Residual Nilpotency of Groups

Abstract

Residually finite groups appear in low dimensional topology. In the latter half of the twentieth century, it is shown that the fundamental groups of the complement of fibered knots are residually finite. Free groups are residually finite, too. Free groups have also other residual properties, i.e., residual nilpotency and residual solvability. Every residually nilpotent group is residually solvable, as every nilpotent group is solvable. Magnus, Karrass and Solitar show these properties of free groups in their book by algebraic methods. In the knot theory, it is well known that the fundamental group of the complement of a knot (= the knot group) is not residually nilpotent.
Recently, Hashiguchi proved that the free products of the order p cyclic group and the order q cyclic group are residually solvable if p and q are relatively prime. This is shown by the fact that the knot groups of torus knots are residually solvable. This method cannot be applied to the case p = q because (p, p)-torus knot does not exist.
In this paper, we will prove that the free product of r copies of the order p cyclic groups is residually nilpotent by applying Magnus-Karrass-Solitar’s method when p is a prime number and r is greater than 1.