On the Conjugacy Separability of Certain Graphs of Groups

Abstract

A group G is said to be conjugacy separable if, whenever x and y are nonconjugate elements of G, there exists a finite quotient group of G in which the images of x and y are not conjugate. The importance of this w x notion was pointed out by Mal’cev, who proved in 7 that if a finitely presented group G is conjugacy separable, then G has solvable the conjugacy problem, that is, there exists an algorithm to decide whether or not any two given elements of G are conjugate. Žw x. Žw x. Polycyclic-by-finite groups 5, 12 , free groups 13, 16 , free-by-finite w x groups 3 are conjugacy separable. It is known that free products of Žw x. conjugacy separable groups are again conjugacy separable 16, 13 . However, the property is not preserved in general by the formation of free products with amalgamation or HNN-extensions. For example, one of the simplest type of HNN-extensions, the Baum2 y1 7 13: slag]Solitar group a, t ¬ t a t s a is not even residually finite; note that if a group G is conjugacy separable then it must be residually finite. w x C. F. Miller in 8 gave examples of HNN-extensions which are residually finite and not conjugacy separable. In this paper we are concerned with the conjugacy separability of fundamental groups of certain graphs of groups. Our main results are: