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Conjugacy Separability and Free Products of Groups with Cyclic Amalgamation
Author(s) -
Ribes L.,
Segal D.,
Zalesskii P. A.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006267
Subject(s) - conjugacy class , free product , separable space , mathematics , quotient , nilpotent , group (periodic table) , pure mathematics , conjugacy problem , product (mathematics) , combinatorics , quotient group , free group , finite group , cyclic group , abelian group , mathematical analysis , physics , geometry , quantum mechanics
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 G 1 and G 2 amalgamating a cyclic subgroup is again conjugacy separable. The main result of [ 15 ] states that this is the case if G 1 and G 2 are free‐by‐finite or finitely generated and nilpotent‐by‐finite. We show here that the same conclusion holds for groups G 1 and G 2 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.)