G ₂ of polycyclic groupsG ₁ andG ₂ by a normal subgroupH, whereH is central inG ₁ orG ₂, thenG is finitely approximable with respect to conjugacy.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 525–535, October, 1995.This work was partially supported by the INTAS-94-3420 Grant Geometric Theory of Groups.