## 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