On polygonal products of finitely generated abelian groups

Abstract

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

Tree Products of Conjugacy Separable Groups

“…By Lemmas 4-6, E satisfies the hypotheses of Lemmas 8,10,13,26, and 32 and this implies that E has the properties (i)-(v). Now by assumption, G n is conjugacy separable and subgroup separable.…”

“…By induction, E is c.s. By Lemmas 4-6, E satisfies the hypotheses of Lemmas 8,10,13,26,and 32. Let x y ∈ G be such that x G y.…”

On the Residual Finiteness of Certain Polygonal Products of Free Groups

“…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 . Allenby [2] showed, using the criterion in [6], that polygonal products of four polycyclic-by-finite groups, amalgamating normal subgroups, are π c .…”

“…Allenby [2] showed, using the criterion in [6], that polygonal products of four polycyclic-by-finite groups, amalgamating normal subgroups, are π c . Subgroup separability of polygonal 462 G. KIM products is also considered in [5]. Hence, for polygonal products of nilpotent groups, most of important residual properties are known.…”

Cyclic Subgroup Separability of Certain Graph Products of Subgroup Separable Groups

“…This paper is motivated by the works of Kim [11], Allenby [1], and Wong and Wong [25]. We will give a generalization of the Allenby's Theorem [1, Theorem C], which is a generalization of the Kim's Theorem [11,Theorem 2.11].…”