Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T00:49:00.075Z Has data issue: false hasContentIssue false
  • English
  • Français

JSJ DECOMPOSITIONS OF DOUBLES OF FREE GROUPS

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  02 May 2019

SIMON HEIL
SIMON HEIL*
Affiliation:
Christian-Albrechts-Universität zu Kiel, Mathematisches Seminar, Kiel, 24098, Germany e-mail: heil@math.uni-kiel.de
Rights & Permissions [Opens in a new window]

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 classify all possible JSJ decompositions of doubles of free groups of rank two, and we also compute the Makanin–Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup separable.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20E05: Free nonabelian groups 20E08: Groups acting on trees
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 367 - 382
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

References

Baumslag, B., Residually free groups, Proc. London Math. Soc. 17(3) (1967), 402–418.10.1112/plms/s3-17.3.402CrossRefGoogle Scholar
Baumslag, G., On generalised free products, Math. Z. 78(1) (1962), 423–438.10.1007/BF01195185CrossRefGoogle Scholar
Cashen, C., Splitting line patterns in free groups, Alg. Geom. Topol. 16(2) (2016), 621–673.10.2140/agt.2016.16.621CrossRefGoogle Scholar
Cashen, C., Manning, J., Virtually geometric. 2014. Computer program, https://bitbucket.org/christopher_cashen/virtuallygeometric.Google Scholar
Dahmani, F. and Guirardel, V., The isomorphism problem for all hyperbolic groups, Geom. Funct. Anal. 21 (2011), 223–300.10.1007/s00039-011-0120-0CrossRefGoogle Scholar
Dunfield, N. and Thurston, W., Finite covers of random 3-manifolds, Invent. Math. 166(3) (2006), 457–521.10.1007/s00222-006-0001-6CrossRefGoogle Scholar
Guirardel, V. and Levitt, G., JSJ decompositions of groups (2016). arXiv:1602.05139v1 [math.GR]Google Scholar
Hall, M., Coset representations in free groups, Trans. Am. Math. Soc. 67 (1949), 421–432.10.1090/S0002-9947-1949-0032642-4CrossRefGoogle Scholar
Ivanov, S. V., On certain elements of free groups, J. Alg. 204 (1998), 394–405.10.1006/jabr.1997.7354CrossRefGoogle Scholar
Kapovich, I. and Weidmann, R., On the structure of two-generated hyperbolic groups, Math. Z. 231 (1999), 783–801.10.1007/PL00004753CrossRefGoogle Scholar
Kapovich, I. and Weidmann, R., Two-generated groups acting on trees, Arch. Math. 73 (1999), 172–181.10.1007/PL00000401CrossRefGoogle Scholar
Karrass, A. and Solitar, D., The free product of two groups with malnormal amalgamated subgroup, Can. J. Math. XXIII(6) (1971), 933–959.10.4153/CJM-1971-102-8CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Elementary theory of free non-abelian groups, J. Alg. 302 (2006), 451–552.10.1016/j.jalgebra.2006.03.033CrossRefGoogle Scholar
Lee, D., On certain C-test words for free groups, J. Alg. 247 (2002), 509–540.10.1006/jabr.2001.9001CrossRefGoogle Scholar
Louder, L. and Touikan, N., Magnus pairs in, and free conjugacy separability of, limit groups, Geometria Dedicata, Published online: 06 January 2018.10.1007/s10711-017-0314-1CrossRefGoogle Scholar
Mal’cev, A. I., On homomorphisms onto finite groups, Am. Math. Soc. Trans. 119(2) (1983), 67–79.Google Scholar
https://mathoverflow.net/questions/163585/can-finitely-generated-subgroups-of-limit-groups-be-detected-in-free-group-quoti.Google Scholar
Rips, E. and Sela, Z., Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. Math. 146 (1997), 53–109.10.2307/2951832CrossRefGoogle Scholar
Scott, P., Subgroups of surface groups are almost geometric, J. London Math. Soc. 17(3) (1978), 555–565.CrossRefGoogle Scholar
Sela, Z., Diophantine Geometry over groups I: Makanin-Razborov diagrams, Publications Mathematiques de l’IHES 93 (2001), 53–104.Google Scholar
Sela, Z., Diophantine geometry over groups VI: The elementary theory of a free group, Geom. Funct. Anal. 16 (2006), 707–730.Google Scholar
Touikan, N., On the one-endedness of graphs of groups, Pac. J. Math. 278(2) (2015), 463–478.10.2140/pjm.2015.278.463CrossRefGoogle Scholar
Touikan, N., The equation w(x, y) = u over free groups: An algebraic approach, J. Group Theory 12(4) (2009), 611–634.10.1515/JGT.2008.100CrossRefGoogle Scholar
Touikan, N., The fully residually F quotients of F*⟨x, y⟩, Groups Geom. Dyn. 6(1) (2012), 155–220.CrossRefGoogle Scholar
Turner, E., Test words for automorphisms of free groups, Bull. London Math. Soc. 28(3) (1996), 255–263.10.1112/blms/28.3.255CrossRefGoogle Scholar
Wilton, H., Hall’s Theorem for limit groups, Geom. Funct. Anal. 18 (2008), 271–303.10.1007/s00039-008-0657-8CrossRefGoogle Scholar