Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-27T06:07:46.956Z Has data issue: false hasContentIssue false
  • English
  • Français

Orders on Trees and Free Products of Left-ordered Groups

Part of: Structure and classification of infinite or finite groups Ordered structures Special aspects of infinite or finite groups

Published online by Cambridge University Press:  31 March 2020

Warren Dicks  and
Zoran Šunić
Warren Dicks
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain Email: dicks@mat.uab.cat
Zoran Šunić
Affiliation:
Department of Mathematics, Hofstra University, 306 Roosevelt Hall, Hempstead, NY 11549, USA Email: zoran.sunic@hofstra.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.

As an application, we provide a short proof (modulo Bass–Serre theory) of Vinogradov’s result that the free product of left-orderable groups is left-orderable.

Keywords

orderable groupsfree productsactions on treesBass–Serre tree

MSC classification

Primary: 06F15: Ordered groups
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E08: Groups acting on trees 20F60: Ordered groups
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 335 - 347
Copyright
© Canadian Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first-named author’s research was partially supported by Spain’s Ministerio de Ciencia e Innovación through Project MTM2011-25955. The second-named author’s research was partially supported by the National Science Foundation under Grant No. DMS-1105520

References

Bergman, George M., Ordering coproducts of groups and semigroups. J. Algebra 133(1990), no. 2, 313–339. https://doi.org/10.1016/0021-8693(90)90272-PCrossRefGoogle Scholar
Bludov, V. V. and Glass, A. M. W., Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup. Proc. Lond. Math. Soc. 99(2009), no. 3, 585–608. https://doi.org/10.1112/plms/pdp008CrossRefGoogle Scholar
Burns, R. G. and Hale, V. W. D., A note on group rings of certain torsion-free groups. Canad. Math. Bull. 15(1972), 441–445. https://doi.org/10.4153/CMB-1972-080-3CrossRefGoogle Scholar
Chiswell, I. M., Right orderability and graphs of groups. J. Group Theory 14(2011), no. 4, 589–601. https://doi.org/10.1515/JGT.2010.065CrossRefGoogle Scholar
Holland, C. and Scrimger, E., Free products of lattice ordered groups. Algebra Universalis 2(1972), 247–254. https://doi.org/10.1007/BF02945034CrossRefGoogle Scholar
Johnson, R. E., Free products of ordered semigroups. Proc. Amer. Math. Soc. 19(1968), 697–700. https://doi.org/10.2307/2035867CrossRefGoogle Scholar
Passman, Donald S., The algebraic structure of group rings. Pure and Applied Mathematics. Wiley-Interscience, New York, 1977.Google Scholar
Šunić, Zoran, Explicit left orders on free groups extending the lexicographic order on free monoids. C. R. Math. Acad. Sci. Paris 351(2013), 507–511. https://doi.org/10.1016/j.crma.2013.07.001CrossRefGoogle Scholar
Šunić, Zoran, Orders on free groups induced by oriented words. arxiv:1309.6070.Google Scholar
Vinogradov, A. A., On the free product of ordered groups. Mat. Sbornik N.S. 25(1949), 67, 163–168.Google Scholar