On the residual finiteness and other properties of (relative) one-relator groups
HTML articles powered by AMS MathViewer
- by Stephen J. Pride PDF
- Proc. Amer. Math. Soc. 136 (2008), 377-386 Request permission
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).References
- R. B. J. T. Allenby and C. Y. Tang, Residual finiteness of certain $1$-relator groups: extensions of results of Gilbert Baumslag, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 2, 225–230. MR 771817, DOI 10.1017/S0305004100062782
- Gilbert Baumslag, Residually finite one-relator groups, Bull. Amer. Math. Soc. 73 (1967), 618–620. MR 212078, DOI 10.1090/S0002-9904-1967-11799-3
- Gilbert Baumslag, Free subgroups of certain one-relator groups defined by positive words, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 2, 247–251. MR 691993, DOI 10.1017/S0305004100060527
- 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
- G. Baumslag, A. Miasnikov and V. Shpilrain, Open problems in combinatorial and geometric group theory, http://zebra.sci.ccny.cuny.edu/web/nygtc/problems/
- W. A. Bogley and S. J. Pride, Aspherical relative presentations, Proc. Edinburgh Math. Soc. (2) 35 (1992), no. 1, 1–39. MR 1150949, DOI 10.1017/S0013091500005290
- O. Bogopolski, A. Martino, O. Maslakova, and E. Ventura, The conjugacy problem is solvable in free-by-cyclic groups, Bull. London Math. Soc. 38 (2006), no. 5, 787–794. MR 2268363, DOI 10.1112/S0024609306018674
- Kenneth S. Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math. 90 (1987), no. 3, 479–504. MR 914847, DOI 10.1007/BF01389176
- V. Egorov, The residual finiteness of certain one-relator groups, Algebraic systems, Ivanov. Gos. Univ., Ivanovo, 1981, pp. 100–121 (Russian). MR 745301
- J. Howie and S. J. Pride, A spelling theorem for staggered generalized $2$-complexes, with applications, Invent. Math. 76 (1984), no. 1, 55–74. MR 739624, DOI 10.1007/BF01388491
- Kourovka Notebook $\mathbf {15}$ (2002).
- W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory (Second Edition), Dover, New York, 1976.
- John Meier, Geometric invariants for Artin groups, Proc. London Math. Soc. (3) 74 (1997), no. 1, 151–173. MR 1416729, DOI 10.1112/S0024611597000063
- C. F. Miller III, On group-theoretic decision problems and their classification, Annals of Mathematics Studies 68, Princeton University Press, 1971.
- Stephen Meskin, Nonresidually finite one-relator groups, Trans. Amer. Math. Soc. 164 (1972), 105–114. MR 285589, DOI 10.1090/S0002-9947-1972-0285589-5
- B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968), 568–571. MR 222152, DOI 10.1090/S0002-9904-1968-12012-9
- Stephen J. Pride, Star-complexes, and the dependence problems for hyperbolic complexes, Glasgow Math. J. 30 (1988), no. 2, 155–170. MR 942986, DOI 10.1017/S0017089500007175
- J.-P. Serre, Trees, Springer-Verlag, Berlin Heidelberg New York, 1980.
- Daniel T. Wise, The residual finiteness of positive one-relator groups, Comment. Math. Helv. 76 (2001), no. 2, 314–338. MR 1839349, DOI 10.1007/PL00000381
- Daniel T. Wise, Residual finiteness of quasi-positive one-relator groups, J. London Math. Soc. (2) 66 (2002), no. 2, 334–350. MR 1920406, DOI 10.1112/S0024610702003538
Additional Information
- Stephen J. Pride
- Affiliation: Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, United Kingdom
- Email: sjp@maths.gla.ac.uk
- Received by editor(s): June 5, 2006
- Published electronically: October 25, 2007
- Communicated by: Jonathan I. Hall
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 377-386
- MSC (2000): Primary 20E26, 20F05; Secondary 20F10, 57M07
- DOI: https://doi.org/10.1090/S0002-9939-07-09160-5
- MathSciNet review: 2358474