Mathematical Proceedings of the Cambridge Philosophical Society

Conjugacy separability of certain Bianchi groups and HNN extensions

J. S. WILSON a1 and P. A. ZALESSKII a2
a1 School of Mathematics and Statistics, University of Birmingham
a2 Institute of Engineering Cybernetics, Academy of Sciences of Belarus, Minsk, Belarus


The Bianchi groups are the groups PSL2([script O]d), where [script O]d denotes the ring of integers of the field [open face Q] ([surd radical]−d) for each square-free positive integer d. These groups have long been of interest, not only because of their intrinsic interest as abstract groups, but also because they arise naturally in number theory and geometry. For a discussion of their algebraic properties we refer the reader to Fine [4]. Among the groups PSL2(R), with R the ring of integers of an algebraic number field, they are distinguished by the nature of their normal subgroup structure. It was shown by Serre [20] that if R is not isomorphic to [open face Z] or [script O]d, then for every normal subgroup K of SL2(R) there is an ideal I of R such that the image in SL2(R)/K of the kernel of the natural map from SL2(R) to SL2(R/I) is central and isomorphic to a subgroup of the group of roots of 1 in R. On the other hand, the group PSL2([open face Z]) and the Bianchi groups have many subgroups of finite index which are not of the above type: this follows easily from the fact that PSL2([open face Z]) is a free product of a group of order 2 and a group of order 3, and the fact, proved by Grunewald and Schwermer [6], that each Bianchi group has a normal subgroup of finite index which can be mapped epimorphically to a non-abelian free group.

(Received June 20 1996)