The free product of residually finite groups amalgamated along retracts is residually finite

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.

The Cyclic Subgroup Separability of Certain Generalized Free Products of Two Groups

“…We note that the criterion of the residual finiteness formulated in this corollary was proved by J. Boler and B. Evans in [3].…”

On a Structural Property of the Groups of Alternating Links

“…Let I be a link in the oriented 3-sphere S 3 and let G = 7n(S 3 -/). If we define a homomorphism from G to the additive group of integers, Z, by sending the homotopy class of any loop to the sum of the linking numbers of this loop with the various components l t of /, then the kernel of this mapping is called the augmentation subgroup of /.…”

“…The class also includes all knot groups from the classical knot tables [14]. Although a product knot need not have this property, it is shown in [3] that the group of a product knot is residually finite if and only if each factor has a residually finite group. If the group of each factor has the above property, so does the group of the product knot, since by H. Schubert's factorization theorem [22], the commutator subgroup of the product knot is a free product of the commutator subgroups of the factors, and it follows from a theorem of K. Gruenberg [8], that the commutator subgroup of such a product knot is residually a finite g-group for almost all primes q.…”

“…Let 5 be a connected surface in S 3 , consisting (as illustrated in Figure 4.1) of a finite number of disks Di, . .…”

“…the spine of 5 is a graph in the plane, i.e., S has a planar graph as a deformation retract, We now consider two primitive flat surfaces S\ and 52 in 5 3 , spanning respective links h and / 2 . We take disks D n and D t 2 from S\ and S 2 respectively, and identify them so that the resulting orientable surface 5 = Si VJ 5 2 spans a link and that S -Si and 5 -5 2 are separated, i.e., so that there exists a 2-sphere T in 5 3 such that T C\ S = D tl (= D i2 ) and each component of S z -T contains points oî S -D n .…”