Geometriae Dedicata

, Volume 94, Issue 1, pp 1–31 | Cite as

Automorphisms of Free Groups and Outer Space

  • Karen Vogtmann

Abstract

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.

automorphisms of free groups outer space 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alperin, R.: Locally compact groups acting on trees and property T, Monatsh. Math. 93(4) (1982), 261–265.Google Scholar
  2. 2.
    Alperin, R. and Bass, H.: Length functions of group actions on L-trees, In: Combinatorial Group Theory and Topology (Alta, Utah, 1984), Ann. of Math. Stud. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265–378.Google Scholar
  3. 3.
    Bass, H. and Lubotzky, A.: Linear-central filltrations on groups, In: W. Abicoff, J. Birman, K. Kuiken (eds), The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions, Contemp. Math. 169, Amer. Math. Soc., Providence, 1993.Google Scholar
  4. 4.
    Bao, Z.: Maximum order of periodic outer automorphisms of a free group, J. Algebra 224(2) (2000), 437–453.Google Scholar
  5. 5.
    Baumslag, G.: Automorphism groups of residually finite groups, J. London Math Soc. 38 (1963), 117–118.Google Scholar
  6. 6.
    Baumslag, G. and Taylor, T.: The centre of groups with one defining relator, Math Ann. 175 (1968), 315–319.Google Scholar
  7. 7.
    Bestvina, M.: Degenerations of the hyperbolic space, Duke Math. J. 56(1) (1988), 143–161.Google Scholar
  8. 8.
    Bestvina, M. and Feighn, M.: Outer limits, Preprint.Google Scholar
  9. 9.
    Bestvina, M. and Feighn, M.: The topology at infinity of Out(F n), Invent. Math. 140(3) (2000), 651–692.Google Scholar
  10. 10.
    Bestvina, M. and Handel, M.: Train tracks and automorphisms of free groups, Ann. of Math. 135(1) (1992), 1–51.Google Scholar
  11. 11.
    Bestvina, M., Feighn, M. and Handel, M.: Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7(2) (1997), 215–244 (Erratum Geom. Funct. Anal. 7(6) (1997), 1143).Google Scholar
  12. 12.
    Bestvina, M., Feighn, M. and Handel, M.: The Tits alternative for Out(F n). I: Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151(2) (2000), 517–623.Google Scholar
  13. 13.
    Bestvina, M., Feighn, M. and Handel, M.: The Tits alternative for Out(F n). II: A Kolchin type theorem, Preprint.Google Scholar
  14. 14.
    Bestvina, M., Feighn, M. and Handel, M.: Solvable subgroups of Out(F n) are virtually abelian, Preprint.Google Scholar
  15. 15.
    Birman, J., Lubotzky, A. and McCarthy, J. D.: Abelian and solvable subgroups of the mapping class groups, Duke Math J. 50 (1983), 1107–1120.Google Scholar
  16. 16.
    Bogopolski, O.: Arboreal decomposability of the group of automorphisms of a free group, Algebra i Logicki 26(2) (1987), 131–149.Google Scholar
  17. 17.
    Boutin, D. L.: When are centralizers of finite subgroups of Out(F n) finite?, In: Groups, Languages and Geometry (South Hadley, MA, 1998), Contemp. Math. 250, Amer. Math. Soc., Providence, RI, 1999, pp. 37–58.Google Scholar
  18. 18.
    Brady, T.: The integral cohomology of Out+(F 3), J. Pure Appl. Algebra 87(2) (1993), 123–167.Google Scholar
  19. 19.
    Brady, T.: Automatic structures on AutF 2, Arch. Math. 63 (1994), 97–102.Google Scholar
  20. 20.
    Bridson, M. R.: Geodesics and curvature in metric simplicial complexes, In: Group Theory from a Geometrical Viewpoint (Trieste, 1990), World Scientific, River Edge, NJ, 1991, pp. 373–463.Google Scholar
  21. 21.
    Bridson, M. R. and Farb, B.: A remark about actions of lattices on free groups, In: Geometric Topology and Geometric Group Theory (Milwaukee, WI, 1997), Topology Appl. 110(7) (2001), 21–24.Google Scholar
  22. 22.
    Bridson, M. R. and Vogtmann, K.: On the geometry of the automorphism group of a free group, Bull. London Math. Soc. 27(6) (1995), 544–552.Google Scholar
  23. 23.
    Bridson, M. R. and Vogtmann, K.: The symmetries of outer space, Duke Math. J. 106(2) (2001), 391–409.Google Scholar
  24. 24.
    Bridson, M. R. and Vogtmann, K.: Automorphisms of automorphism groups of free groups, J. Algebra 229(2) (2000), 785–792.Google Scholar
  25. 25.
    Bridson, M. R. and Vogtmann, K.: Homomorphisms of automorphism groups of free groups, To appear in Bull. London Math. Soc. Google Scholar
  26. 26.
    Brown, K. S.: Cohomology of Groups, Springer-Verlag, New York, 1982.Google Scholar
  27. 27.
    Chandler, B. and Magnus, W.: The History of Combinatorial Group Theory. A Case Study in the History of Ideas, Stud. Hist. Math. Phys. Sci. 9, Springer-Verlag, New York, 1982.Google Scholar
  28. 28.
    Yuqing Chen: Thesis, Ohio State University, 1998.Google Scholar
  29. 29.
    Cohen, F. R. and Peterson, F. P.: Some remarks on the space Im J, In: Algebraic Topology (Arcata, CA, 1986), Lecture Notes in Math. 1370, Springer, New York, 1989, pp. 117–125.Google Scholar
  30. 30.
    Cohen, M. M. and Lustig, M.: On the dynamics and the fixed subgroup of a free group automorphism, Invent. Math. 96(3) (1989), 613–638.Google Scholar
  31. 31.
    Cohen, M. M. and Lustig, M.: Very small group actions on R-trees and Dehn twist automorphisms, Topology 34(3) (1995), 575–617.Google Scholar
  32. 32.
    Cohen, M. M. and Lustig, M.: The conjugacy problem for Dehn twist automorphisms of free groups, Comment. Math. Helv. 74(2) (1999), 179–200.Google Scholar
  33. 33.
    Collins, D. J. and Miller, C. F. III, The conjugacy problem and subgroups of finite index, Proc. London Math. Soc. (3) 34(3) (1977), 535–556.Google Scholar
  34. 34.
    Collins, D. J. and Turner, E. C.: All automorphisms of free groups with maximal rank fixed subgroups, Math. Proc. Cambridge Philos. Soc. 119(4) (1996), 615–630.Google Scholar
  35. 35.
    Cooper, D.: Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111(2) (1987), 453–456.Google Scholar
  36. 36.
    Culler, M.: Finite groups of outer automorphisms of a free group, In: Contributions to Group Theory, Contemp. Math. 33, Amer. Math. Soc., Providence, R.I., 1984, pp. 197–207.Google Scholar
  37. 37.
    Culler, M. and Morgan, J. W.: Group actions on R-trees, Proc. London Math. Soc. 55 (1987), 571–604.Google Scholar
  38. 38.
    Culler, M. and Vogtmann, K.: Moduli of graphs and automorphisms of free groups, Invent. Math. 84(1) (1986), 91–119.Google Scholar
  39. 39.
    Culler, M. and Vogtmann, K.: The boundary of outer space in rank two, In: Arboreal Group Theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ. 19, Springer, New York, 1991, pp. 189–230.Google Scholar
  40. 40.
    Culler, M. and Vogtmann, K.: A group-theoretic criterion for property FA, Proc. Amer Math. Soc. 124(3) (1996), 677–683.Google Scholar
  41. 41.
    Dyer, J. L. and Formanek, E.: The automorphism group of a free group is complete, J. London Math. Soc. 11 (1975), 181–190.Google Scholar
  42. 42.
    Dyer, J. L., Formanek, E. and Grossman, E. K.: On the linearity of automorphism groups of free groups, Arch. Math. 38 (1982), 404–409.Google Scholar
  43. 43.
    Dyer, J. L. and Scott, G. P.: Periodic automorphisms of free groups, Comm. Algebra 31 (1975), 195–201.Google Scholar
  44. 44.
    Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F. Paterson, M. S. and Thurston, W. P.: Word Processing in Groups, Jones and Bartlett, Boston, MA, 1992.Google Scholar
  45. 45.
    Formanek, E. and Procesi, C.: The automorphism group of a free group is not linear, J. Algebra 149(2) (1992), 494–499.Google Scholar
  46. 46.
    Gaboriau, D. and Levitt, G.: The rank of actions on R-trees, Ann. Sci. é cole Norm. Sup. (4) 28(5) (1995), 549–570.Google Scholar
  47. 47.
    Gaboriau, D., Jaeger, A., Levitt, G. and Lustig, M.: An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93(3) (1998), 425–452.Google Scholar
  48. 48.
    Gaboriau, D. Levitt, G. and Lustig, M.: A dendrological proof of the Scott conjecture for automorphisms of free groups, Proc. Edinburgh Math. Soc. (2) 41(2) (1998), 325–332.Google Scholar
  49. 49.
    Geogehegan, R. and Mihalik, M.: Free abelian cohomology of groups and ends of universal covers, J. Pure Appl. Algebra 36 (1985), 123–137.Google Scholar
  50. 50.
    Geogehegan, R. and Mihalik, M.: The fundamental group at infinity, Topology 35(3) (1996), 655–669.Google Scholar
  51. 51.
    Gersten, S. M.: The automorphism group of a free group is not a CAT(0) group, Proc. Amer. Math. Soc. 121(4) (1994), 999–1002.Google Scholar
  52. 52.
    Gersten, S. M.: A presentation for the special automorphism group of a free group, J. Pure Appl. Algebra 33(3) (1984), 269–279.Google Scholar
  53. 53.
    Gersten, S. M.: Fixed points of automorphisms of free groups, Adv. in Math. 64(1) (1987), 51–85.Google Scholar
  54. 54.
    Gersten, S. M. and Stallings, J. R.: Irreducible outer automorphisms of a free group, Proc. Amer. Math. Soc. 111(2) (1991), 309–314.Google Scholar
  55. 55.
    Glover, H. H. and Mislin, G.: On the p-primary cohomology of Out(F n) in the p-rank one case, J. Pure Appl. Algebra 153(1) (2000), 45–63.Google Scholar
  56. 56.
    Glover, H. H., Mislin, G. and Voon, S. N.: The p-primary Farrell cohomology of Out(F p-1), Geometry and Cohomology in Group Theory (Durham, 1994), London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press, Cambridge, 1998, pp. 161–169.Google Scholar
  57. 57.
    Goldstein, R. Z. and Turner, E. C.: Fixed subgroups of homomorphisms of free groups, Bull. London Math. Soc. 18, (1986), 468–470.Google Scholar
  58. 58.
    Goldstein, R. Z. and Turner, E. C.: Automorphisms of free groups and their fixed points, Invent. Math. 78(1) (1984), 1–12.Google Scholar
  59. 59.
    Goldstein, R. Z. and Turner, E. C.: Monomorphisms of finitely generated free groups have finitely generated equalizers, Invent. Math. 82(2) (1985), 283–289.Google Scholar
  60. 60.
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York-Berlin, 1987, pp. 75–263.Google Scholar
  61. 61.
    Grossman, E. K.: On the residual finiteness of certain mapping class groups, J. London Math. Soc. 9 (1974), 160–164.Google Scholar
  62. 62.
    Hatcher, A.: Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995), 129–137.Google Scholar
  63. 63.
    Hatcher A. and Vogtmann, K.: Isoperimetric inequalities for automorphisms of free groups, Pacific J. Math. 173(2) (1996), 425–442.Google Scholar
  64. 64.
    Hatcher, A. and Vogtmann, K.: Cerf theory for graphs, J. London Math. Soc. (2) 58(3) (1998), 633–655.Google Scholar
  65. 65.
    Hatcher, A. and Vogtmann, K.: The complex of free factors of a free group, Quart. J. Math. Oxford Ser. (2) 49(196) (1998), 459–468.Google Scholar
  66. 66.
    Hatcher, A. and Vogtmann, K.: Rational homology of Aut(F n), Math. Res. Lett. 5(6) (1998), 759–780.Google Scholar
  67. 67.
    Ivanov, N.: Algebraic properties of the Teichmuller modular group, Dokl. Akad. Nauk SSSR 275 (1984), 786–789.Google Scholar
  68. 68.
    Jensen, C. A.: Cohomology of Aut(F n) in the p-rank two case, J. Pure Appl. Algebra 158(1) (2001), 41–81.Google Scholar
  69. 69.
    Kalajdzžievski, S.: Automorphism group of a free group: centralizers and stabilizers, J. Algebra 150(2) (1992), 435–502.Google Scholar
  70. 70.
    Karrass, A., Pietrowski, A. and Solitar, D.: Finite and infinite cyclic extensions of free groups, Collection of articles dedicated to the memory of Hanna Neumann, IV, J. Austral. Math. Soc. 16 (1973), 458–466.Google Scholar
  71. 71.
    Khramtsov, D. G.: Finite groups of automorphisms of free groups, Mat. Zam. 38(3) (1985), 386–392, 476.Google Scholar
  72. 72.
    Khramtsov, D. G.: Completeness of groups of outer automorphisms of free groups, In: Group-Theoretic Investigations (Russian), Akad. Nauk SSSR Ural. Otdel., Sverdlovsk, 1990, pp. 128–143.Google Scholar
  73. 73.
    Kontsevich, M.: Feynman diagrams and low-dimensional topology, In: First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. in Math. 120, Birkhaäuser, Basel, 1994, pp. 97–121.Google Scholar
  74. 74.
    Krammer, D.: The braid group B4 is linear, Invent. Math. 142(3) (2000), 451–486.Google Scholar
  75. 75.
    Krstić, S.: Actions of finite groups on graphs and related automorphisms of free groups, J. Algebra 124(1) (1989), 119–138.Google Scholar
  76. 76.
    Krstić, S.: A uniqueness decomposition theorem for actions of finite groups on free groups, J. Pure Appl. Algebra 61(1) (1989), 29–48.Google Scholar
  77. 77.
    Krstić, S.: Finitely generated virtually free groups have finitely presented automorphism group, Proc. London Math. Soc. (3) 64(1) (1992), 49–69.Google Scholar
  78. 78.
    Krstić, S., Lustig, M. and Vogtmann, K.: An equivariant Whitehead algorithm and conjugacy for roots of Dehn twist automorphisms, Proc. Edinburgh Math. Soc. 44 (2001), 117–141.Google Scholar
  79. 79.
    Krstić, S. and McCool, J.: The non-finite presentability of IA(F 3) and GL(Z[t, t(in-1]), Invent. Math. 129(3) (1997), 595–606.Google Scholar
  80. 80.
    Krstić, S. and Vogtmann, K.: Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68(2) (1993), 216–262.Google Scholar
  81. 81.
    Kulkarni, R.: Lattices on trees, automorphisms of graphs, free groups and surfaces, Preprint, CUNY, 1988.Google Scholar
  82. 82.
    Levitt, G.: Constructing free actions on R-trees, Duke Math. J. 69(3) (1993), 615–633.Google Scholar
  83. 83.
    Levitt, G. and Nicolas, J.-L.: On the maximum order of torsion elements in GL(n, Z) and Aut(F n), J. Algebra 208(2) (1998), 630–642.Google Scholar
  84. 84.
    Los, J. E.: On the conjugacy problem for automorphisms of free groups, With an addendum by the author, Topology 35(3) (1996), 779–808.Google Scholar
  85. 85.
    Lubotzky, A. and Pak, I.: The product replacement algorithm and Kazhdan's property (T), J. Amer. Math. Soc. 14(2) (2001), 347–363 (electronic).Google Scholar
  86. 86.
    Lustig, M.: Structure and conjugacy for automorphisms of free groups I, Preprint, Max-Planck-Institut für Mathematik, 2000.Google Scholar
  87. 87.
    Lustig, M.: Structure and conjugacy for automorphisms of free groups II, Preprint, Max-Planck-Institut für Mathematik, 2001.Google Scholar
  88. 88.
    Lyndon, R. C.: Problems in combinatorial group theory, In: Combinatorial GroupTheory and Topology, Ann. of Math. Stud. 111, Princeton Univ. Press, NJ, 1987.Google Scholar
  89. 89.
    Lyndon, R. C. and Schupp, P. E.: Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.Google Scholar
  90. 90.
    Magnus, W.: Über n-dimensional Gittertransformationen, Acta Math. 64 (1934), 353–367.Google Scholar
  91. 91.
    Magnus, W., Karass, A. and Solitar, D.: Combinatorial Group Theory, Wiley, New York, 1966.Google Scholar
  92. 92.
    Mazurov, V. D.: Finite groups of outer automorphisms of free groups, Siberian Math. J. 32 (1991), 796–811.Google Scholar
  93. 93.
    McCarthy, J. D.: A ‘Tits-alternative’ for subgroups of mapping class groups, Trans. Amer. Math. Soc. 291 (1985), 583–612.Google Scholar
  94. 94.
    McCool, J.: The automorphism groups of finite extensions of free groups, Bull. London Math. Soc. 20(2) (1988), 131–135.Google Scholar
  95. 95.
    McCool, J.: A presentation for the automorphism group of a free group of finite rank, J. London Math. Soc. (2) 8 (1974), 259–266.Google Scholar
  96. 96.
    McCool, J.: A faithful polynomial representation of Out F3, Math. Proc. CambridgePhilos. Soc. 106(2) (1989), 207–213.Google Scholar
  97. 97.
    McCullough, D. and Miller, A.: Symmetric automorphisms of free products, Mem. Amer. Math. Soc. 122 (1996), No. 582.Google Scholar
  98. 98.
    Morita, S.: Structure of the mapping class groups of surfaces: a survey and a prospect, In: Proce. Kirbyfest (Berkeley, CA, 1998), (electronic), Geom. Topol. Monogr. 2, Geom. Topol., Coventry, 1999, pp. 349–406.Google Scholar
  99. 99.
    Nielsen, J.: Die isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924), 169–209.Google Scholar
  100. 100.
    Nielsen, J.: Die Gruppe der dreidimensionalen Gittertransformationen, Danske Vid. Selsk. Mat.-Fys. Medd. 12 (1924), 1–29.Google Scholar
  101. 101.
    Nielsen, J. and Nielsen, J.: Collected Mathematical Papers, Vols. 1-2, Edited and with a preface by Vagn Lundsgaard Hansen, Contemp. Mathematicians. Birkhôuser, Boston, Mass., 1986.Google Scholar
  102. 102.
    Paulin, F.: The Gromov topology of R-trees, Topology Appl. 32 (1989), 197–221.Google Scholar
  103. 103.
    Paulin, F.: A dynamical system approach to free actions on R-trees: a survey with complements, In: Geometric Topology (Haifa, 1992), Contemp. Math. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 187–217.Google Scholar
  104. 104.
    Paulin, F.: Actions de groupes sur les arbres, Seiminaire Bourbaki, Vol. 1995/96. Astirisque No. 241 (1997), Exp. No. 808, 3, 97–137.Google Scholar
  105. 105.
    Pettet, M.: Virtually free groups with finitely many outer automorphisms, Trans. Amer. Math. Soc. 349(11) (1997), 4565–4587.Google Scholar
  106. 106.
    Quillen, D.: Homotopy properties of the poset of p-subgroups of a finite group, Adv. in Math. 28 (1978), 129–137.Google Scholar
  107. 107.
    Rickert, J.: On the simple connectivity at infinity of Out F*, Thesis, Vanderbilt University, 1995.Google Scholar
  108. 108.
    Sela, Z.: The isomorphism problem for hyperbolic groups, I, Ann. of Math. (2) 141(2) (1995), 217–283.Google Scholar
  109. 109.
    Sela, Z.: The Nielsen-Thurston classi.cation and automorphisms of a free group I, Duke Math. J. 84 (2) (1996), 379–397.Google Scholar
  110. 110.
    Sela, Z.: The Nielsen-Thurston classification and automorphisms of a free group II, Preprint 1996.Google Scholar
  111. 111.
    Serre, J.-P.: Arbres, amalgames, SL2, Astérisque No. 46 (1977).Google Scholar
  112. 112.
    Shalen, P. B.: Dendrology and its applications, In: Group Theory from a GeometricalViewpoint (Trieste, 1990), World Scientific, River Edge, NJ, 1991, pp. 543–616.Google Scholar
  113. 113.
    Skora, R.: Deformations of length functions in groups, Preprint, Columbia University, 1989.Google Scholar
  114. 114.
    Smillie, J. and Vogtmann, K.: Automorphisms of graphs, p-subgroups of Out(F n) and the Euler characteristic of Out(F n), J. Pure Appl. Algebra 49 (1987), 187–200.Google Scholar
  115. 115.
    Smillie, J. and Vogtmann, K.: A generating function for the Euler characteristic of Out(F n), J. Pure Appl. Algebra 44 (1987), 329–348.Google Scholar
  116. 116.
    Smillie, J. and Vogtmann, K.: Length functions and outer space, Michigan Math J. 39(3) (1992), 485–493.Google Scholar
  117. 117.
    Spanier, E. H.: Algebraic Topology, McGraw-Hill, New York, 1966.Google Scholar
  118. 118.
    Stallings, J. R.: Topology of finite graphs, Invent. Math. 71(3) (1983), 551–565.Google Scholar
  119. 119.
    Stallings, J. R.: Topologically unrealizable automorphisms of free groups, Proc. Amer. Math. Soc. 84(1) (1982), 21–24.Google Scholar
  120. 120.
    Stallings, J. R.: A graph-theoretic lemma and group-embeddings, In: Combinatorial Group Theory and Topology (Alta, Utah, 1984), Ann. of Math. Stud. 111, Princeton Univ. Press, Princeton, N.J., 1987, pp. 145–155.Google Scholar
  121. 121.
    Stallings, J. R.: Graphical theory of automorphisms of free groups, In: Combinatorial Group Theory and Topology (Alta, Utah, 1984), Ann. of Math. Stud. 111, Princeton Univ. Press, Princeton, N.J., 1987, pp. 79–105.Google Scholar
  122. 122.
    Stallings, J. R.: Foldings of G-trees, In: Arboreal Group Theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ. 19, Springer, New York, 1991, pp. 355–368.Google Scholar
  123. 123.
    Stallings, J. R. and Gersten, S. M.: Casson's idea about 3-manifolds whose universal cover is R3, Internat. J. Algebra Comput. 1(4) (1991), 395–406.Google Scholar
  124. 124.
    Steiner, M.: Gluing data and group actions on R-trees, Thesis, Columbia University, 1988.Google Scholar
  125. 125.
    Vogtmann, K.: End invariants of the group of outer automorphisms of a free group, Topology 34(3) (1995), 533–545.Google Scholar
  126. 126.
    Vogtmann, K.: Local structure of some Out(F n)-complexes, Proc. Edinburgh Math. Soc. (2) 33(3) (1990), 367–379.Google Scholar
  127. 127.
    Wang, S. C. and Zimmermann, B.: The maximum order of finite groups of outer automorphisms of free groups, Math. Z. 216 (1) (1994), 83–87.Google Scholar
  128. 128.
    Watatani, Y.: Property T of Kazhdan implies property FA of Serre, Math. Japon. 27(1) (1982), 97–103.Google Scholar
  129. 129.
    White, T.: Fixed points of finite groups of free group automorphisms, Proc. Amer. Math. Soc. 118(3) (1993), 681–688.Google Scholar
  130. 130.
    Whitehead, J. H. C.: On certain sets of elements in a free group, Procf. London Math. Soc. 41 (1936), 48–56.Google Scholar
  131. 131.
    Zieschang, H., Vogt, E. and Coldewey, H.-D.: Surfaces and Planar Discontinuous Groups, Transl. from the German by John Stillwell, Lecture Notes in Math 835, Springer, Berlin, 1980.Google Scholar
  132. 132.
    Zimmermann, B.: Uber Homvomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv. 56(3) (1981), 474–486.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Karen Vogtmann
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaU.S.A.

Personalised recommendations