Local indicability in ordered groups: Braids and elementary amenable groups

Authors:
Akbar Rhemtulla and Dale Rolfsen

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2569-2577

MSC (2000):
Primary 20F36; Secondary 20F60, 06F15

Published electronically:
February 12, 2002

MathSciNet review:
1900863

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Groups which are locally indicable are also right-orderable, but not conversely. This paper considers a characterization of local indicability in right-ordered groups, the key concept being a property of right-ordered groups due to Conrad. Our methods answer a question regarding the Artin braid groups which are known to be right-orderable. The subgroups of pure braids enjoy an ordering which is invariant under multiplication on both sides, and it has been asked whether such an ordering of could extend to a right-invariant ordering of . We answer this in the negative. We also give another proof of a recent result of Linnell that for elementary amenable groups, the concepts of right-orderability and local indicability coincide.

**1.**Emil Artin,*Theorie der Zöpfe*, Abh. Math. Sem. Univ. Hamburg**4**(1925), 47-72.**2.**George M. Bergman,*Right orderable groups that are not locally indicable*, Pacific J. Math.**147**(1991), no. 2, 243–248. MR**1084707****3.**S. D. Brodskiĭ,*Equations over groups, and groups with one defining relation*, Sibirsk. Mat. Zh.**25**(1984), no. 2, 84–103 (Russian). MR**741011****4.**R. G. Burns and V. W. D. Hale,*A note on group rings of certain torsion-free groups*, Canad. Math. Bull.**15**(1972), 441–445. MR**0310046****5.**I. M. Chiswell and P. H. Kropholler,*Soluble right orderable groups are locally indicable*, Canad. Math. Bull.**36**(1993), no. 1, 22–29. MR**1205890**, 10.4153/CMB-1993-004-2**6.**Ching Chou,*Elementary amenable groups*, Illinois J. Math.**24**(1980), no. 3, 396–407. MR**573475****7.**Paul Conrad,*Right-ordered groups*, Michigan Math. J.**6**(1959), 267–275. MR**0106954****8.**Patrick Dehornoy,*From large cardinals to braids via distributive algebra*, J. Knot Theory Ramifications**4**(1995), no. 1, 33–79. MR**1321290**, 10.1142/S0218216595000041**9.**R. Fenn, M. T. Greene, D. Rolfsen, C. Rourke, and B. Wiest,*Ordering the braid groups*, Pacific J. Math.**191**(1999), no. 1, 49–74. MR**1725462**, 10.2140/pjm.1999.191.49**10.**A. M. W. Glass,*Partially ordered groups*, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR**1791008****11.**E. A. Gorin and V. Ja. Lin,*Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids*, Mat. Sb. (N.S.)**78 (120)**(1969), 579–610 (Russian). MR**0251712****12.**Djun M. Kim and Dale Rolfsen,*Ordering groups of pure braids and hyperplane arrangements*, preprint.**13.**Valeriĭ M. Kopytov and Nikolaĭ Ya. Medvedev,*Right-ordered groups*, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1996. MR**1393199****14.**Peter A. Linnell,*Left ordered amenable and locally indicable groups*, J. London Math. Soc. (2)**60**(1999), no. 1, 133–142. MR**1721820**, 10.1112/S0024610799007462**15.**Patrizia Longobardi, Mercede Maj, and Akbar Rhemtulla,*When is a right orderable group locally indicable?*, Proc. Amer. Math. Soc.**128**(2000), no. 3, 637–641. MR**1694872**, 10.1090/S0002-9939-99-05534-3**16.**Roberta Botto Mura and Akbar Rhemtulla,*Orderable groups*, Marcel Dekker, Inc., New York-Basel, 1977. Lecture Notes in Pure and Applied Mathematics, Vol. 27. MR**0491396****17.**L. P. Neuwirth,*The status of some problems related to knot groups*, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Springer, Berlin, 1974, pp. 209–230. Lecture Notes in Math., Vol. 375. MR**0431127****18.**Dale Rolfsen,*Knots and links*, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR**1277811****19.**Dale Rolfsen and Jun Zhu,*Braids, orderings and zero divisors*, J. Knot Theory Ramifications**7**(1998), no. 6, 837–841. MR**1643939**, 10.1142/S0218216598000425**20.**Stan Wagon,*The Banach-Tarski paradox*, Cambridge University Press, Cambridge, 1993. With a foreword by Jan Mycielski; Corrected reprint of the 1985 original. MR**1251963**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20F36,
20F60,
06F15

Retrieve articles in all journals with MSC (2000): 20F36, 20F60, 06F15

Additional Information

**Akbar Rhemtulla**

Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
ar@ualberta.ca

**Dale Rolfsen**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Email:
rolfsen@math.ubc.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06413-4

Received by editor(s):
February 16, 2001

Received by editor(s) in revised form:
April 26, 2001

Published electronically:
February 12, 2002

Additional Notes:
The authors thank NSERC for partial financial support

Communicated by:
Stephen D. Smith

Article copyright:
© Copyright 2002
American Mathematical Society