Dendrology of Groups: An Introduction

  • Peter B. Shalen
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 8)


The study of group actions on “generalized trees” or “ℝ-trees” has recently been attracting the attention of mathematicians in several different fields. This subject had its beginnings in the work of R. Lyndon [L] and I. Chiswell [Chi], and—from a different point of view—in the work of J. Tits [Ti]. The link between the points of view of these authors was provided by R. Alperin and K. Moss in [AM]. J. Morgan and I, in [MoS1,Mo2], established connections of this theory with hyperbolic geometry and with W. Thurston’s theory of measured laminations. The picture has been developed further by the above-mentioned people and also by H. Bass, M. Bestvina, M. Culler, H. Gillet, M. Gromov, W. Parry, F. Rimlinger and J. Stallings, among others.


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  1. [AB]
    R.C. Alperin and H. Bass, “Length functions of group actions on Λ-trees,” preprint.Google Scholar
  2. [AM]
    R.C. Alperin and K.N. Moss, “Complete trees for groups with a real-valued length function,” J. London Math. Soc. (2), 31 (1985), 55–68.MathSciNetMATHCrossRefGoogle Scholar
  3. [Be]
    M. Bestvina, to appear.Google Scholar
  4. [Ca]
    A. Casson and S. Bleiler, “Automorphisms of surfaces after Nielsen and Thurston,” London Math. Soc. Student Text Series, to appear.Google Scholar
  5. [Chi]
    I.M. Chiswell, “Abstract length functions in groups,” Math. Proc. Camb. Phil. Soc. 80 (1976), 451–463.MathSciNetMATHCrossRefGoogle Scholar
  6. [Chu]
    V. Chuckrow, “On Schottky groups with applications to Kleinian groups,” Ann. of Math. 88 (1968), 42–58.MathSciNetCrossRefGoogle Scholar
  7. [CuM]
    M. Culler and J.W. Morgan, “Group actions on ℝ-trees,” MSRI preprint 06512–85. (To appear in Proc. London Math. Soc).Google Scholar
  8. [CuV]
    M. Culler and K. Vogtmann, “Moduli of graphs and outer automorphisms of free groups,” Invent. Math. 84 (1986), 91–119.MathSciNetMATHCrossRefGoogle Scholar
  9. [FaLP]
    A. Fathi, F. Laudenbach and V. Poenaru, “Travaux de Thurston sur les surfaces,” Astérisque 66–67 (1979).Google Scholar
  10. [FlO]
    W. Floyd and U. Oertel, “Incompressible surfaces via branched surfaces,” Topology 23 (1984), 117–125.MathSciNetMATHCrossRefGoogle Scholar
  11. [Ge]
    S. Gersten, “Topology of the automorphism group of a free group,” to appear.Google Scholar
  12. [GiS]
    H. Gillet and P.B. Shalen, “Trees defined over rank-two subgroups of ℝ,” in preparation.Google Scholar
  13. [Hk]
    W. Haken, “Theorie der Normalflächen,” Acta Math. 105 (1961), 245–375.MathSciNetMATHCrossRefGoogle Scholar
  14. [HrdW]
    G.H. Hardy and E.M. Wright, “An introduction to the theory of numbers,” (4th ed.), Oxford, the Clarendon Press, 1960.MATHGoogle Scholar
  15. [Hrr]
    N. Harrison, “Real length functions in groups,” Trans. Amer. Math. Soc. 174 (1972), 77–106.MathSciNetCrossRefGoogle Scholar
  16. [Ja]
    W. Jaco, “Finitely presented subgroups of three-manifold groups,” Invent. Math. 13 (1971), 335–346.MathSciNetMATHCrossRefGoogle Scholar
  17. [JaS]
    W. Jaco and P.B. Shalen, “Seifert fibered spaces in 3-manifolds,” Memoirs Amer. Math. Soc. 21 (1979), No. 220.Google Scholar
  18. [Jo]
    K. Johannson, “Homotopy equivalences of 3-manifolds with boundaries,” Lecture Notes in Mathematics, 761, Springer-Verlag, 1979.MATHGoogle Scholar
  19. [JoM]
    D. Johnson and J.J. Millson, “Deformation spaces associated to compact hyperbolic manifolds,” to appear in Discrete Subgroups in Geometry and Analysis, proceedings of a conference held in honor of G.D. Mostow on his 60th birthday.Google Scholar
  20. [KS]
    A. Karrass and D. Solitar, “Subgroup theorems in the theory of groups given by defining relations,” Comm. Pure and Appl. Math. 11 (1958), 547–571.MathSciNetMATHCrossRefGoogle Scholar
  21. [L]
    R.C. Lyndon, “Length functions in groups,” Math. Scand., 12 (1963), 209–234.MathSciNetMATHGoogle Scholar
  22. [Ma]
    M.S. Massey, “Algebraic topology: an introduction,” Graduate Texts in Mathematics, No. 56, Springer-Verlag, 1977.Google Scholar
  23. [Mo1]
    J.W. Morgan, “On Thurston’s uniformization theorem for 3-dimensional manifolds,” The Smith Conjecture, J.W. Morgan and H. Bass eds., Academic Press, 1984.Google Scholar
  24. [Mo2]
    J.W. Morgan, “Group actions on trees and conjugacy classes of SO(n,1) representations,” Topology, to appear.Google Scholar
  25. [Mo3]
    J.W. Morgan, “Deformation and degeneration of geometric structures,” CBMS Regional Conference Series in Mathematics, to appear.Google Scholar
  26. [MoO]
    J.W. Morgan and J.P. Otal, “Non-archimedean measures laminations and degenerations of surfaces,” to appear.Google Scholar
  27. [MoS1]
    J.W. Morgan and P.B. Shalen, “Valuations, trees, and degenerations of hyperbolic structures, I.,” Ann. of Math. 120 (1984), 401–476.MathSciNetMATHCrossRefGoogle Scholar
  28. [MoS2]
    J.W. Morgan and P.B. Shalen, “Valuations, trees and degenerations of hyperbolic structures, II.,” preprint.Google Scholar
  29. [MoS3]
    J.W. Morgan and P.B. Shalen, “Actions of surface groups on trees,” in preparation.Google Scholar
  30. [P]
    W. Parry, “Pseudo-length functions on groups,” preprint.Google Scholar
  31. [Sc]
    G.P. Scott, “Finitely generated 3-manifold groups are finitely presented,” J. London Math. Soc. 6 (1973), 437–440.MathSciNetMATHCrossRefGoogle Scholar
  32. [ScW]
    G.P. Scott and C.T.C. Wall, “Topological methods in group theory,” Homological Group Theory, C.T.C. Wall ed., London Math. Soc. Lecture Note Series, No. 36, Cambridge Univ. Press, 1979.Google Scholar
  33. [Se]
    J.P. Serre, “Trees,” Springer-Verlag, 1980.MATHCrossRefGoogle Scholar
  34. [Ti]
    J. Tits, “A theorem of Lie-Kolchin for trees,” Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin (Academic Press, 1977), 377–388.Google Scholar
  35. [Tr]
    M. Tretkoff, “A topological approach to the theory of groups acting on trees,” J. Pure Appl. Algebra 16 (1980), 323–333.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Peter B. Shalen
    • 1
  1. 1.University of Illinois at ChicagoUSA

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