Group Theory/Dynamical Systems
Explicit left orders on free groups extending the lexicographic order on free monoids
Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 507-511.

For every finitely generated free group, we construct an explicit left order extending the lexicographic order on the free monoid generated by the positive letters. The order is defined by a left, free action on the orbit of 0 of a free group of piecewise linear homeomorphisms of the line. The membership in the positive cone is decidable in linear time in the length of the input word. The positive cone forms a context-free language closed under word reversal.

Pour tout groupe libre fini engendré, nous construisons explicitement un ordre à gauche qui étend lʼordre lexicographique sur le monoïde libre engendré par les lettres positives. Cet ordre est défini par une action à gauche, libre, sur lʼorbite de 0 dʼun groupe libre dʼhoméomorphismes de la droite linéaires par morceaux. Lʼappartenance au cône positif est décidable en temps linéaire par rapport à la longueur du mot. Le cône positif forme un langage non contextuel fermé par image miroir.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.001
Šunić, Zoran 1

1 Dept. of Mathematics, Texas A&M University, MS-3368, College Station, TX 77843-3368, USA
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Šunić, Zoran. Explicit left orders on free groups extending the lexicographic order on free monoids. Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 507-511. doi : 10.1016/j.crma.2013.07.001. http://www.numdam.org/articles/10.1016/j.crma.2013.07.001/

[1] Bergman, George M. Ordering coproducts of groups and semigroups, J. Algebra, Volume 133 (1990) no. 2, pp. 313-339

[2] Magnus, Wilhelm Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann., Volume 111 (1935) no. 1, pp. 259-280

[3] McCleary, Stephen H. Free lattice-ordered groups represented as o-2 transitive l-permutation groups, Trans. Am. Math. Soc., Volume 290 (1985) no. 1, pp. 69-79

[4] Navas, Andrés On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble), Volume 60 (2010) no. 5, pp. 1685-1740

[5] Neumann, B.H. On ordered division rings, Trans. Am. Math. Soc., Volume 66 (1949), pp. 202-252

[6] Neumann, B.H. On ordered groups, Am. J. Math., Volume 71 (1949), pp. 1-18

[7] Šunić, Zoran Free subgroups acting properly discontinuously, Topol. Appl., Volume 160 (2013) no. 10, pp. 1108-1114

[8] Vinogradov, A.A. On the free product of ordered groups, Mat. Sb. N.S., Volume 25 (1949) no. 67, pp. 163-168

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