Advertisement

Geometriae Dedicata

, Volume 194, Issue 1, pp 99–129 | Cite as

Divergence spectra and Morse boundaries of relatively hyperbolic groups

  • Hung Cong TranEmail author
Original Paper
  • 82 Downloads

Abstract

We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence spectra of relatively hyperbolic groups. We show the existence of an infinite collection of right-angled Coxeter groups which all have exponential divergence but they all have different divergence spectra. We also study Morse boundaries of relatively hyperbolic groups and examine their connection with Bowditch boundaries.

Keywords

Divergence spectra Morse boundaries Relatively hyperbolic groups 

Mathematics Subject Classification (2000)

20F67 20F65 

Notes

Acknowledgements

I would like to thank Prof. Ruth Charney for her suggestion to study the quasi-isometry invariant divergence spectra and her encouragement to publish this paper. I also thank Prof. Kim Ruane for helpful conversations about the connection between contracting boundaries and Bowditch boundaries that gave me motivation for studying Morse boundaries of relatively hyperbolic groups. I want to thank Prof. Christopher Hruska, Prof. Pallavi Dani, Hoang Thanh Nguyen and Kevin Schreve for their very helpful conversations and suggestions. I also thank the referee for advice that improved the exposition of the paper.

References

  1. 1.
    Behrstock, J., Charney, R.: Divergence and quasimorphisms of right-angled Artin groups. Math. Ann. 352(2), 339–356 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Behrstock, J., Druţu, C., Mosher, L.: Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344(3), 543–595 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Behrstock, J., Hagen, M.F., Sisto, A.: Thickness, relative hyperbolicity, and randomness in coxeter groups. With an appendix written jointly with Pierre-Emmanuel Caprace. arXiv:1312.4789 (preprint)
  4. 4.
    Bowditch, B.H.: Relatively hyperbolic groups. Int. J. Algebra Comput. 22(3), 1250016 (2012). doi: 10.1142/S0218196712500166
  5. 5.
    Caprace, P.-E.: Erratum to “Buildings with isolated subspaces and relatively hyperbolic Coxeter groups” [MR2665193]. Innov. Incid. Geom. 14, 77–79 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cordes, M.: Morse boundaries of proper geodesic metric spaces. arXiv:1502.04376 (preprint)
  7. 7.
    Charney, R., Sultan, H.: Contracting boundaries of \(\text{ CAT }(0)\) spaces. J. Topol. 8(1), 93–117 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dani, P., Thomas, A.: Divergence in right-angled Coxeter groups. Trans. Am. Math. Soc. 367(5), 3549–3577 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davis, M.W.: The Geometry and Topology of Coxeter Groups, Volume 32 of London Mathematical Society Monographs Series. Princeton University Press, Princeton (2008)Google Scholar
  10. 10.
    Druţu, C., Sapir, M.: Tree-graded spaces and asymptotic cones of groups. Topology 44(5), 959–1058 (2005). (With an appendix by Denis Osin and Sapir)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Druţu, C., Mozes, S., Sapir, M.: Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Am. Math. Soc. 362(5), 2451–2505 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Duchin, M., Rafi, K.: Divergence of geodesics in Teichmüller space and the mapping class group. Geom. Funct. Anal. 19(3), 722–742 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gersten, S.M.: Divergence in \(3\)-manifold groups. Geom. Funct. Anal. 4(6), 633–647 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gersten, S.M.: Quadratic divergence of geodesics in \(\text{ CAT }(0)\) spaces. Geom. Funct. Anal. 4(1), 37–51 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hruska, G.C.: Relative hyperbolicity and relative quasiconvexity for countable groups. Algebr. Geom. Topol. 10(3), 1807–1856 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Macura, N.: CAT(0) spaces with polynomial divergence of geodesics. Geom. Dedicata 163, 361–378 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sisto, A.: On metric relative hyperbolicity. arXiv:1210.8081 (preprint)
  18. 18.
    Tran, H.C.: Relations between various boundaries of relatively hyperbolic groups. Int. J. Algebr. Comput. 23(7), 1551–1572 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tran, H.C.: Relative divergence of finitely generated groups. Algebr. Geom. Topol. 15(3), 1717–1769 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tran, H.C.: Divergence of Morse geodesics. Geom. Dedicata 180, 385–397 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe University of GeorgiaAthensUSA

Personalised recommendations