Cutoff on hyperbolic surfaces

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 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of SL 2 (Z) and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). 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 ([7]), who asked under what general phenomena cutoff exists.

A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$

“…Although Iwaniec's theorem has been generalized in various directions [Hux86,Sar87,Hum18], as far as we know, Iwaniec's result has not been directly improved, so speaking about density of eigenvalues, Theorem 1.8 establishes for random covers the best result known in the arithmetic setting for eigenvalues above the Kim-Sarnak bound 975 4096 [Kim03]. Density estimates such as Theorem 1.8 have applications to the cutoff phenomenon on hyperbolic surfaces by work of Golubev and Kamber [GK19].…”

On Sarnak's Density Conjecture and its Applications

“…The reason is that the entire continuous spectrum is contained in V 0 ( [42]). For more details for hyperbolic surfaces, see [27].…”

“…Recently, Sarnak made a density conjecture which is an approximation to the very naive Ramanujan property and should serve as a replacement of it for applications. Some instances of this general idea were previously given for hyperbolic surfaces ( [52,27]) and for graphs ( [6,35]). Our goal here is to give a general framework for the proof of similar density conjectures and their use in applications.…”

From Ramanujan graphs to Ramanujan complexes

“…Finally, we mention that in [35] a different direction is taken: replacing PGL 2 (Q p ) with PGL 2 (R), the authors suggest the notion of Ramanujan surfaces, which are hyperbolic Riemann surfaces which spectrally behave like their universal cover, the hyperbolic plane. It is then shown that a discrete random walk with constant-length steps on these surfaces exhibits L 1 -cut-off.…”