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One-Relator Groups: An Overview

Published online by Cambridge University Press:  15 April 2019

By
Gilbert Baumslag ,
Benjamin Fine  and
Gerhard Rosenberger
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
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Summary

In 1985, at Groups St Andrews, Gilbert Baumslag gave a short course on one-relator groups which provided a look at the subject up to that point. In this paper we partially update the massive amount of work done over the past three decades. For the most part we concentrate on areas and results to which the authors have made contributions. We look at the important connections with surface groups and elementary theory, and describe the surface group conjecture and the Gromov conjecture on surface subgroups. We look at the solution by Wise of Baumslag’s residual finiteness conjecture and discuss a new Baumslag conjecture on virtually free-by-cyclic groups. We examine various amalgam decompositions of one-relator groups and the Baumslag-Shalen conjectures. We then look at a series of open problems in one-relator group theory and their status. Finally we introduce a concept called plainarity based on the Magnus breakdown of a one-relator group which might provide a systematic approach to the solution of problems in one-relator groups.

Type
Chapter
Information
Groups St Andrews 2017 in Birmingham , pp. 119 - 157
Publisher: Cambridge University Press
Print publication year: 2019

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References

Aab, M. and Rosenberger, G., Subgroup separable free products with cyclic amalgamtion, Results in Math. 28 (1995), 185–194.CrossRefGoogle Scholar
Ackermann, P., Fine, B. and Rosenberger, G., Surface groups: Motivating examples in combinatorial group theory, Groups St Andrews 2005, London Math. Soc. Lecture Notes Series 339 (2007), 96–130.CrossRefGoogle Scholar
Allenby, R., Mosner, L. E., and Tang, C. Y., The residual finiteness of certain onerelator groups, Proc. Amer. Math. Soc. 75 (1) (1980), 8–10.Google Scholar
Anshel, I., On two relator groups, in Topology and Combinatorial Group Theory, Springer Lecture Notes in Mathematics 1440 (1990), 1–21.CrossRefGoogle Scholar
Baumslag, B., Residually free groups, Proc. London Math. Soc. 17 (1967), 402–418.Google Scholar
Baumslag, G., On generalised free products, Math. Z. 78 (1962), 423-438.CrossRefGoogle Scholar
Baumslag, G., Groups with the same lower central sequence as a relatively free group. I. The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321.Google Scholar
Baumslag, G., Residually finite one-relator groups, Bull. Amer. Math. Soc. 73 (1967), 618–620.CrossRefGoogle Scholar
Baumslag, G., Finitely generated cyclic extensions of free groups are residually finite, Bull. Austral. Math. Soc. 5 (1971), 87–94.CrossRefGoogle Scholar
Baumslag, G., A survey of groups with a single defining relation, Groups St Andrews 1985 (Cambridge University Press, 1986), 30–56.Google Scholar
Baumslag, G., Musings on Magnus, in The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math. 169 (Amer. Math. Soc., Providence, RI, 1994), 99–106.Google Scholar
Baumslag, G., Embedding wreath-like products in finitely presented groups, Contemp. Math. 372 (2005), 197–206.CrossRefGoogle Scholar
Baumslag, G. and Cleary, S., Parafree one-relator groups, J. Group Theory 9 (2006), 191–201.CrossRefGoogle Scholar
Baumslag, G., Fine, B., Miller, C. and Troeger, D., Virtual properties of cyclically pinched one-relator groups, Int. J. Alg. Comp. 19 (2009), 1–15.CrossRefGoogle Scholar
Baumslag, G., Gersten, S., Shapiro, M. and Short, H., Automatic groups and amalgams, J. Pure Appl. Algebra 76 (3) (1991), 229–316.CrossRefGoogle Scholar
Baumslag, G., Miller, C. and Troeger, D., Reflections on the residual finiteness of one-relator groups, Groups Geom. Dyn. 1 (2007), 209–219.CrossRefGoogle Scholar
Baumslag, G., Morgan, J. and Shalen, P., Generalized triangle groups, Math. Proc. Camb. Phil. Soc. 102 (1987), 25–31.CrossRefGoogle Scholar
Baumslag, G., Myasnikov, A. G. and Remeslennikov, V. N., Algebraic geometry over groups. I. Ideals and algebraic sets, J. Algebra 219 (1999), 16–79.CrossRefGoogle Scholar
Baumslag, G., Myasnikov, A. G. and Remeslennikov, V. N., Discriminating completions of hyperbolic groups, Geom. Dedicata 92 (2002), 115–143.CrossRefGoogle Scholar
Baumslag, G., Myasnikov, A. G. and Romankov, V., Two theorems on equationally Noetherian groups, J. Algebra 194 (1994), 654–664.Google Scholar
Baumslag, G. and Shalen, P., Amalgamated products and finitely presented groups, Comment. Math. Helv. 65 (1990), 243–254.CrossRefGoogle Scholar
Baumslag, G. and Solitar, D., Some two-generator one-relator non-hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201.CrossRefGoogle Scholar
Baumslag, G. and Troeger, D., Virtually free-by-cyclic groups I, in Aspects of Infinite Groups (World Scientific Press, 2009).Google Scholar
Benyash-Krivets, V. V., Decomposing one-relator products of cyclic groups into free products with amalgamation, Mat. Sb. 189 (1998), 13–26.Google Scholar
Seminar, Bernstein, University, Cornell, Exploring the Works of Gilbert Baumslag – One relator Groups 111 – Wise’s Solution to Baumslag Conjecture, Online.Google Scholar
Bestvina, M. and Feighn, M., A combination theorem for negatively curved groups, J. Diff. Geom. 35 (1992), 85–101.Google Scholar
Bogley, W. A., An identity theorem for multi-relator groups, Math. Proc. Camb. Phil. Soc. 109 (1991), 313–321.CrossRefGoogle Scholar
Bogopolski, O., A surface analogue of a theorem of Magnus, Contemp. Math. 352 (2005), 55–89.CrossRefGoogle Scholar
Bogopolski, O. and Sviridov, K., A Magnus theorem for some one-relator groups, in The Zieschang Gedenkschrift 14 (2008), 63–73.Google Scholar
Bousfield, A. K. and Kan, D. M., Homotopy limits, completion and localization, Lecture Notes in Mathematics 304 (Springer-Verlag, 1972).Google Scholar
Bridson, M., and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathemtischen Wissenschaften 319 (Springer-Verlag, Berlin, 1999).Google Scholar
Brown, K. and Dror, E., The Artin-Rees property and homology, Israel J. Math. 22 (1975), 93–109.CrossRefGoogle Scholar
Brunner, A. M., Burns, R. G. and Solitar, D., The subgroup separability of free products with cyclic amalgamation, Contemp. Math. 33 (1984), 90–115.CrossRefGoogle Scholar
Bumagin, I., Kharlampovich, O. and Myasnikov, A., The isomorphism problem for fully residually free groups, J. Pure Appl. Alg. 20 (2007), 961–977.Google Scholar
Button, J. O. and Kropholler, R. P., Non-hyperbolic free-by-cyclic and one-relator groups, ArXiv: 1503.01989, v1.Google Scholar
Camps, T., grosse Rebel, V. and Rosenberger, G., Einführung in die kombinatorische und die geometrische Gruppentheorie (Heldermann-Verlag, 2008).Google Scholar
Chandler, B. and Magnus, W., The History of Combinatorial Group Theory: A Case Study in the History of Ideas (Springer-Verlag, New York, 1982).CrossRefGoogle Scholar
Ciobanu, L., Fine, B. and Rosenberger, G., The surface group conjecture: cyclically pinched and conjugacy pinched one-relator groups, Results in Mathematics 64 (2013), 175–184.CrossRefGoogle Scholar
Cohen, M. and Lustig, M., Very small actions on R-trees and Dehn twist automorphisms, Topology 34 (1985), 575–617.Google Scholar
Curran, P. M., Subgroups of finite index in certain classes of finitely presented groups, J. Algebra 122 (1989), 118–129.CrossRefGoogle Scholar
Dahmani, F. and Guiradel, V., The isomorphism problem for all hyperbolic groups, Geometric and Functional Analysis 21 (2011), 223–300.CrossRefGoogle Scholar
Dehn, M., Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1912), 116–144.Google Scholar
Dyer, J. L., Separating conjugates in free products with amalgamation and HNN extension, J. Austral. Math. Soc. 29 (1980), 35–51.Google Scholar
Edmonds, J., Maximum matching and a polyhedron with 0, 1-vertices, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 125–130.CrossRefGoogle Scholar
Epstein, D. B. A., Cannon, J., Holt, D., Levy, S., Paterson, M. and Thurston, W., Word Processing in Groups (James and Bartlett Publishers, Boston, London, 1992).CrossRefGoogle Scholar
Feighn, M. and Handel, M., Mapping tori of free group automorphisms are coherent, Ann. Math. 149 (1999), 1061–1077.CrossRefGoogle Scholar
Fine, B., Gaglione, A., Myasnikov, A., Rosenberger, G. and Spellman, D., A classification of fully residually free groups, J. Algebra 200 (1998), 571–605.CrossRefGoogle Scholar
Fine, B., Gaglione, A., Myasnikov, A., Rosenberger, G. and Spellman, D., The Elementary Theory of Groups (De Gruyter, 2016).Google Scholar
Fine, B., Gaglione, A., Rosenberger, G. and Spellman, D., Something for Nothing: Some consequences of the solution to the Tarski problems, Proc. Groups St Andrews, 2013 (Cambridge University Press, 2015).Google Scholar
Fine, B., Gaglione, A., Rosenberger, G. and Spellman, D., Elementary free groups, Cont. Math. 633 (2015), 41–58.CrossRefGoogle Scholar
Fine, B., Howie, J. and Rosenberger, G., One-relator quotients and free products of cyclics, Proc. Amer. Math. Soc. 102 (1988), 1–6.CrossRefGoogle Scholar
Fine, B., Kharlampovich, O., Myasnikov, A., Remeslennikov, V. and Rosenberger, G., On the surface group conjecture, Scient. Series A (2007).Google Scholar
Fine, B., Kharlampovich, O., Myasnikov, A. and Remeslennikov, V., Tame automorphisms of elementary free groups, Comm. Algebra 42 (2014), 3386–3394.CrossRefGoogle Scholar
Fine, B., Kreuzer, M. and Rosenberger, G., Faithful real representations of cyclically pinched one-relator groups, Internat. J. Group Theory 3 (1) (2014), 1–8.Google Scholar
Fine, B., Levin, F. and Rosenberger, G., Subgroups and decompositions of one-relator products of cyclics: Part 2: Normal torsion-free subgroups, J. Indian Math. Soc. 49 (1989), 237–247.Google Scholar
Fine, B., Moldenhauer, A. and Rosenberger, G., Faithful real representations of groups of F-type, to appear.Google Scholar
Fine, B., Myasnikov, A., grosse Rebel, V. and Rosenberger, G., A classification of CSA, commutative transitive and restricted Gromov one-relator groups, Result. Math. 50, 2007, 183–193.Google ScholarGoogle Scholar
Fine, B. and Peluso, A., Amalgam decompositions of one-relator groups, J. Pure Appl. Algebra 141 (1999), 1–11.CrossRefGoogle Scholar
Fine, B., Röhl, F. and Rosenberger, G., On HNN groups whose three generator subgroups are free, Inf. Groups and Group Rings, World Scientific 1 (1993), 13–36.CrossRefGoogle Scholar
Fine, B., Rosenberger, A. and Rosenberger, G., A note on Lyndon properties in onerelator Groups, Results Math. 59 (2012), 239–250.Google Scholar
Fine, B. and Rosenberger, G., Generalizing algebraic properties of Fuchsian groups, Groups St Andrews 1989 – London Math. Soc. Lecture Notes Series 159 (1990), 124–148.Google Scholar
Fine, B. and Rosenberger, G., On Restricted Gromov Groups, Comm. Algebra 20 (8) (1992), 2171–2182.CrossRefGoogle Scholar
Fine, B. and Rosenberger, G., The Freiheitssatz of Magnus and its extensions, Cont. Math. 169 (1994), 213–252.CrossRefGoogle Scholar
Fine, B. and Rosenberger, G., Algebraic generalizations of discrete groups (Marcel Dekker, 2000).Google Scholar
Fine, B. and Rosenberger, G., Surface groups within Baumslag doubles, Proc. Edinburgh Math. Soc. 54 (1) (2011), 91–97.CrossRefGoogle Scholar
Fine, B., Rosenberger, G. and Stille, M., The isomorphism problem for a class of parafree groups, Proc. Edinburgh Math. Soc. 40 (1997), 541–549.CrossRefGoogle Scholar
Fine, B., Rosenberger, G. and Stille, M., Conjugacy pinched and cyclically pinched one-relator groups, Revista Math. Madrid 10 (1997), 207–227.Google Scholar
Fisher, J., Karrass, A. and Solitar, D., On one-relator groups having elements of finite order, Proc. Amer. Math. Soc. 33 (1972), 297–301.Google Scholar
Fricke, R. and Klein, F., Vorlesungen über die Theorie der automorphen Funktionen, vols. 1, (1897) and 2 (1912), Teubner Leipzig (Reprinted: Johnson, New York, 1965).Google Scholar
Gaglione, A., Lipschutz, S. and Spellman, D., Almost locally free groups and a theorem of Magnus, J. Groups, Complexity and Cryptology 1 (2009), 181–198.CrossRefGoogle Scholar
Gaglione, A.M. and Spellman, D., Even more model theory of free groups, in Infinite Groups and Group Rings (World Scientific Press, 1993), 37–40.Google Scholar
Gardam, G. and Woodhouse, D.J., The geometry of one-relator groups satisfying a polynomial isoperimatric inequality, ArXiv: 1711,08155v2.Google Scholar
Gildenhuys, D., Kharlampovich, O. and Myasnikov, A., CSA groups and separated free constructions, Bull. Austral. Math. Soc. 52 (1995), 63–84.CrossRefGoogle Scholar
Gordon, C. and Wilton, H., Surface subgroups of doubles of free groups, ArXiv 0902.3693v1.Google Scholar
Gromov, M., Hyperbolic groups, in Essays in Group Theory (MSRI Publications, Springer-Verlag, 1987), 75–263.Google Scholar
Guba, V., Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Mat. Zametki 40 (1986), 321–324.Google Scholar
Gutierrez, M., Homology and completions of groups, J. Algebra 51 (1978), 354–366.CrossRefGoogle Scholar
Howie, J., On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 280 (1985), 281–302.Google Scholar
Ivanov, S. and Schupp, P., On the hyperbolicity of small cancellation groups and one-relator groups, Trans. Amer. Math. Soc. 350 (1998), 1851–1894.CrossRefGoogle Scholar
Juhasz, A., Small cancellation theory with a unified small cancellation condition, J. London Math. Soc. 4 (1989), 57–80.Google Scholar
Juhasz, A., Solution of the conjugacy problem in one-relator groups, in Algorithms and classification in combinatorial group theory, (Baumslag, G. and Miller, C. F. (eds.)), MSRI (1992), 60–81.Google Scholar
Juhasz, A., Some applications of small cancellation theory to one-relator groups and one-relator products. in Geometric group theory, Vol.1, London Math. Soc. Lecture Notes Ser. 181 (Cambridge U. Press, Cambridge, 1993), 132–137.Google Scholar
Juhasz, A. and Rosenberger, G., On the combinatorial curvature of groups of F-type and other one-relator products of cyclics, Cont. Math. 169 (1994), 373–384.CrossRefGoogle Scholar
Haring-Smith, R.H., Groups and simple languages, Trans. Amer. Math. Soc. 279 (1983), 337–356.CrossRefGoogle Scholar
Kapovich, I. and Schupp, P., Genericity, the Arzhantseva-Olshanskii method and the isomorphism problem for one-relator groups, Math. Ann. 331 (2005), 1–19.CrossRefGoogle Scholar
Karrass, A., Magnus, W. and Solitar, D., Elements of finite order in a group with a single defining relation, Comm. Pure Appl. Math. 13 (1960), 57–66.CrossRefGoogle Scholar
Karrass, A. and Solitar, D., The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 214–224.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Irreducible affine varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), 472–516.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Irreducible affine varieties over a free group: II. Systems in triangular quasi-quadratic form and a description of residually free groups, J. Algebra 200 (1998), 517–569.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (2) (1998), 571–613.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Description of fully residually free groups and irreducible affine varieties over free groups, Summer school in Group Theory in Banff, 1996, CRM Proceedings and Lecture notes 17 (1999), 71-81.Google Scholar
Kharlampovich, O. and Myasnikov, A., Implicit function theorem for free groups, J. Algebra 290 (2005), 1–203.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Effective JSJ decompostions, Contemp. Math. 378 (2005), 87–212.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Elementary theory for free nonabelian groups, J. Algebra 302 (2006), 451–552.CrossRefGoogle Scholar
Kharlampovich, O. and Myasnikov, A., Algebraic Geometry over Free Groups, to appear.Google Scholar
Kharlampovich, O., Myasnikov, A., Remeslennikov, V. and Serbin, D., Subgroups of fully residually free groups: algorithmic problems, Contemp. Math. 360 (2004).CrossRefGoogle Scholar
Kim, S. H. and Oum, S. I., Hyperbolic surface subgroups of one-ended doubles of free groups, J. Topology 7 (4) (2014), 927–947.CrossRefGoogle Scholar
Kim, S. and Wilton, H., Surface subgroups of doubles of free groups, preprint.Google Scholar
Lazard, M., Sur les groupes nilpotents et les anneaux de Lie, Annales Sci. L’Ecole Normale Superieure 3 (71) (1954), 101–190.Google Scholar
Leary, I., Niblo, G. and Wise, D., Some free-by-cyclic groups, Groups St Andrews 1997 in Bath, II, London Math. Soc. Lecture Note Ser. 261 (Cambridge Univ. Press, Cambridge, 1999), 512–516.Google Scholar
Lipschutz, S., The conjugacy problem and cyclic amalgamation, Bull. Amer. Math. Soc. 81 (1975), 114–116.CrossRefGoogle Scholar
Liriano, S., The Nonisomorphism of Two One-Relator Groups, Ph.D. Thesis CUNY, (1993).Google Scholar
Lossov, K. I., SQ-universality of free products with amalgamated finite subgroups, Sibirsk. Mat. Zh. 27 (6) (1986), 128–139.Google Scholar
Louder, L. and Wilton, H., Stackings and the W-cycles conjectures, Can. Math. Bull. 60 (3) (2017), 604–612.CrossRefGoogle Scholar
Louder, L. and Wilton, H., One-relator Groups with Torsion are Coherent, preprint.Google Scholar
Lubotzky, A., A group theoretic characterization of linear groups, J. Algebra 113 (1988), 207–214.CrossRefGoogle Scholar
Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory (Springer-Verlag, 1977).Google Scholar
Magnus, W., Über diskontinuierliche Gruppen mit einer definierden Relation (Der Freheitssatz), J. Reine Angew. Math. 163 (1930), 141–165.Google Scholar
Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory: Presentations of groups in terms of generators and relations (Interscience Publishers, New York, London, Sydney, 1966).Google Scholar
Malcev, A. I., On faithful representations of infinite groups by matrices, Amer. Math. Soc. Translations 45 (1965), 1–18.CrossRefGoogle Scholar
Mendelsohn, N. and Ree, R., Free subgroups of groups with a single defining relation, Arch. Math. 19 (1968), 577–580.Google Scholar
Merzlyakov, Y. I., ed., Kourovka Notebook – Unsolved Problems in Group Theory (1980).Google Scholar
Moldavanskii, D. I., Certain subgroups of groups with one defining relation, Sibirsk. Math. Zh. 8 (1967), 1370–1384.Google Scholar
Myasnikov, A. G. and Remeslennikov, V. N., Exponential groups 1: Foundations of the theory and tensor completion, Siberian Mat. J. 5 (1994), 1106–1118.Google Scholar
Newman, B. B., Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968), 568–571.CrossRefGoogle Scholar
Ponzoni, G., On the surface group conjecture (Thesis, Univ. Milan-Bicocca, 2014).Google Scholar
Pride, S. J., The isomorphism for two generator one-relator groups with torsion is solvable, Trans. Amer. Math. Soc. 227 (1977), 109–139.CrossRefGoogle Scholar
Rapaport, E. S., Proof of a conjecture of Papakyriakopoulos, Ann. Math. 2 (1964), 608–613.Google Scholar
Remeslennikov, V. N., ∃-free groups, Siberian Mat. J. 30 (1989), 998–1001.Google Scholar
Rips, E. and Sela, Z., Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. Math. 146 (1) (1997), 53–109.CrossRefGoogle Scholar
Rosenberger, G., Zum Isomorphieproblem für Gruppen mit einer definierenden Relation, Ill. J. Math. 20 (1976), 614–621.Google Scholar
Rosenberger, G., Gleichungen in freien Produkten mit Amalgam, Math. Z. 173 (1980), 1–12:Google ScholarGoogle Scholar
Rosenberger, G., Eine Bemerkung zu einer Arbeit von R. C. Lyndon, Archiv. der Math. 40 (1983), 200–207.CrossRefGoogle Scholar
Rosenberger, G., The isomorphism problem for cyclically pinched one-relator groups, J. Pure Appl. Algebra 95 (1994), 75–86.CrossRefGoogle Scholar
Sacerdote, G. and Schupp, P., SQ-universality in HNN groups and one-relator groups, J. London Math. Soc. 7 (1974), 733–740.Google Scholar
Sela, Z., The isomorphism problem for hyperbolic groups I, Ann. Math. 141 (1995), 217–283.CrossRefGoogle Scholar
Sela, Z., Diophantine geometry over groups I: Makanin-Razborov diagrams, Publ. Math. de IHES 93 (2001), 31–105.CrossRefGoogle Scholar
Sela, Z., Diophantine geometry over groups II: Completions, closures and formal solutions, Israel J. Math. 104 (2003), 173–254.Google Scholar
Sela, Z., Diophantine geometry over groups IV: An iterative procedure for validation of a sentence, Israel J. Math. 143 (2004), 17–130.CrossRefGoogle Scholar
Sela, Z., Diophantine geometry over groups III: Rigid and solid solutions, Israel J. Math. 147 (2005), 1–73.CrossRefGoogle Scholar
Sela, Z., Diophantine geometry over groups V: Quantifier elimination, Israel J. Math. 150 (2005), 1–97.Google Scholar
Sela, Z., Diophantine geometry over groups VI: The elementary theory of a free group, GAFA 18 (2006), 707–730.Google Scholar
Selberg, A., On discontinuous groups in higher dimensional symmetric spaces, Int. Colloq. Function Theory (Tata Institute, Bombay, 1960), 147–164.Google Scholar
Shalen, P., Linear representations of certain amalgamated products, J. Pure Appl. Algebra 15 (1979), 187–197.CrossRefGoogle Scholar
Shpilrain, V., Recognizing automorphisms of the free groups, Arch. Math. 62 (1994), 385–392.CrossRefGoogle Scholar
Stallings, J., Homology and central series of groups, J. Algebra 2 (1965), 170–181.CrossRefGoogle Scholar
Turner, E. C., Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), 255–263.CrossRefGoogle Scholar
Wehfritz, B. A. F., Generalized free products of linear groups, Proc. London Math. Soc. 27 (3) (1973), 402–424.Google Scholar
Wilton, H., One ended subgroups of graphs of free groups with cyclic edge groups, Geom. Topol. 16 (2012), 665–683.CrossRefGoogle Scholar
Wise, D., Residual finiteness of positive one-relator groups, Comment. Math. Helv. 76 (2001), 314–338.CrossRefGoogle Scholar
Wise, D., The residual finiteness of quasi positive one-relator groups with torsion, J. Algebra 66 (2002), 334–350.Google Scholar
Wise, D., The coherence of one-relator groups with torsion, and the Hanna Neumann Conjecture, Bull. London Math. Soc. 37 (5) (2005), 697–705.CrossRefGoogle Scholar
Wise, D., The structure of groups with a quasiconvex hierarchy, Electronic Research Announcement in the Mathematical Sciences, AIMS 16 (2009), 44–55.Google Scholar
Zieschang, H., Automorphismen ebener discontinuierlicher Gruppen, Math. Ann. 166 (1966), 148–167.CrossRefGoogle Scholar
Zieschang, H., Vogt, E. and Coldewey, H.-D., Surfaces and planar discontinuous groups, Lecture Notes in Math. 835 (1980).Google Scholar

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