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Free subgroups of certain one-relator groups defined by positive words

  • Gilbert Baumslag (a1)

Let ℒ be the class of those groups G which can be presented in the form

where u and v are positive words in the given generators. Here a word w is termed positive if only non-negative powers of a, b,…, c occur in w. If each generator occurs with exponent sum zero in uv-1, we term the ℒ-group G a -group. This class contains, in particular, the class X of those groups G which can be presented in the form

where u and v are positive words, and where [u, v] is the commutator uvu-1v-1.

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(1) G. Baumslag On generalised free products. Math. Z. 78 (1962), 423438.

(2) G. Baumslag On the residual finiteness of generalised free products of nilpotent groups. Trans. Amer. Math. Soc. 106 (1963), 193209.

(3) G. Baumslag Positive one-relator groups. Trans. Amer. Math. Soc. 156 (1971), 165183.

(4) G. Baumslag , E. Dyer and A. Heller The topology of discrete groups. J. Pure and Appl. Algebra, 16 (1980), 147.

(5) K. N. Frederick The hopfian property for a class of fundamental groups. Comm. Pure Appl. Math. 16 (1963), 18.

(9) W. Magnus Das Identitatsproblem fur Gruppen mit einer definierenden Relation. Math Annalen, 106 (1932), 295307.

(10) W. Magnus Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring. Math. Annalen, 111 (1935), 259280.

(11) W. Magnus Uber freie Faktorgruppen und freie Untergruppen gegebener Gruppen. Monatsh. Math. 47 (1939), 105115.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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