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The outer automorphism groups of two-generator, one-relator groups with torsion


Author: Alan D. Logan
Journal: Proc. Amer. Math. Soc. 144 (2016), 4135-4150
MSC (2010): Primary 20F28, 20E99; Secondary 20F65, 20F67
DOI: https://doi.org/10.1090/proc/13063
Published electronically: April 25, 2016
MathSciNet review: 3531167
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Abstract: The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani-Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.


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  • [Bog00] O. Bogopolski, Classification of automorphisms of the free group of rank 2 by ranks of fixed-point subgroups, J. Group Theory 3 (2000), no. 3, 339-351. MR 1772010 (2001f:20046), https://doi.org/10.1515/jgth.2000.027
  • [BT67] Gilbert Baumslag and Tekla Taylor, The centre of groups with one defining relator, Math. Ann. 175 (1968), 315-319. MR 0222144 (36 #5196)
  • [Car15] Mathieu Carette, Virtually splitting the map from $ {\rm Aut}(G)$ to $ {\rm Out}(G)$, Proc. Amer. Math. Soc. 143 (2015), no. 2, 543-554. MR 3283643, https://doi.org/10.1090/S0002-9939-2014-12278-7
  • [Cla06] Matt Clay, Deformation spaces of G-trees, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)-The University of Utah. MR 2708594
  • [DG11] François Dahmani and Vincent Guirardel, The isomorphism problem for all hyperbolic groups, Geom. Funct. Anal. 21 (2011), no. 2, 223-300. MR 2795509 (2012e:20097), https://doi.org/10.1007/s00039-011-0120-0
  • [FKS72] J. Fischer, A. Karrass, and D. Solitar, On one-relator groups having elements of finite order, Proc. Amer. Math. Soc. 33 (1972), 297-301. MR 0311780 (47 #342)
  • [FR40] D. I. Fouxe-Rabinovitch, Über die Automorphismengruppen der freien Produkte. I, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 265-276 (Russian, with German summary). MR 0003413 (2,215b)
  • [FR93] Benjamin Fine and Gerhard Rosenberger, Classification of all generating pairs of two generator Fuchsian groups, Groups '93 Galway/St. Andrews, Vol. 1 (Galway, 1993) London Math. Soc. Lecture Note Ser., vol. 211, Cambridge Univ. Press, Cambridge, 1995, pp. 205-232. MR 1342792 (96i:20065), https://doi.org/10.1017/CBO9780511629280.019
  • [FR94] Benjamin Fine and Gerhard Rosenberger, The Freiheitssatz and its extensions, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 213-252. MR 1292902 (95i:20049), https://doi.org/10.1090/conm/169/01657
  • [FT15] S. Friedl and S. Tillmann, Two-generator one-relator groups and marked polytopes, arXiv:1501.03489 (2015).
  • [GHMR00] N. D. Gilbert, J. Howie, V. Metaftsis, and E. Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000), no. 2, 213-223. MR 1753479 (2001a:20039), https://doi.org/10.1515/jgth.2000.017
  • [Gil87] N. D. Gilbert, Presentations of the automorphism group of a free product, Proc. London Math. Soc. (3) 54 (1987), no. 1, 115-140. MR 872253 (87m:20076), https://doi.org/10.1112/plms/s3-54.1.115
  • [HP84] J. Howie and S. J. Pride, A spelling theorem for staggered generalized $ 2$-complexes, with applications, Invent. Math. 76 (1984), no. 1, 55-74. MR 739624 (85k:20103), https://doi.org/10.1007/BF01388491
  • [IT15] Kazuhiro Ichihara and Yuki Temma, Non-left-orderable surgeries and generalized Baumslag-Solitar relators, J. Knot Theory Ramifications 24 (2015), no. 1, 1550003, 8. MR 3319680, https://doi.org/10.1142/S0218216515500030
  • [KM98] O. Kharlampovich and A. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (1998), no. 2, 571-613. MR 1390041 (98d:20041), https://doi.org/10.1090/S0002-9947-98-01773-5
  • [KS71a] A. Karrass and D. Solitar, The free product of two groups with a malnormal amalgamated subgroup, Canad. J. Math. 23 (1971), 933-959. MR 0314992 (47 #3541)
  • [KS71b] A. Karrass and D. Solitar, Subgroups of $ {\rm HNN}$ groups and groups with one defining relation, Canad. J. Math. 23 (1971), 627-643. MR 0301102 (46 #260)
  • [KT09] Goansu Kim and C. Y. Tang, Residual finiteness of outer automorphism groups of certain 1-relator groups, Sci. China Ser. A 52 (2009), no. 2, 287-292. MR 2491728 (2010a:20065), https://doi.org/10.1007/s11425-008-0151-7
  • [KT10] Goansu Kim and C. Y. Tang, Outer automorphism groups of certain 1-relator groups, Sci. China Math. 53 (2010), no. 6, 1635-1641. MR 2658619 (2011f:20066), https://doi.org/10.1007/s11425-010-3085-9
  • [KW99a] Ilya Kapovich and Richard Weidmann, On the structure of two-generated hyperbolic groups, Math. Z. 231 (1999), no. 4, 783-801. MR 1709496 (2000f:20068), https://doi.org/10.1007/PL00004753
  • [KW99b] Ilya Kapovich and Richard Weidmann, Two-generated groups acting on trees, Arch. Math. (Basel) 73 (1999), no. 3, 172-181. MR 1705011 (2001d:20022), https://doi.org/10.1007/PL00000401
  • [Lev05] Gilbert Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005), 49-70. MR 2174093 (2006m:20051), https://doi.org/10.1007/s10711-004-1492-1
  • [Log14] Alan D. Logan, The outer automorphism groups of three classes of groups, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)-University of Glasgow (United Kingdom). MR 3389465
  • [Log16] Alan D. Logan, The JSJ-decompositions of one-relator groups with torsion, Geom. Dedicata 180 (2016), 171-185. MR 3451463, https://doi.org/10.1007/s10711-015-0097-1
  • [LS77] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064 (58 #28182)
  • [Mag30] Wilhelm Magnus, Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz), J. Reine Angew. Math. 163 (1930), 141-165 (German). MR 1581238, https://doi.org/10.1515/crll.1930.163.141
  • [Mag32] W. Magnus, Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann. 106 (1932), no. 1, 295-307 (German). MR 1512760, https://doi.org/10.1007/BF01455888
  • [MKS04] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of generators and relations. MR 2109550 (2005h:20052)
  • [New73] B. B. Newman, The soluble subgroups of a one-relator group with torsion, J. Austral. Math. Soc. 16 (1973), 278-285. Collection of articles dedicated to the memory of Hanna Neumann, III. MR 0338188 (49 #2954)
  • [Pri77a] Stephen J. Pride, The isomorphism problem for two-generator one-relator groups with torsion is solvable, Trans. Amer. Math. Soc. 227 (1977), 109-139. MR 0430085 (55 #3092)
  • [Pri77b] Stephen J. Pride, The two-generator subgroups of one-relator groups with torsion, Trans. Amer. Math. Soc. 234 (1977), no. 2, 483-496. MR 0466325 (57 #6205)
  • [Wis12] Daniel T. Wise, From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, vol. 117, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012. MR 2986461
  • [Zim96] Bruno Zimmermann, Finite groups of outer automorphisms of free groups, Glasgow Math. J. 38 (1996), no. 3, 275-282. MR 1417356 (97i:20044), https://doi.org/10.1017/S0017089500031700

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Additional Information

Alan D. Logan
Affiliation: School of Mathematics and Statistics, University of Glasgow, University Gardens, G12 8QW, Scotland
Email: Alan.Logan@glasgow.ac.uk

DOI: https://doi.org/10.1090/proc/13063
Keywords: One-relator groups, outer automorphism groups, Nielsen equivalence
Received by editor(s): March 31, 2015
Received by editor(s) in revised form: December 11, 2015
Published electronically: April 25, 2016
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2016 American Mathematical Society

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