Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T19:57:59.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Virtual geometricity is rare

Part of:Low-dimensional topologyStructure and classification of infinite or finite groupsProbabilistic methods in group theory

Published online by Cambridge University Press:  01 July 2015

Christopher H. Cashen and
Jason F. Manning
Christopher H. Cashen
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria email christopher.cashen@univie.ac.at
Jason F. Manning
Affiliation:
Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853, USA email jfmanning@math.cornell.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We also prove this fact.

MSC classification

Primary:20E05: Free nonabelian groups20P05: Probabilistic methods in group theory57M10: Covering spaces

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015, pp. 444 - 455
Copyright
© The Author(s) 2015 

References

Arzhantseva, G. N. and Ol’shanskiĭ, A. Yu., ‘Generality of the class of groups in which subgroups with a lesser number of generators are free’, Mat. Zametki 59 (1996) no. 4, 489–496, 638, translated in Math. Notes 59 (1996), no. 3–4, 350–355, doi:10.1007/BF02308683; MR 1445193 (98k:20040).Google Scholar
Berge, J., Documentation for the program Heegaard, circa 1990, available at http://www.math.uic.edu/∼t3m.Google Scholar
Cashen, C. H., Splitting line patterns in free groups, Preprint, 2010, arXiv:1009.2492v4.Google Scholar
Cashen, C. H. and Macura, N., ‘Line patterns in free groups’, Geom. Topol. 15 (2011) no. 3, 1419–1475, doi:10.2140/gt.2011.15.1419; MR 2825316.CrossRefGoogle Scholar
Cashen, C. H. and Manning, J. F., virtuallygeometric, 2014, computer program, available athttps://bitbucket.org/christopher_cashen/virtuallygeometric.Google Scholar
Gordon, C. and Wilton, H., On surface subgroups of doubles of free groups, Preprint, 2009,arXiv:0902.3693v1.Google Scholar
Gordon, C. and Wilton, H., ‘On surface subgroups of doubles of free groups’, J. Lond. Math. Soc. (2) 82 (2010) no. 1, 17–31, doi:10.1112/jlms/jdq007; MR 2669638 (2011k:20085).CrossRefGoogle Scholar
Grigorchuk, R. and de la Harpe, P., ‘On problems related to growth, entropy, and spectrum in group theory’, J. Dyn. Control Syst. 3 (1997) no. 1, 51–89, doi:10.1007/BF02471762; MR 1436550 (98d:20039).Google Scholar
Gromov, M., ‘Asymptotic invariants of infinite groups’, Geometric group theory, Vol. 2 (Sussex, 1991) , London Mathematical Society Lecture Note Series 182 (Cambridge University Press, Cambridge, 1993) 1–295; MR 1253544 (95m:20041).Google Scholar
Guirardel, V., ‘Approximations of stable actions on R -trees’, Comment. Math. Helv. 73 (1998) no. 1, 89–121, doi:10.1007/s000140050047; MR 1610591 (99e:20037).Google Scholar
Kapovich, I. and Lustig, M., ‘Intersection form, laminations and currents on free groups’, Geom. Funct. Anal. 19 (2010) no. 5, 1426–1467, doi:10.1007/s00039-009-0041-3; MR 2585579 (2011g:20052).Google Scholar
Kapovich, I., Schupp, P. and Shpilrain, V., ‘Generic properties of Whitehead’s algorithm and isomorphism rigidity of random one-relator groups’, Pacific J. Math. 223 (2006) no. 1, 113–140; doi:10.2140/pjm.2006.223.113; MR 2221020 (2007e:20068).Google Scholar
Lind, D. and Marcus, B., An introduction to symbolic dynamics and coding (Cambridge University Press, Cambridge, 1995) doi:10.1017/CBO9780511626302; MR 1369092 (97a:58050).Google Scholar
Mackay, J. M., ‘Conformal dimension and random groups’, Geom. Funct. Anal. 22 (2012) no. 1, 213–239, doi:10.1007/s00039-012-0153-z; MR 2899687.CrossRefGoogle Scholar
Manning, J. F., ‘Virtually geometric words and Whitehead’s algorithm’, Math. Res. Lett. 17 (2010) no. 5, 917–925, doi:10.4310/MRL.2010.v17.n5.a9; MR 2727618 (2012c:20114).Google Scholar
Otal, J.-P., ‘Certaines relations d’equivalence sur l’ensemble des bouts d’un groupe libre’, J. Lond. Math. Soc. s2–46 (1992) no. 1, 123–139, doi:10.1112/jlms/s2-46.1.123; MR 1180888 (93h:20041).Google Scholar
Papakyriakopoulos, C. D., ‘On Dehn’s lemma and the asphericity of knots’, Ann. of Math. (2) 66 (1957) 1–26, MR 0090053 (19,761a).CrossRefGoogle Scholar
Pérez, F. and Granger, B. E., ‘IPython: a system for interactive scientific computing’, Comput. Sci. Eng. 9 (2007) no. 3, 21–29, doi:10.1109/MCSE.2007.53.Google Scholar
Scott, P., ‘Subgroups of surface groups are almost geometric’, J. Lond. Math. Soc. (2) 17 (1978) no. 3, 555–565, MR 0494062 (58 #12996).Google Scholar
Solie, B. B., ‘Genericity of filling elements’, Int. J. Algebra Comput. 22 (2012) no. 2, doi:10.1142/S0218196711006741; MR 2903743.Google Scholar
Stallings, J. R., ‘Whitehead graphs on handlebodies’, Geometric group theory down under (Canberra, 1996) (de Gruyter, Berlin, 1999) 317–330; MR 1714852 (2001i:57028).Google Scholar
Stong, R., ‘Diskbusting elements of the free group’, Math. Res. Lett. 4 (1997) no. 2, 201–210; MR 1453054 (98h:20049).Google Scholar
Whitehead, J. H. C., ‘On equivalent sets of elements in a free group’, Ann. of Math. (2) 37 (1936) no. 4, 782–800, http://www.jstor.org/stable/1968618; MR 1503309.Google Scholar