Conjugacy separability of HNN-extensions of abelian groups

Abstract

!. Introduction. A group G is said to be conjugacy separable if, for each pair of nonconjugate elements x, y ~ G, there exists a finite homomorphic image G of G such that the images of x, y in G are not conjugate in G. Conjugacy separability of groups is related to the conjugacy problem in the study of groups. In fact, Mostowski [7] proved that finitely presented conjugacy separable groups have solvable conjugacy problem. Since Stebe [9] and Dyer [2] studied the conjugacy separability of generalized free products of free or nilpotent groups, the conjugacy separability of generalized free products of various groups has been studied in [3, 4, 5, 8, I0, 11]. But the conjugacy separability of HNNextensions was not much known, because one of the simplest type of HNN-extensions, the Baumslag-Solitar group, (b, t : t l b Z t = b 3) is not even residually finite. However Andreadakis, Raptis and Varsos [1] characterized the residual finiteness of HNN-extensions of abelian groups. In [5] Kim, McCarron and Tang characterized the conjugacy separability of l-relator groups of the form (b, t : t -~bPP= b~). Hence the conjugacy separability of HNN-extensions of cyclic groups is known (Theorem 2.6). In [6] Kim and Tang gave necessary and sufficient conditions for HNN-extensions of abelian groups with cyclic associated subgroups to be residually finite and ~ . The purpose of this paper is to study the conjugacy separability of those HNN-extensions, We show that if either the associated subgroups intersect trivially or the associated subgroups have a common subgroup of the same index then the HNN-extensions are conjugacy separable. Applying this result we completely characterize the coniugacy separability of HNN-extensions of abelian groups with cyclic associated subgroups (Theorem 3,6).