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EXOTIC LEFT-ORDERINGS OF THE FREE GROUPS FROM THE DEHORNOY ORDERING

Published online by Cambridge University Press:  01 April 2011

ADAM CLAY*
Affiliation:
CIRGET, Université du Québec à Montréal, Case postale 8888, Succursale Centre-ville, Montréal QC, Canada H3C 3P8 (email: [email protected])
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Abstract

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We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left-ordering of the free group on k generators, k>1, that has no convex subgroups.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Botto Mura, R. and Rhemtulla, A., Orderable Groups, Lecture Notes in Pure and Applied Mathematics, 27 (Marcel Dekker Inc., New York, 1977).Google Scholar
[2]Conrad, P., ‘Right-ordered groups’, Michigan Math. J. 6 (1959), 267275.CrossRefGoogle Scholar
[3]Dehornoy, P., ‘Braid groups and left distributive operations’, Trans. Amer. Math. Soc. 345(1) (1994), 115150.CrossRefGoogle Scholar
[4]Dehornoy, P., Dynnikov, I., Rolfsen, D. and Wiest, B., Ordering Braids, Surveys and Monographs, 148 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
[5]Fenn, R., Rolfsen, D. and Zhu, J., ‘Centralisers in the braid group and singular braid monoid’, Enseign. Math. (2) 42(1–2) (1996), 7596.Google Scholar
[6]Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Classics in Mathematics (Springer, Berlin, 2001), reprint of the 1977 edition.CrossRefGoogle Scholar
[7]McCleary, S. H., ‘Free lattice-ordered groups represented as o-2 transitive l-permutation groups’, Trans. Amer. Math. Soc. 290(1) (1985), 6979.Google Scholar
[8]Mulholland, J. and Rolfsen, D., ‘Local indicability and commutator subgroups of artin groups’, Preprint, available via http://arxiv.org/abs/math/0606116.Google Scholar
[9]Navas, A., ‘On the dynamics of (left) orderable groups’, Ann. Inst. Fourier (Grenoble) 60(5) (2010), 16851740.CrossRefGoogle Scholar
[10]Navas, A. and Wiest, B., ‘Nielsen–Thurston orderings and the space of braid orderings’, Preprint, available via http://arxiv.org/pdf/0906.2605.Google Scholar
[11]Rotman, J. J., An Introduction to the Theory of Groups, 4th edn, Graduate Texts in Mathematics, 148 (Springer, New York, 1995).CrossRefGoogle Scholar