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A tree-free group that is not orderable

Author: Shane O Rourke
Journal: Proc. Amer. Math. Soc. 143 (2015), 41-43
MSC (2010): Primary 20E08; Secondary 20F60
Published electronically: August 22, 2014
MathSciNet review: 3272730
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Abstract: I. M. Chiswell has asked whether every group that admits a free isometric action (without inversions) on a $ \Lambda $-tree is orderable. We give an example of a multiple HNN extension $ \Gamma $ which acts freely on a $ \mathbb{Z}^2$-tree but which has non-trivial generalised torsion elements. The existence of such elements implies that $ \Gamma $ is not orderable.

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  • [1] Hyman Bass, Group actions on non-Archimedean trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 69-131. MR 1105330 (93d:57003),
  • [2] Stephen J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471-486 (electronic). MR 1815219 (2002a:20043),
  • [3] I. M. Chiswell, Non-standard free groups, Model theory of groups and automorphism groups (Blaubeuren, 1995) London Math. Soc. Lecture Note Ser., vol. 244, Cambridge Univ. Press, Cambridge, 1997, pp. 153-165. MR 1689867 (2000f:03110),
  • [4] Ian Chiswell, Introduction to $ \Lambda $-trees, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR 1851337 (2003e:20029)
  • [5] I. M. Chiswell, Locally invariant orders on groups, Internat. J. Algebra Comput. 16 (2006), no. 6, 1161-1179. MR 2286427 (2007k:20083),
  • [6] I. M. Chiswell, Right orderability and graphs of groups, J. Group Theory 14 (2011), no. 4, 589-601. MR 2818951 (2012f:20084),
  • [7] John Crisp and Bert Wiest, Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups, Algebr. Geom. Topol. 4 (2004), 439-472. MR 2077673 (2005e:20052),
  • [8] C. Droms, Graph groups, Ph.D. thesis, Syracuse University, 1983, available from dromscg/vita/thesis/.
  • [9] Michael Falk and Richard Randell, Pure braid groups and products of free groups, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 217-228. MR 975081 (90d:20070),
  • [10] O. Kharlampovich, A. Myasnikov, and D. Serbin, Groups acting freely on $ {\Lambda }$-trees, arXiv:0911.0209v4, November 2011.
  • [11] -, Actions, length functions, and non-archimedean words, Internat. J. Algebra 23 (2013), no. 2, 325-455. MR 3038860
  • [12] Daan Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131-156. MR 1888796 (2003c:20040),
  • [13] Shane O Rourke, Affine actions on non-Archimedean trees, Internat. J. Algebra Comput. 23 (2013), no. 2, 217-253. MR 3038858
  • [14] Dale Rolfsen, New developments in the theory of Artin's braid groups, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop ``Invariants of Three-Manifolds'' (Calgary, AB, 1999), 2003, pp. 77-90. MR 1953321 (2004f:20070),
  • [15] Daniel T. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci. 16 (2009), 44-55. MR 2558631 (2011c:20052),
  • [16] Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, available from, 2011.

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Additional Information

Shane O Rourke
Affiliation: Department of Mathematics, Cork Institute of Technology, Rossa Avenue, Cork, Ireland

Received by editor(s): November 29, 2012
Received by editor(s) in revised form: March 1, 2013
Published electronically: August 22, 2014
Additional Notes: The author would like to thank Ian Chiswell for helpful conversations.
Communicated by: Kevin Whyte
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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