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A bound for diameter of arithmetic hyperbolic orbifolds

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Abstract

Let \({\mathcal {O}}\) be a closed n-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of \({\mathcal {O}}\) is bounded above by

$$\begin{aligned} \frac{c_1\log \mathrm {vol}({\mathcal {O}}) + c_2}{h({\mathcal {O}})}, \end{aligned}$$

where \(h({\mathcal {O}})\) is the Cheeger constant of \({\mathcal {O}}\), \(\mathrm {vol}({\mathcal {O}})\) is its volume, and constants \(c_1\), \(c_2\) depend only on n.

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References

  1. Agol, I., Belolipetsky, M., Storm, P., Whyte, K.: Finiteness of arithmetic hyperbolic reflection groups. Groups Geom. Dyn. 2, 481–498 (2008)

    Article  MathSciNet  Google Scholar 

  2. Alpert, H., Belolipetsky, M.: Thickness of skeletons of arithmetic hyperbolic orbifolds. arXiv:1811.05280 [math.DG]

  3. Belolipetsky, M.: Geodesics, volumes and Lehmer’s conjecture. Oberwolfach Rep. 7, 2136–2139 (2010)

    MathSciNet  Google Scholar 

  4. Belolipetsky, M.: Counting maximal arithmetic subgroups. Duke Math. J. 140, 1–33 (2007)

    Article  MathSciNet  Google Scholar 

  5. Belolipetsky, M., Gelander, T., Lubotzky, A., Shalev, A.: Counting arithmetic lattices and surfaces. Ann. Math. 172, 2197–2221 (2010)

    Article  MathSciNet  Google Scholar 

  6. Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Super. Pisa 4(8), 1–33 (1981)

    MathSciNet  MATH  Google Scholar 

  7. Borel, A.: Harish-Chandra, arithmetic subgroups of algebraic groups. Ann. Math. 2(75), 485–535 (1962)

    Article  Google Scholar 

  8. Borzellino, J.: Riemannian geometry of orbifolds, Ph.D. Thesis, UCLA (1992)

  9. Brooks, R.: Some relations between spectral geometry and number theory, en Topology’90 (Columbus, OH, Ohio State Univ. Math. Res. Inst. Publ., Vol. 1), de Gruyter. Berlin 1992, 61–75 (1990)

  10. Burger, M., Sarnak, P.: Ramanujan duals II. Invent. Math. 106, 1–11 (1991)

    Article  MathSciNet  Google Scholar 

  11. Burger, M., Schroeder, V.: Volume, diameter and the first eigenvalue of locally symmetric spaces of rank one. J. Differ. Geom. 26, 273–284 (1987)

    Article  MathSciNet  Google Scholar 

  12. Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Super. 15, 213–230 (1982)

    Article  MathSciNet  Google Scholar 

  13. Chinburg, T., Friedman, E.: The smallest arithmetic hyperbolic three-orbifold. Invent. Math. 86, 507–527 (1986)

    Article  MathSciNet  Google Scholar 

  14. Clozel, L.: Démonstration de la conjecture \(\tau \). Invent. Math. 151, 297–328 (2003)

    Article  MathSciNet  Google Scholar 

  15. Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34, 391–401 (1979)

    Article  MathSciNet  Google Scholar 

  16. Federer, H.: Geometric Measure Theory, Grundlehren 153. Springer-Verlag, Berlin (1969)

    Google Scholar 

  17. Gelander, T.: Homotopy type and volume of locally symmetric manifolds. Duke Math. J. 124, 459–515 (2004)

    Article  MathSciNet  Google Scholar 

  18. Goldman, W.M.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (1999)

    Google Scholar 

  19. Greenberg, L.: Discrete subgroups of the Lorentz group. Math. Scand. 10, 85–107 (1962)

    Article  MathSciNet  Google Scholar 

  20. Otsu, Y., Shioya, T.: The Riemannian structure of Alexandrov spaces. J. Differ. Geom. 39, 629–658 (1994)

    Article  MathSciNet  Google Scholar 

  21. Samet, I.: Betti numbers of finite volume orbifolds. Geom. Topol. 17, 1113–1147 (2013)

    Article  MathSciNet  Google Scholar 

  22. Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. USA 42, 359–363 (1956)

    Article  MathSciNet  Google Scholar 

  23. Wang, H.C.: Topics on totally discontinuous groups. In: Symmetric Spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., vol. 8, pp. 459–487

  24. Witte Morris, D.: Introduction to Arithmetic Groups. Deductive Press, San Rafael (2015)

    MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank the referee for careful proofreading of the manuscript and helpful comments. The author is partially supported by CNPq and FAPERJ research grants.

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  1. IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, 22460-320, Brazil

    Mikhail Belolipetsky

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  1. Mikhail Belolipetsky
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Correspondence to Mikhail Belolipetsky.

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Belolipetsky, M. A bound for diameter of arithmetic hyperbolic orbifolds. Geom Dedicata 214, 295–302 (2021). https://doi.org/10.1007/s10711-021-00616-z

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