Skip to main content
SpringerLink
Log in

Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

An Erratum to this article was published on 30 March 2017

Abstract

Let \(\mathcal {X}_S\) denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let \(\mathcal {C}_S\) denote the set of fundamental groups of spaces in \(\mathcal {X}_S\). In this paper, we characterize the abstract commensurability classes within \(\mathcal {C}_S\) in terms of the ratio of the Euler characteristic of the surfaces identified and the topological type of the curves identified. We prove that all groups in \(\mathcal {C}_S\) are quasi-isometric by exhibiting a bilipschitz map between the universal covers of two spaces in \(\mathcal {X}_S\). In particular, we prove that the universal covers of any two such spaces may be realized as isomorphic cell complexes with finitely many isometry types of hyperbolic polygons as cells. We analyze the abstract commensurability classes within \(\mathcal {C}_S\): we characterize which classes contain a maximal element within \(\mathcal {C}_S\); we prove each abstract commensurability class contains a right-angled Coxeter group; and, we construct a common CAT(0) cubical model geometry for each abstract commensurability class.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Belen’kiĭ, A.S., Burago, Y.D.: Bi-Lipschitz-equivalent Aleksandrov surfaces. I. Algebra i Analiz 16(4), 24–40 (2004)

    MathSciNet  Google Scholar 

  2. Bogopolski, O., Bux, K.-U.: Subgroup conjugacy separability for surface groups, pp. 1–22. arXiv:1401.6203 (2012)

  3. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)

    Google Scholar 

  4. Behrstock, J.A., Januszkiewicz, T., Neumann, W.D.: Commensurability and QI classification of free products of finitely generated abelian groups. Proc. Am. Math. Soc. 137(3), 811–813 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Behrstock, J.A., Neumann, W.D.: Quasi-isometric classification of graph manifold groups. Duke Math. J. 141(2), 217–240 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cashen, C.H.: Quasi-isometries between tubular groups. Groups Geom. Dyn. 4(3), 473–516 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crisp, J., Paoluzzi, L.: Commensurability classification of a family of right-angled Coxeter groups. Proc. Am. Math. Soc. 136(7), 2343–2349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Dani, P., Thomas, A.: Quasi-isometry classification of certain right-angled coxeter groups,. arXiv:1402.6224 (2014)

  10. Edmonds, A.L., Kulkarni, R.S., Stong, R.E.: Realizability of branched coverings of surfaces. Trans. Am. Math. Soc. 282(2), 773–790 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Farb, B., Margalit, D.: A Primer on Mapping Class Groups, volume 49 of Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)

    Google Scholar 

  12. Frigerio, R.: Commensurability of hyperbolic manifolds with geodesic boundary. Geom. Dedic. 118, 105–131 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Green, E.: Graph products of groups. Thesis (Ph.D.)—The University of Leeds (1990)

  14. Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, volume 8 of Math. Sci. Res. Inst. Publ., pp. 75–263. Springer, New York (1987)

  15. Husemoller, D.H.: Ramified coverings of Riemann surfaces. Duke Math. J. 29, 167–174 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  16. Haglund, F., Wise, D.T.: Special cube complexes. Geom. Funct. Anal. 17(5), 1551–1620 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Michael, K.: Hyperbolic Manifolds and Discrete Groups. Modern Birkhäuser Classics. Birkhäuser Boston, Inc, Boston (2009). Reprint of the 2001 edition

    Google Scholar 

  18. Lafont, J.-F.: Diagram rigidity for geometric amalgamations of free groups. J. Pure Appl. Algebra 209(3), 771–780 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Malone, W.: Topics in geometric group theory. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)—The University of Utah (2010)

  20. Margulis, G.A.: Discrete groups of motions of manifolds of nonpositive curvature. In: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 2, pp. 21–34. Canad. Math. Congress, Montreal, Que. (1975)

  21. McMullen, C.T.: Kleinian groups. http://www.math.harvard.edu/~ctm/programs/index.html

  22. Mosher, L., Sageev, M., Whyte, K.: Quasi-actions on trees I. Bounded valence. Ann. Math. 158(1), 115–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Neumann, W.D.: Commensurability and virtual fibration for graph manifolds. Topology 36(2), 355–378 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Paoluzzi, L.: The notion of commensurability in group theory and geometry. RIMS Kkyroku 1836, 124–137 (2013)

    Google Scholar 

  25. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149, second edn. Springer, New York (2006)

    MATH  Google Scholar 

  26. Sageev, M.: CAT(0) cube complexes and groups. In: Geometric Group Theory, volume 21 of IAS/Park City Math. Ser., pp. 7–54. Am. Math. Soc., Providence (2014)

  27. Schwartz, R.E.: The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. 82, 133–168 (1995). (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. 17(3), 555–565 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  30. Scott, P., Wall, T.: Topological methods in group theory. In: Homological Group Theory Proc. Sympos., Durham, 1977), volume 36 of London Math. Soc. Lecture Note Ser., pp. 137–203. Cambridge Univ. Press, Cambridge (1979)

  31. Walsh, GS.: Orbifolds and commensurability. In: Interactions between Byperbolic Geometry, Quantum Topology and Number Theory, volume 541 of Contemp. Math., pp. 221–231. Am. Math. Soc., Providence (2011)

  32. Whyte, K.: Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture. Duke Math. J. 99(1), 93–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is deeply grateful for many discussions with her Ph.D. advisor Genevieve Walsh. The author wishes to thank Pallavi Dani for pointing out a gap in an earlier version of this paper, and her peers at Tufts University for helpful conversations throughout this work. The author is thankful for very useful comments and corrections from an anonymous referee. This material is partially based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0806676.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Haifa, 199 Abba Khoushy Avenue, Mount Carmel, 3498838, Haifa, Israel

    Emily Stark

Authors
  1. Emily Stark
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emily Stark.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s10711-017-0237-x.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stark, E. Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams. Geom Dedicata 186, 39–74 (2017). https://doi.org/10.1007/s10711-016-0179-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-016-0179-8

Keywords

Mathematics Subject Classification

Search

Navigation