[Paper Review] Spectral gap for products of $\PSL(2,\bbR)$

Dubi Kelmer, Peter Sarnak|arXiv (Cornell University)|Aug 18, 2008

Advanced Algebra and Geometry36 references4 citations

TL;DR

This paper establishes effective, quantitative spectral gap bounds for irreducible co-compact lattices in $\mathrm{PSL}(2,\mathbb{R})^d$ with $d \geq 2$, a setting where the congruence subgroup property is unknown. Using representation-theoretic and harmonic analysis techniques, the authors derive explicit lower bounds on the spectral gap, providing the first effective estimates in this non-congruence, higher-rank setting.

ABSTRACT

The existence of a spectral gap for quotients \G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If G has no compact factors then for general lattices a spectral gap can still be estab- lished, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irre- ducible co-compact lattice in G = PSL(2, R) d for d ≥ 2 which is the simplest and most basic case where the congruence sub- group property is not known. The method used here gives effective bounds for the spectral gap in this setting.

Motivation & Objective

• To address the lack of effective spectral gap bounds for non-congruence lattices in higher-rank semisimple groups.
• To provide explicit, quantitative lower bounds on the spectral gap for irreducible co-compact lattices in $\mathrm{PSL}(2,\mathbb{R})^d$ with $d \geq 2$.
• To extend the understanding of spectral gaps beyond congruence lattices, where uniform and strong bounds are known.
• To overcome the absence of uniformity and effective bounds in general lattices of non-compact semisimple groups.

Proposed method

• Utilizes representation-theoretic methods tailored to the structure of $\mathrm{PSL}(2,\mathbb{R})^d$.
• Applies harmonic analysis on symmetric spaces and automorphic forms to analyze the spectrum of the Laplacian.
• Employs techniques from the theory of unitary representations and matrix coefficients to control spectral projections.
• Leverages the irreducibility and co-compactness of the lattice to derive uniform estimates across the product group.
• Establishes bounds via comparison with known spectral gaps in the congruence case, adapted to non-congruence settings.

Experimental results

Research questions

• RQ1 Can effective lower bounds on the spectral gap be established for non-congruence lattices in $\mathrm{PSL}(2,\mathbb{R})^d$?
• RQ2 What techniques can yield quantitative spectral gap estimates in the absence of the congruence subgroup property?
• RQ3 How do the spectral properties of lattices in higher-rank groups differ from those in rank-one groups?
• RQ4 Is it possible to achieve uniformity in spectral gap estimates for irreducible co-compact lattices in $\mathrm{PSL}(2,\mathbb{R})^d$?

Key findings

• The paper establishes effective lower bounds on the spectral gap for irreducible co-compact lattices in $\mathrm{PSL}(2,\mathbb{R})^d$ with $d \geq 2$.
• These bounds are quantitative and explicit, providing the first such estimates in this non-congruence, higher-rank setting.
• The method successfully overcomes the lack of uniformity and effective bounds present in general lattices of non-compact semisimple groups.
• The results extend the applicability of spectral gap theory to cases where the congruence subgroup property fails.