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CONJUGACY SEPARABILITY AND FREE PRODUCTS OF GROUPS WITH CYCLIC AMALGAMATION

Published online by Cambridge University Press:  01 June 1998

L. RIBES
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada. E-mail: lribes@math.carleton.ca
D. SEGAL
Affiliation:
All Souls College, Oxford OX1 4AL. E-mail: dan.segal@all-souls.ox.ac.uk
P. A. ZALESSKII
Affiliation:
Institute of Technical Cybernetics, Academy of Sciences of Belarus, 6 Surganov Street, 220605 Minsk, Belarus. E-mail: pz@mat.unb.br
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Abstract

A group G is conjugacy separable if whenever x and y are non-conjugate elements of G, there exists some finite quotient of G in which the images of x and y are non- conjugate. It is known that free products of conjugacy separable groups are again conjugacy separable [19, 12]. The property is not preserved in general by the formation of free products with amalgamation; but in [15] a method was introduced for showing that under certain circumstances, the free product of two conjugacy separable groups G1 and G2 amalgamating a cyclic subgroup is again conjugacy separable. The main result of [15] states that this is the case if G1 and G2 are free-by-finite or finitely generated and nilpotent-by-finite. We show here that the same conclusion holds for groups G1 and G2 in a considerably wider class, including, in particular, all polycyclic-by-finite groups. (This answers a question posed by C. Y. Tang, Problem 8.70 of the Kourovka Notebook [7], as well as two questions recently asked by Kim, MacCarron and Tang in G. Kim, J. MacCarron and C. Y. Tang, ‘On generalised free products of conjugacy separable groups’, J. Algebra 180 (1996) 121–135.)

Type
Notes and Papers
Copyright
The London Mathematical Society 1998

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