The Free Product of Residually Finite Groups Amalgamated Along Retracts is Residually Finite

Abstract

Abstract.It is shown that residual finiteness is preserved by the generalized free product provided that the amalgamated subgroups are retracts of their respective factors. This result is applied to knot groups. The outcome is that the question of residual finiteness for knot groups need only be answered for prime knots.

Cyclic Subgroup Separability of Generalized Free Products

“…Next result is a generalization of Boler and Evans' result [5] and Allenby and Gregorac [1] mentioned the result for the generalized free product of two TT C groups amalgamating a retract. A subgroup H of a group G is called a retract if there exists G\ < G such that G = G\H and Gi H // = 1.…”

Virtual retraction properties in groups

“…, and the group G, given by ( 3), is naturally isomorphic to the semidirect product (K 1 * K 2 ) R, where R normalizes each K i , i = 1, 2 (the action of R on K 1 is induced from P , and the action of R on K 2 comes from the composition of ϕ with the action of S on K 2 in Q). This isomorphism was first observed by Boler and Evans [11], who used it to prove that G is residually finite whenever P and Q are residually finite. Both decompositions of G as an amalgamated free product and as the semidirect product will be useful for us below.…”

The root class residuality of the tree product of groups with amalgamated retracts

“…Болер и Эванс [20] установили, что свободное произведение двух финитно аппроксимируемых групп с объединенными ретрактами является финитно аппроксимируемой группой. В [21] получен аналогичный результат для случая аппроксимируемости конечными p-группами.…”