Groups, Geometry, and Dynamics
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Published online: 2007-09-30
Reflections on the residual finiteness of one-relator groupsGilbert Baumslag, Charles F. Miller III and Douglas Troeger (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