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On the residual finiteness and other properties of (relative) one-relator groups

Author: Stephen J. Pride
Journal: Proc. Amer. Math. Soc. 136 (2008), 377-386
MSC (2000): Primary 20E26, 20F05; Secondary 20F10, 57M07
Published electronically: October 25, 2007
MathSciNet review: 2358474
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Abstract: A relative one-relator presentation has the form $ \mathcal{P} = \langle \mathbf{x}, H; R \rangle$ where $ \mathbf{x}$ is a set, $ H$ is a group, and $ R$ is a word on $ \mathbf{x}^{\pm 1} \cup H$. We show that if the word on $ \mathbf{x}^{\pm 1}$ obtained from $ R$ by deleting all the terms from $ H$ has what we call the unique max-min property, then the group defined by $ \mathcal{P}$ is residually finite if and only if $ H$ is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form $ \mathcal{P}$ (Theorem 6).

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  • 1. R. B. J. T. Allenby and C. Y. Tang, Residual finiteness of certain 1-relator groups: extensions of results of Gilbert Baumslag, Math. Proc. Camb. Phil. Soc. $ \mathbf{97}$ (1985), 225-230. MR 0771817 (86k:20029)
  • 2. G. Baumslag, Residually finite one-relator groups, Bull. Amer. Math. Soc. $ \mathbf{73}$ (1967), 618-620. MR 0212078 (35:2953)
  • 3. G. Baumslag, Free subgroups of certain one-relator groups defined by positive words, Math. Proc. Camb. Phil. Soc. $ \mathbf{93}$ (1985), 247-251. MR 691993 (84i:20028)
  • 4. G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. $ \mathbf{68}$ (1962), 199-201. MR 0142635 (26:204)
  • 5. G. Baumslag, A. Miasnikov and V. Shpilrain, Open problems in combinatorial and geometric group theory,
  • 6. W. A. Bogley and S. J. Pride, Aspherical relative presentations, Proc. Edin. Math. Soc. $ \mathbf{35}$ (1992), 1-39. MR 1150949 (93d:57004)
  • 7. O. Bogopolski, A. Martino, O. Maslakova and E. Ventura, The conjugacy problem is solvable for free-by-cyclic groups, Bull. London Math. Soc. $ \mathbf{38}$ (2006), 787-794. MR 2268363
  • 8. K. S. Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math. $ \textbf{90}$, (1987), 479-504. MR 914847 (89e:20060)
  • 9. V. Egorov, The residual finiteness of certain one-relator groups, Algebraic Systems, Ivanov. Gos. Univ., Ivanovo (1981), 100-121. MR 745301 (85i:20035)
  • 10. J. Howie and S. J. Pride, A spelling theorem for staggered generalized 2-complexes, with applications, Invent. Math. 76 (1984), 55-74. MR 739624 (85k:20103)
  • 11. Kourovka Notebook $ \mathbf{15}$ (2002).
  • 12. W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory (Second Edition), Dover, New York, 1976.
  • 13. J. Meier, Geometric invariants for Artin groups, Proc. London Math. Soc. (3) 74 (1997), 151-173. MR 1416729 (97h:20049)
  • 14. C. F. Miller III, On group-theoretic decision problems and their classification, Annals of Mathematics Studies 68, Princeton University Press, 1971.
  • 15. S. Meskin, Nonresidually finite one-relator groups, Trans. Amer. Math. Soc. $ \mathbf{164}$ (1972), 105-114. MR 0285589 (44:2807)
  • 16. B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. $ \mathbf{74}$ (1968), 568-571. MR 0222152 (36:5204)
  • 17. S. J. Pride, Star-complexes, and the dependence problems for hyperbolic complexes, Glasgow Math. J. 30 (1988), 155-170. MR 942986 (89k:20049)
  • 18. J.-P. Serre, Trees, Springer-Verlag, Berlin Heidelberg New York, 1980.
  • 19. D. Wise, The residual finiteness of positive one-relator groups, Comment. Math. Helv. $ \mathbf{76}$ (2001), 314-338. MR 1839349 (2002d:20043)
  • 20. D. Wise, Residual finiteness of quasi-positive one-relator groups, J. London Math. Soc. (2) $ \mathbf{66}$ (2002), 334-350. MR 1920406 (2003f:20043)

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Additional Information

Stephen J. Pride
Affiliation: Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, United Kingdom

Keywords: Residual finiteness, one-relator group, relative presentation, (power) conjugacy problem, asphericity, unique max-min property, 2-complex of groups, covering complex
Received by editor(s): June 5, 2006
Published electronically: October 25, 2007
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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