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Branch Points and Free Actions On ℝ-Trees

  • Renfang Jiang
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 19)

Abstract

Actions of groups on ℝ-trees are natural generalizations of actions of groups on ℤ-trees (simplicial trees). The latter, known as Bass-Serre Theory, has become a standard tool in combinatorial group theory. This theory says that a group action on a ℤ-tree is equivalent to a combinatorial presentation of the group. On the other hand, actions of groups on ℝ-trees arise from groups of isometries of hyperbolic spaces, and have significant applications in the study of hyperbolic manifolds. Morgan, Shalen, Culler and others define and analyze compactification of representation space R(G) of a group G in SL(2, ℂ) and SO (n,1). For a valuation of the function field of a subvariety of R(G), they define an ℝ-tree on which G acts. This is applied notably when G = π 1(M), M a hyperbolic manifold. The tree and action then give a codimension-1 measured lamination on M. Applications include the theory of incompressible surfaces in 3-manifolds and the description of a natural boundary for Teichmüller space. The points at infinity correspond to πl (M)-actions on ℝ-trees.

Keywords

Branch Point Free Product Free Action Hyperbolic Manifold Free Abelian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AB]
    R. Alperin and H. Bass, Length functions of group actions on A-trees, Combinatorial group theory and topology, Ann. of Math. Studies III (1987), Princeton Univ. Press.Google Scholar
  2. [AM]
    R. Alperin and K. Moss, Complete trees for groups with a real-valued length function,J. London Math. Soc. (2) 31 (1985).MathSciNetGoogle Scholar
  3. [B]
    H. Bass, Group actions on non-archimedean trees, Proceedings of the Workshop on Arboreal Group Theory, MSRI, Sept., 1988, to appear.Google Scholar
  4. [BF]
    M. Bestvina and M. Feighn, Bounding the complexity of simplicial group actions on trees, preprint.Google Scholar
  5. [CM]
    M. Culler and J.W. Morgan, Group actions on R-trees, Proc. London Math. Soc. (3) 55 (1987), 451–463.MathSciNetGoogle Scholar
  6. [CV]
    M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91–119.MathSciNetMATHGoogle Scholar
  7. [GS]
    H. Gillet and P.B. Shalen, Dendrology of groups in low Q-ranks, preprint, J. Dif. Geom. (1988) (to appear).Google Scholar
  8. [J]
    R. Jiang, On free actions on a-trees, Dissertation, Columbia University.Google Scholar
  9. [L]
    R.C. Lyndon, Length functions in groups, Math Scand. 12 (1963), 209–234MathSciNetMATHGoogle Scholar
  10. [Ml]
    J.W. Morgan, Ergodic theory and free group actions on R-trees, preprint.Google Scholar
  11. [M2]
    J.W. Morgan, UCLA lecture notes, CBMS Series in Mathematics (to appear).Google Scholar
  12. [M3]
    J.W. MorganGroup actions on trees and the compactifications of the space of classes of SO(n, 1)-representations, Topology 25 (1986), 1–34.MathSciNetMATHCrossRefGoogle Scholar
  13. [MS1]
    J.W. Morgan and P.B. Shalen, Valuations, trees, and degenerations of hyperbolic structure,I, Ann. of Math. 120 (1984), 401–476.MathSciNetMATHCrossRefGoogle Scholar
  14. [MS2]
    J.W. Morgan and P.B. Shalen, Degenerations of hyperbolic structures III,Ann. of Math. 127 (1988), 457–519.MathSciNetMATHCrossRefGoogle Scholar
  15. [P]
    W. Parry, Pseudo-length functions on groups, preprint.Google Scholar
  16. [SB]
    J.P. Serre and H. Bass, Arbres, Amalgames, SL2, Astérisque 46 (1977)Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • Renfang Jiang
    • 1
  1. 1.Department of MathematicsColumbia UniversityColumbiaUSA

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