Generalized Torsion Elements and Bi-orderability of 3-manifold Groups

Motegi, Kimihiko ; Teragaito, Masakazu

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 830-844

Résumé

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2,m)\,(m>2)$ is a generalized torsion element.

Détail

DOI : 10.4153/CMB-2017-008-8

Mots-clés : 57M25, 57M05, 06F15, 20F05, generalized torsion element, bi-ordering, h-manifold group

Bibliographie

[1] [1] Aschenbrenner, M., S. Friedl, and H. Wilton, 3-manifold groups. EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zurich, 2015. Google Scholar | DOI

[2] [2] Bardakov, V. G. and Vesnin, A. Y., On a generalization of Fibonacci groups. (Russian) Algebra Logika 42(2003 ), no. 2,131-160, 255; translation in Algebra Logic 42(2003), no. 2, 73–91. Google Scholar | DOI

[3] [3] Bludov, V. V. and Lapshina, E. S., On ordering groups with a nilpotent commutant. (Russian), Sibirsk. Mat. Zh. 44( 2003 ), no. 3, 513– 520 ; translation in Siberian Math. J. 44(2003), no. 3, 405-410. Google Scholar | DOI

[4] [4] Bonahon, F., Geometric structures on 3-manifolds. In: Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 93–164. Google Scholar

[5] [5] Boyer, S., D. Rolfsen, and B. Wiest, Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble) 55(2005), no. 1, 243–288. Google Scholar | DOI

[6] [6] Chiswell, I., A. Glass, and J. Wilson, Residual nilpotence and ordering in one-relator groups and knot groups. Math. Proc. Cambridge Philos. Soc. 158(2015), no. 2, 275–288. Google Scholar | DOI

[7] [7] Clay, A., C. Desmarais, and P. Naylor, Testing bi-orderability of knot groups. Canad. Math. Bull. 59(2016), no. 3, 472–482. Google Scholar | DOI

[8] [8] Conway, J. H., Advanced problem 5327. Amer. Math. Monthly 72(1965), 915. Google Scholar

[9] [9] Dabkowski, M., J. Przytycki, and A. Togha, Non-left-orderable 3-manifold groups. Canad. Math. Bull. 48(2005), no. 1, 32–40. Google Scholar | DOI

[10] [10] Gomez-Larrafiaga, J., W. Heil, and F. Gonzalez-Acufia, 3-manifolds that are covered by two open bundles. Bol. Soc. Mat. Mexicana (3) 10(2004), Special Issue, 171–179. Google Scholar

[11] [11] Helling, H., A. Kim, and J. Mennicke, A geometric study of Fibonacci groups. J. Lie Theory 8(1998), no. 1,1-23. Google Scholar

[12] [12] Hempel, J., 3-Manifolds. Ann. of Math. Studies, 86, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1976. Google Scholar

[13] [13] Hilden, H., M. Lozano, and J. Montesinos-Amilibia, The arithmeticity of the figure eight knot orbifolds. In: Topology ‘90 (Columbus, OH, 1990), Ohio State Univ. Math. Res. Inst. Publ., 1, de Gruyter, Berlin, 1992, pp. 169–183. Google Scholar

[14] [14] Howie, J. and G. Williams, Fibonacci type presentations and 3-manifolds. Topology Appl. 215(2017), 24–34. Google Scholar | DOI

[15] [15] Jaco, W., Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43, American Mathematical Society, Providence, RI, 1980. Google Scholar

[16] [16] Jaco, W. and P. Shalen, Seifertfibered spaces in 3-manifolds. Mem. Amer. Math. Soc. 21(1979), no. 220. Google Scholar | DOI

[17] [17] Johnson, D. L., J. W Wamsley, and D. Wright, The Fibonacci groups. Proc. London Math. Soc. (3) 29(1974), 577–592. Google Scholar | DOI

[18] [18] Kokorin, A. I. and Kopytov, V. M., Linearly ordered groups. (Russian), Monographs in Contemporary Algebra. Izdat. “Nauka”, Moscow, 1972. Google Scholar

[19] [19] Longobardi, P., M. Maj, and A. Rhemtulla, On solvable R*-groups. J. Group Theory 6(2003), no. 4, 499–503. Google Scholar | DOI

[20] [20] Magnus, W, A. Karrass, and D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations. Second ed., Dover Publications, Inc., Mineola, NY, 2004. Google Scholar

[21] [21] Mura, R. and A. Rhemtulla, Solvable R*-groups. Math. Z. 142(1975), 293–298. Google Scholar | DOI

[22] [22] Mura, R., Orderable groups. Lecture Notes in Pure and Applied Mathematics, 27. Marcel Dekker, Inc., New York-Basel, 1977. Google Scholar

[23] [23] Naylor, G. and D. Rolfsen, Generalized torsion in knot groups. Canad. Math. Bull. 59(2016), no. 1, 182–189. Google Scholar | DOI

[24] [24] Newman, M. F., Proving a group infinite. Arch. Math. 54(1990), 209-211. Google Scholar | DOI

[25] [25] Orlik, P., Seifert manifolds. Lecture Notes in Mathematics, 291, Springer-Verlag, Berlin-New York, 1972. Google Scholar

[26] [26] Peters, T., On L-spaces and non left-orderable 3-manifold groups. arxiv:0903.4495 Google Scholar

[27] [27] Roberts, R., J. Shareshian, and M. Stein, Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation. J. Amer. Math. Soc. 16(2003), no. 3, 639–679. Google Scholar | DOI

[28] [28] Scott, P., The geometries of 3-manifolds. Bull. London Math. Soc. 15(1983), no. 5, 401–487. Google Scholar | DOI

[29] [29] Teragaito, M., Fourfold cyclic branched covers of genus one two-bridge knots are L-spaces. Bol. Soc. Mat. Mex. (3) 20(2014), no. 2, 391–403. Google Scholar | DOI

[30] [30] Teragaito, M., Generalized torsion elements in the knot groups of twist knots. Proc. Amer. Math. Soc. 14(2016), no. 6, 2677–2682. Google Scholar | DOI

[31] [31] Thomas, R. M., The Fibonacci groups revisited. Groups St. Andrews 1989, 2, 445–454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991. Google Scholar | DOI

[32] [32] Vinogradov, A. A., On the free product of ordered groups. Mat. SbornikN. S. 25(67)(1949), 163–168. Google Scholar