Groups, Geometry, and Dynamics
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Reflections on the residual finiteness of one-relator groups
Gilbert Baumslag (1), Charles F. Miller III (2) and Douglas Troeger (3)
(1) Department of Mathematics, The City College of CUNY, Convent Avenue and 138th Street, NY 10031, NEW YORK, UNITED STATES(2) Department of Mathematics, University of Melbourne, 3052, PARKVILLE, VIC, AUSTRALIA
(3) Department of Computer Science, The City College of CUNY, Convent Avenue and 138th Street, NY 10031, NEW YORK, UNITED STATES
Let G = < a, b, … | r = 1 > be a one-relator group equipped with at least two generators. For all w which do not commute with r in the ambient free group on the generators a, b, …, the groups G(r,w) = < a,b, … | r r w = r 2 > are not residually finite and have the same finite images as G. The existence of this family of one-relator groups which are not residually finite reinforces what is becoming more obvious with time, that one-relator groups can be extremely complicated. This not only serves to underline the complexity of one-relator groups but provides us with the opportunity to raise a number of problems about these groups in the hope that they will stimulate further work on the conjugacy and isomorphism problems for one-relator groups as a whole.
Keywords: One-relator group, residually finite, isomorphism problem