Conjugacy separability of HNN-extensions of abelian groups

摘要

1. Introduction. A group G is said to be conjugacy separable if, for each pair ofnoncon-jugate elements x, y e 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, 8,10,11]. But the conjugacy separability of HNN-extensions was not much known, because one of the simplest type of HNN-extensions, the Banmslag-Solitar group, (tb,t:t~1b2t = b2) is not even residually finite. However Andreadakis, Raptis and Varsos [1] characterized the residual finiteness of HNN-exten-sions of abelian groups. In [5] Kim, McCarron and Tang characterized the conjugacy separability of 1-relator groups of the form <6, t: t~abft'1 = 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 yclic associated subgroups to be residually finite and nc. The purpose of this paper is to tudy the conjugacy separability of those HNN-extensions. We show that if either the ssociated subgroups intersect trivially or the associated subgroups have a common bgroup of the same index then the HNN-extensions are conjugacy separable. Applying is result we completely characterize the conjugacy separability of HNN-extensions of lian groups with cyclic associated subgroups (Theorem 3.6).