Conjugacy separability of certain free products with amalgamation

Abstract

Let G be a group. An element g of G is called conjugacy distinguished or c.d. in G if and only if given any element h of G either h is conjugate to g or there is a homomorphism ξ \xi from G onto a finite group such that ξ ( h ) \xi (h) and ξ ( g ) \xi (g) are not conjugate in ξ ( G ) \xi (G) . Following A. Mostowski, a group G is conjugacy separable or c.s. if and only if every element of G is c.d. in G. In this paper we prove that every element conjugate to a cyclically reduced element of length greater than 1 in the free product of two free groups with a cyclic amalgamated subgroup is c.d. We also prove that a group formed by adding a root of an element to a free group is c.s.