On the Residual Finiteness of Polygonal Products of Nilpotent Groups

Abstract

In general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

Cyclic Subgroup Separability of Generalized Free Products

“…Finally, we apply our result to a special kind of generalized free products, known as polygonal products of groups, and we generalize some results found in [2], [11].…”

On the Residual Finiteness of Certain Polygonal Products of Free Groups

“…The proofs of next two results are very similar to Lemma 4.5 and Theorem 4.6 in [8]. Here we use Z i (G) to denote the i-th term of the upper central series of G with Z 1 (G), the center of G.…”

“…And they gave an example of a polygonal product of finitely generated nilpotent -or free-groups which is not RF . However, in [6,8], Tang and Kim showed that certain polygonal products of finitely generated nilpotent groups are RF or π c . Then, Allenby [1] constructed polygonal products of nilpotent groups which are not RF , hence untidy conditions in [8] can not be removed.…”

“…However, in [6,8], Tang and Kim showed that certain polygonal products of finitely generated nilpotent groups are RF or π c . Then, Allenby [1] constructed polygonal products of nilpotent groups which are not RF , hence untidy conditions in [8] can not be removed. In [5,7], Kim proved that polygonal products of more than three polycyclicby-finite groups amalgamating central subgroups with trivial intersections are π c and conjugacy separable, hence they are RF .…”

On polygonal products of finitely generated abelian groups

“…In [3], Allenby and Tang proved that polygonal products of four finitely generated free abelian groups, amalgamating cyclic subgroups with trivial intersections, is residually finite. Kim and Tang [9] showed that certain polygonal products of four nilpotent groups, amalgamating cyclic subgroups with trivial intersections, are residually finite. In this paper, we prove that polygonal products of more than four polycyclic-by-finite groups amalgamating any subgroups, contained in the centres of their vertex groups, with trivial intersections are 7r c (Theorem 2.11), hence they are residually finite.…”