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JSJ DECOMPOSITIONS OF DOUBLES OF FREE GROUPS

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

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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

Information

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