×

Free group automorphisms, invariant orderings and topological applications. (English) Zbl 0985.57006

The authors call a group \(G\) endowed with a strict total ordering \(<\) of its elements left-ordered provided \(x<y\) if and only if \(zx<zy\) for every \(x,y,z\) in \(G\), and bi-ordered provided the ordering is also right-invariant. The goal of the paper is to establish certain orderability results for several families of groups which arise in topology. It is shown that: (1) The fundamental group of every closed surface is bi-orderable, with the exceptions of the fundamental groups of the projective plane and the Klein bottle. (2) The pure braid groups associated with nonorientable surfaces distinct from the projective plane are left-orderable. (3) The fundamental groups of certain punctured-torus bundles over the circle are bi-orderable, for example the figure eight knot group.

MSC:

57M05 Fundamental group, presentations, free differential calculus
06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
20F36 Braid groups; Artin groups
57M07 Topological methods in group theory
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML EMIS

References:

[1] G Baumslag, On generalised free products, Math. Z. 78 (1962) 423 · Zbl 0104.24402
[2] B H Bowditch, A variation on the unique product property, J. London Math. Soc. \((2)\) 62 (2000) 813 · Zbl 1033.20040
[3] R Botto Mura, A Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker (1977) · Zbl 0358.06038
[4] S Boyer, D Rolfsen, B Wiest, Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005) 243 · Zbl 1068.57001
[5] J González-Meneses, Ordering pure braid groups on compact, connected surfaces, Pacific J. Math. 203 (2002) 369 · Zbl 1059.20033
[6] D M Kim, D Rolfsen, An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math. 55 (2003) 822 · Zbl 1047.20027
[7] G Levitt, private communication
[8] D D Long, Planar kernels in surface groups, Quart. J. Math. Oxford Ser. \((2)\) 35 (1984) 305 · Zbl 0556.57006
[9] W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers, New York-London-Sydney (1966) · Zbl 0138.25604
[10] L P Neuwirth, Knot groups, Annals of Mathematics Studies 56, Princeton University Press (1965) · Zbl 0184.48903
[11] L P Neuwirth, The status of some problems related to knot groups, Springer (1974) · Zbl 0288.55003
[12] B Perron, D Rolfsen, Ordering certain fibred knot groups, preprint · Zbl 1046.57008
[13] H Short, B Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. \((2)\) 46 (2000) 279 · Zbl 1023.57013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.