Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nonresidually finite one-relator groups


Author: Stephen Meskin
Journal: Trans. Amer. Math. Soc. 164 (1972), 105-114
MSC: Primary 20.10
DOI: https://doi.org/10.1090/S0002-9947-1972-0285589-5
MathSciNet review: 0285589
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Gilbert Baumslag, On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193-209. MR 26 #2489. MR 0144949 (26:2489)
  • [2] Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199-201. MR 26 #204. MR 0142635 (26:204)
  • [3] P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595-622. MR 22 #1618. MR 0110750 (22:1618)
  • [4] Graham Higman, A finitely related group with an isomorphic proper factor group, J. London Math. Soc. 25 (1951), 59-61. MR 12, 390. MR 0038347 (12:390b)
  • [5] A. G. Kuroš, Theory of groups, 3rd ed., ``Nauka", Moscow, 1967; English transl. of 2nd ed., Chelsea, New York, 1955. MR 17, 124; MR 40 #2740. MR 0249495 (40:2740)
  • [6] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Appl. Math., vol. 13, Interscience, New York, 1966. MR 34 #7617. MR 0207802 (34:7617)
  • [7] Paul E. Schupp, On the substitution problem for free groups, Proc. Amer. Math. Soc. 23 (1969), 421-423. MR 39 #6963. MR 0245657 (39:6963)
  • [8] Peter Stebe, Residual finiteness of a class of knot groups, Comm. Pure Appl. Math. 21 (1968), 563-583. MR 38 #5902. MR 0237621 (38:5902)
  • [9] M. J. Wicks, Commutators in free products, J. London Math. Soc. 37 (1962), 433-444. MR 26 #179. MR 0142610 (26:179)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20.10

Retrieve articles in all journals with MSC: 20.10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0285589-5
Keywords: One-relator groups, residually finite, algorithm, free groups, word form
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society