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Geometriae Dedicata

, Volume 203, Issue 1, pp 225–255 | Cite as

Cutoff on hyperbolic surfaces

  • Konstantin GolubevEmail author
  • Amitay Kamber
Original Paper

Abstract

In this paper, we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least \(\nicefrac {1}{4}\), then the distances on the surface are highly concentrated around the minimal possible value of the diameter, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres (Geom Funct Anal 26(4):1190–1216, 2016.  https://doi.org/10.1007/s00039-016-0382-7) from the setting of graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from Sarnak and Xue (Duke Math J 64(1):207–227, 1991), we are able to show that the results apply to congruence subgroups of \(SL_{2}\left( \mathbb {Z}\right) \) and other arithmetic lattices, without relying on the well-known conjecture of Selberg (Proc Symp Pure Math 8:1–15, 1965), thus relaxing the condition on the Laplace spectrum of a surface. Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis (Proc Natl Acad Sci 93(4):1659–1664, 1996), who asked under what general phenomena cutoff exists.

Keywords

First keyword Second keyword More Hyperbolic surfaces Random Walks Cutoff 

Notes

Acknowledgements

We are grateful to Elon Lindenstrauss, Alex Lubotzky, Shahar Mozes and Józef Dodziuk for fruitful discussions. The first author is supported by the SNF Grant 200020-169106 at ETH Zurich. A big part of this work was done while the first author was on a joint postdoc at the Bar-Ilan University and the Weizmann Institute of Science, supported by the ERC Grant 336283. This work is part of the Ph.D. thesis of the second author at the Hebrew University of Jerusalem, under the guidance of Prof. Alex Lubotzky, and is supported by the ERC Grant 692854. A substantial part of this work was carried out in the cafè Bread&Co in Tel-Aviv, to which we are thankful for its coffee and hospitality.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathemathicsETH ZurichZurichSwitzerland
  2. 2.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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