Groups, Geometry, and Dynamics

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Volume 1, Issue 3, 2007, pp. 209–219
DOI: 10.4171/GGD/11

Published online: 2007-09-30

Reflections on the residual finiteness of one-relator groups

Gilbert Baumslag, Charles F. Miller III[1] and Douglas Troeger[2]

(1) University of Melbourne, Melbourne, Australia
(2) The City College of 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

Baumslag Gilbert, Miller III Charles, Troeger Douglas: Reflections on the residual finiteness of one-relator groups. Groups Geom. Dyn. 1 (2007), 209-219. doi: 10.4171/GGD/11