Advertisement

Splittable Fuchsian Groups

  • Sang-hyun Kim
  • Thomas Koberda
  • Mahan Mj
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2231)

Abstract

In this chapter, we study deformations of (possibly indiscrete) faithful representations of Fuchsian groups such that almost all points on the deformations are still faithful. Let L be a splittable Fuchsian group, which includes all Fuchsian groups with Euler characteristic at most − 1; see Definition 4.3. We will prove that an arbitrarily small deformation of a given representation can be chosen so that the new trace spectrum is almost disjoint from the original one (Theorem 4.1). Then we show Xproj(L) contains at least one indiscrete representation (Lemma 4.10). Moreover, if an open set U contains at least one indiscrete representation in Xproj(L), then U contains uncountably many pairwise inequivalent indiscrete representation in Xproj(L) (Theorem 4.2).

References

  1. 4.
    J. Barlev, T. Gelander, Compactifications and algebraic completions of limit groups. J. Anal. Math. 112, 261–287 (2010). MR 2763002MathSciNetCrossRefGoogle Scholar
  2. 7.
    M. Bestvina, M. Feighn, Notes on Sela’s work: limit groups and Makanin-Razborov diagrams, in Geometric and Cohomological Methods in Group Theory. London Mathematical Society Lecture Note Series, vol. 358 (Cambridge University Press, Cambridge, 2009), pp. 1–29. MR 2605174Google Scholar
  3. 12.
    A. Borel, On free subgroups of semisimple groups. Enseign. Math. (2) 29(1–2), 151–164 (1983). MR 702738Google Scholar
  4. 14.
    K. Bou-Rabee, M. Larsen, Linear groups with Borel’s property. J. Eur. Math. Soc. 19(5), 1293–1330 (2017). MR 3635354MathSciNetCrossRefGoogle Scholar
  5. 18.
    E. Breuillard, D. Guralnick, B. Green, T. Tao, Strongly dense free subgroups of semisimple algebraic groups. Israel J. Math. 192(1), 347–379 (2012)MathSciNetCrossRefGoogle Scholar
  6. 51.
    W.M. Goldman, Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988). MR 952283MathSciNetCrossRefGoogle Scholar
  7. 78.
    K. Mann, Rigidity and flexibility of group actions on the circle, in Handbook of Group Actions (2015, to appear)Google Scholar
  8. 79.
    K. Mann, Spaces of surface group representations. Invent. Math. 201(2), 669–710 (2015). MR 3370623MathSciNetCrossRefGoogle Scholar
  9. 84.
    S. Matsumoto, Some remarks on foliated S 1 bundles. Invent. Math. 90(2), 343–358 (1987). MR 910205Google Scholar
  10. 105.
    H. Wilton, Solutions to Bestvina & Feighn’s exercises on limit groups, in Geometric and Cohomological Methods in Group Theory. London Mathematical Society Lecture Note Series, vol. 358 (Cambridge University Press, Cambridge, 2009), pp. 30–62. MR 2605175 (2011g:20037)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sang-hyun Kim
    • 1
  • Thomas Koberda
    • 2
  • Mahan Mj
    • 3
  1. 1.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  3. 3.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations