Skip to main content
SpringerLink
Log in

A limit set intersection theorem for graphs of relatively hyperbolic groups

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Let G be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and edge groups satisfy the limit set intersection property for conical limit points (refer to Definition 3 and Definition 23 for the definitions of conical limit points and limit set intersection property respectively). This result is motivated by the work of Sardar for graph of hyperbolic groups [16].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson J W, Intersections of analytically and geometrically finite subgoups of Kleinian groups, Trans. Amer. Math. Soc.343(1) (1994) 87–98

    Article  MathSciNet  Google Scholar 

  2. Anderson J W, Intersections of topologically tame subgoups of Kleinian groups, J. Anal. Math.65 (1995) 77–94

    Article  MathSciNet  Google Scholar 

  3. Anderson J W, The limit set intersection theorem for finitely generated Kleinian groups, Math. Res. Lett.3(5) (1996) 675–692

    Article  MathSciNet  Google Scholar 

  4. Anderson J W, Limit set intersection theorems for Kleinian groups and a conjecture of Susskind, Comput. Methods Funct. Theory14(2-3) (2014) 453–464

    Article  MathSciNet  Google Scholar 

  5. Bowditch B H, Relatively hyperbolic groups, Internat. J. Algebra Comput.22(3) (2012) 1250016-1

    Article  MathSciNet  Google Scholar 

  6. Bridson M R and Haefliger A, Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1999) (Berlin: Springer-Verlag)

  7. Das T and Simmons D, Intersecting limit sets of Kleinian subgroups and Susskinds question 2019. arXiv:1802.07654v4

  8. Farb B, Relatively hyperbolic groups, Geom. Funct. Anal.8(5) (1998) 810–840

    Article  MathSciNet  Google Scholar 

  9. Gromov M, Hyperbolic Groups, in: Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75–263 (1987) (New York: Springer)

  10. Kapovich I and Benakli N, Boundaries of hyperbolic groups, in: Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) volume 296 of Contemp. Math., pages 39–93, Amer. Math. Soc., Providence, RI (2002)

  11. Mitra M, Cannon–Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom.48(1) (1998) 135–164

    Article  MathSciNet  Google Scholar 

  12. Minsky Y, Bounded geometry for Kleinian groups, Invent. Math.146(1) (2001) 143–192

    Article  MathSciNet  Google Scholar 

  13. Mahan Mj and Pal A, Relative hyperbolicity, trees of spaces and Cannon–Thurston maps, Geom. Dedicata151 (2011 59–78

  14. Mahan Mj and Reeves L, A combination theorem for strong relative hyperbolicity, Geom. Topol.12(3) (2008) 1777–1798

  15. Pal A, Cannon–Thurston maps and relative hyperbolicity Ph.D. Thesis (2009) (Kolkata: Indian Statistical Institute)

  16. Sardar P, Graphs of hyperbolic groups and a limit set intersection theorem, Proc. Amer. Math. Soc.146(5) (2018) 1859–1871

    Article  MathSciNet  Google Scholar 

  17. Sardar P, Corrigendum to “Graphs of hyperbolic groups and a limit set intersection theorem”, arXiv:1909.01823

  18. Serre J-P, Trees, Springer Monographs in Mathematics (2003) (Berlin: Springer-Verlag), translated from the French original by John Stillwell, corrected 2nd printing of the 1980 English translation

  19. Susskind P D and Swarup G A, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math.114(2) (1992) 233–250

    Article  MathSciNet  Google Scholar 

  20. Susskind P D, On Kleinian groups with intersecting limit sets, ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis (1982) (State University of New York at Stony Brook)

  21. Scott P and Wall T, Topological methods in group theory, in: Homological group theory (Proc. Sympos., Durham, 1977). pp. 137–203, London Mathematical Society Lecture Notes Series, vol. 36 (1979) (Cambridge, New York: Cambridge University Press); ISBN 0-521-22729-1

  22. Yang Wen-yuan, Limit sets of relatively hyperbolic groups, Geom. Dedicata13 (2009) 76–90

    Google Scholar 

Download references

Acknowledgements

The author is extremely grateful to Dr. Pranab Sardar for not only suggesting the problem but also for his invaluable comments, discussions and corrections. The author would like to thank Dr. Sushil Bhunia for proofreading the paper. The author would also like to thank the referee for the helpful suggestions and comments that improved the readability of the paper. Finally, the author would like to thank IISER, Mohali for the financial support towards this work.

Author information

Authors and Affiliations

  1. Indian Institute of Science Education and Research Mohali, Mohali, Punjab, 140 306, India

    Swathi Krishna

Authors
  1. Swathi Krishna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swathi Krishna.

Additional information

Communicated by Mj Mahan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Krishna, S. A limit set intersection theorem for graphs of relatively hyperbolic groups. Proc Math Sci 130, 36 (2020). https://doi.org/10.1007/s12044-020-00563-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12044-020-00563-x

Keywords

Mathematics Subject Classification

Search

Navigation