Nonresidually finite one-relator groups

Meskin, Stephen

doi:10.1090/S0002-9947-1972-0285589-5

Nonresidually finite one-relator groups
HTML articles powered by AMS MathViewer

by Stephen Meskin PDF

Trans. Amer. Math. Soc. 164 (1972), 105-114 Request permission

Abstract:

The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form ${u^{ - 1}}{v^l}u{v^m}$ which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.

References

Gilbert Baumslag, On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193–209. MR 144949, DOI 10.1090/S0002-9947-1963-0144949-8
Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201. MR 142635, DOI 10.1090/S0002-9904-1962-10745-9
P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595–622. MR 110750, DOI 10.1112/plms/s3-9.4.595
Graham Higman, A finitely related group with an isomorphic proper factor group, J. London Math. Soc. 26 (1951), 59–61. MR 38347, DOI 10.1112/jlms/s1-26.1.59
A. G. Kuroš, Teoriya grupp, Third augmented edition, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0249495
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Paul E. Schupp, On the substitution problem for free groups, Proc. Amer. Math. Soc. 23 (1969), 421–423. MR 245657, DOI 10.1090/S0002-9939-1969-0245657-6
Peter Stebe, Residual finiteness of a class of knot groups, Comm. Pure Appl. Math. 21 (1968), 563–583. MR 237621, DOI 10.1002/cpa.3160210605
N. J. Wicks, Commutators in free products, J. London Math. Soc. 37 (1962), 433–444. MR 142610, DOI 10.1112/jlms/s1-37.1.433