On the Residual Finiteness of Certain Polygonal Products

Abstract

We give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R, and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R.

On polygonal products of finitely generated abelian groups

“…Then G is X * Y-separable whenever H is 1i.T. We note that the above two results are generalisations of Theorem 3.4 in [3].…”

“…Using their result, Brunner, Frame, Lee and Wielenberg [5] determined all torsionfree subgroups of finite index in the Picard group PSL{2, Z [i]). 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.…”

“…We also find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable (Theorem 3.2). Briefly polygonal products of groups can be described as follows [3]: Let P be a polygon. Assign a group G v to each vertex v and a group G c to each edge e of P.…”

On the Residual Finiteness of Certain Polygonal Products of Free Groups

“…Briefly, polygonal products of groups can be described as follows [3]: Let P be a polygon. Assign a group G v for each vertex v and a group G e for each edge e of P .…”

“…Since a polygonal product can appear as a subgroup of a group, and then the residual properties of the polygonal product determine the residual properties of the whole group [6, Example 1.1], we are interested in the residual properties of polygonal products. In [3], Allenby and Tang proved that polygonal products of four finitely generated free abelian groups, amalgamating cyclic subgroups with trivial intersections, are residually finite (RF ). And they gave an example of a polygonal product of finitely generated nilpotent -or free-groups which is not RF .…”