A tree-free group that is not orderable

O Rourke, Shane

doi:10.1090/S0002-9939-2014-12191-5

A tree-free group that is not orderable
HTML articles powered by AMS MathViewer

by Shane O Rourke PDF

Proc. Amer. Math. Soc. 143 (2015), 41-43 Request permission

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.

References

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, DOI 10.1007/978-1-4612-3142-4_{3}
Stephen J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471–486. MR 1815219, DOI 10.1090/S0894-0347-00-00361-1
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, DOI 10.1017/CBO9780511629174.011
Ian Chiswell, Introduction to $\Lambda$-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1851337, DOI 10.1142/4495
I. M. Chiswell, Locally invariant orders on groups, Internat. J. Algebra Comput. 16 (2006), no. 6, 1161–1179. MR 2286427, DOI 10.1142/S0218196706003463
I. M. Chiswell, Right orderability and graphs of groups, J. Group Theory 14 (2011), no. 4, 589–601. MR 2818951, DOI 10.1515/JGT.2010.065
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, DOI 10.2140/agt.2004.4.439
C. Droms, Graph groups, Ph.D. thesis, Syracuse University, 1983, available from http://educ.jmu.edu/ dromscg/vita/thesis/.
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, DOI 10.1090/conm/078/975081
O. Kharlampovich, A. Myasnikov, and D. Serbin, Groups acting freely on ${\Lambda }$-trees, arXiv:0911.0209v4, November 2011.
Olga Kharlampovich, Alexei Myasnikov, and Denis Serbin, Actions, length functions, and non-Archimedean words, Internat. J. Algebra Comput. 23 (2013), no. 2, 325–455. MR 3038860, DOI 10.1142/S0218196713400031
Daan Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131–156. MR 1888796, DOI 10.2307/3062152
Shane O. Rourke, Affine actions on non-Archimedean trees, Internat. J. Algebra Comput. 23 (2013), no. 2, 217–253. MR 3038858, DOI 10.1142/S0218196713400018
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, DOI 10.1016/S0166-8641(02)00054-8
Daniel T. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci. 16 (2009), 44–55. MR 2558631, DOI 10.3934/era.2009.16.44
Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, available from http://www.math.mcgill.ca/wise/papers.html, 2011.