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Compactification of Spaces of Representations After Culler, Morgan and Shalen

Part of the Lecture Notes in Mathematics book series (LNM,volume 2119)

Abstract

We give an account on Morgan and Shalen’s work on the compactification of complex affine varieties using valuation spaces and its applications to the geometry of the character variety of a finitely generated group.

Keywords

  • Fundamental Group
  • Irreducible Component
  • Valuation Ring
  • Faithful Representation
  • Partial Axis

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. M. Bestvina, Degenerations of the hyperbolic space. Duke Math. J. 56(1), 143–161 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. M. Bestvina, M. Feighn, Stable actions of groups on real trees. Invent. Math. 121(2), 287–321 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. N. Bourbaki, Algèbre Commutative, Chapitres 5–7 (Masson, 1985)

    Google Scholar 

  4. R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Universitext (Springer, Berlin, 1991)

    Google Scholar 

  5. M. Culler, Lifting representations to covering groups. Adv. Math. 59, 54–70 (1986)

    CrossRef  MathSciNet  Google Scholar 

  6. R. Canary, D. Epstein, P. Green, Notes on notes of Thurston, in Analytical and Geometrical Aspects of Hyperbolic Spaces (Cambridge University Press, Cambridge, 1987), pp. 3–92

    Google Scholar 

  7. I. Chiswell, Abstract length functions in groups. Proc. Cambridge Phil. Soc. 80, 451–463 (1976)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. I. Chiswell, Introduction to \(\Lambda \) -Trees (World Scientific, Singapore, 2001)

    Google Scholar 

  9. I. Chiswell, Non standard analysis and the Morgan-Shalen compactification. Quart. J. Math. Oxford Ser. (2) 42(167), 257–270 (1991)

    Google Scholar 

  10. V. Chuckrow, On Schottky groups with applications to Kleinian groups. Ann. Math. 88, 47–61 (1968)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. M. Culler, C. Gordon, J. Luecke, P. Shalen, Dehn surgery on knots. Ann. Math. (2) 125, 237–300 (1987)

    Google Scholar 

  12. M. Culler, J. Morgan, Group actions on \(\mathbb{R}\)-trees. Proc. Lond. Math. Soc. 55, 571–604 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. M. Culler, J. Morgan, Varieties of group representations and splittings of 3-manifolds. Ann. Math. 117, 109–146 (1983)

    CrossRef  MATH  Google Scholar 

  14. L. Ein, R. Lazarsfeld, K. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields. Am. J. Math. 125(2), 409–440 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque, pp. 66–67 (1979), SMF Paris

    Google Scholar 

  16. W. Floyd, U. Oertel, Incompressible surfaces via branched surfaces. Topology 23, 117–125 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. D. Gaboriau, G. Levitt, F. Paulin, Pseudogroups of isometries of \(\mathbb{R}\) and Rips’ theorem on free actions on \(\mathbb{R}\)-trees. Isr. J. Math. 87, 403–428 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. M. Gromov, Hyperbolic groups. in Essays in Group Theory, ed. by S.M. Gersten (ed.), MSRI Publ., vol. 8 (Springer, Berlin, Heidelberg, New York, 1987), pp. 75–263

    Google Scholar 

  19. R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977)

    Google Scholar 

  20. R. Lyndon, Length functions in groups. Math. Scand. 12, 209–234 (1963)

    MATH  MathSciNet  Google Scholar 

  21. J. Morgan, Group actions on trees and the compactification of the space of classes of \(\mathop{SO}\nolimits (n,1)\)-representations. Topology 25, 1–33 (1986)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. J. Morgan, \(\Lambda \)-trees and their applications. Bull. Am. Math. Soc. 26, 87–112 (1992)

    Google Scholar 

  23. J. Morgan, J.-P. Otal, Relative growth rates of closed geodesics on closed surfaces. Comment. Math. Helvitici 68, 171–208 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. J. Morgan, P. Shalen, Degenerations of hyperbolic structures, I: Valuations, trees and surfaces. Ann. Math. 120, 401–476 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. J. Morgan, P. Shalen, Degenerations of hyperbolic structures II: Measured laminations in 3-manifolds. Ann. Math. 127, 403–456 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. J. Morgan, P. Shalen, Degenerations of hyperbolic structures III: Actions of 3-manifold groups on trees and Thurston’s compactness theorem. Ann. Math. 127, 457–519 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. J. Morgan, P. Shalen, Free actions of surface groups on \(\mathbb{R}\)-trees. Topology 30(2), 143–154 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. D. Mumford, Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34 (Springer, New York, 1965)

    Google Scholar 

  29. D. Mumford, Algebraic Geometry I: Complex Projective Varieties. Grundlehren der Math. Wiss., vol. 221 (Springer, New York, 1976)

    Google Scholar 

  30. J.-P. Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque no. 235 (Soc. Math. France, Paris, 1996)

    Google Scholar 

  31. F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Inv. Math. 94, 53–80 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. F. Paulin, Actions de groupes sur les arbres. Séminaire Bourbaki 1995–96, exp. 808, Astérisque no. 241 (Soc. Math. France, Paris, 1997)

    Google Scholar 

  33. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds. Graduate Texts in Math., vol. 149 (Springer, New York, 1994)

    Google Scholar 

  34. J.-P. Serre, Arbres, Amalgames, \(\mathop{SL}\nolimits (2)\). Astérisque no. 46 (Soc. Math. France, Paris, 1977)

    Google Scholar 

  35. P. Shalen, Dendrology of groups: An introduction. in Essays in Group Theory, ed. by S.M. Gersten. Mathematical Sciences Research Institute Publications, vol. 8 (Springer, New York, 1987)

    Google Scholar 

  36. P. Shalen, Dendrology and its applications, in Group Theory from a Geometrical Viewpoint (Trieste, 1990) (World Scientific, River Edge, NJ, 1991), pp. 543–616

    Google Scholar 

  37. P. Shalen, Representations of 3-manifolds groups, in Handbook of 3-Manifold Groups (North-Holland, Amsterdam, 2002), pp. 955–1044

    Google Scholar 

  38. R. Skora, Geometric actions of surface groups on \(\Lambda \)-trees. Comment. Math. Helvetici 65, 519–533 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  39. R. Skora, Splittings of surfaces. J. Am. Math. Soc. 9(2), 605–616 (1996)

    CrossRef  MATH  MathSciNet  Google Scholar 

  40. J. Stallings, A topological proof of Grushko’s theorem on free products. Math. Zeit. 90, 1–8 (1965)

    CrossRef  MATH  MathSciNet  Google Scholar 

  41. W. Thurston, The Geometry and Topology of 3-Manifolds. Princeton Lecture Notes (1978–1981), http://library.msri.org/books/gt3m/

  42. W. Thurston, Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds. Ann. Math. 124, 203–246 (1986)

    CrossRef  MATH  MathSciNet  Google Scholar 

  43. W. Thurston, Three-Dimensional Geometry and Topology (Princeton University Press, Princeton, NJ, 1995)

    Google Scholar 

  44. M. Vaquié, Valuations and local uniformization. Singularity theory and its applications, pp. 477–527. Adv. Stud. Pure Math., vol. 43 (Math. Soc. Japan, Tokyo, 2006)

    Google Scholar 

  45. M. Vaquié, Extension d’une valuation. Trans. Am. Math. Soc. 359(7), 3439–3481 (2007)

    CrossRef  MATH  Google Scholar 

  46. A. Weil, On discrete subgroups of Lie groups. Ann. Math. (2) 72, 369–384 (1960)

    Google Scholar 

  47. O. Zariski, Foundations of a general theory of birational correspondences. Trans. Am. Math. Soc. 53, 490–542 (1943)

    CrossRef  MATH  MathSciNet  Google Scholar 

  48. O. Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions. Bull. Am. Math. Soc. 50, 683–691 (1944)

    CrossRef  MATH  MathSciNet  Google Scholar 

  49. O. Zariski, P. Samuel, Commutative Algebra, vol. 2. Graduate Texts in Mathematics no 29 (Springer, New York-Heidelberg-Berlin, 1975)

    Google Scholar 

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Acknowledgements

I would like to thank Charles Favre, Jan Kiwi, and Juan Rivera-Letelier for the invitation to give a course during the “Ultrametric Dynamics Days” in Santiago de Chile in January 2008. This paper owes a lot to Charles who encouraged me to develop my set of notes to an actual paper and made several important suggestions. The referee, through his constant criticism, also helped to bring the paper to its actual form.

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Otal, JP. (2015). Compactification of Spaces of Representations After Culler, Morgan and Shalen. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_7

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