[Paper Review] On the resonances of convex co-compact subgroups of arithmetic groups

Dmitry Jakobson, Frédéric Naud|arXiv (Cornell University)|Nov 29, 2010

Advanced Algebra and Geometry21 references4 citations

TL;DR

This paper establishes effective lower bounds on the real parts of resonances for convex co-compact Fuchsian groups within arithmetic groups, proving that resonances lie in a strip extending to at least $\frac{\delta}{2} - \frac{1}{4}$ when $\delta > \frac{1}{2}$, and $\frac{\delta(1-2\delta)}{2}$ when $\delta \leq \frac{1}{2}$. These results imply infinitely many resonances in such strips and yield improved error terms in hyperbolic lattice point counting via spectral analysis.

ABSTRACT

Let $Λ$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=Λ\backslash\H$ has infinitely many resonances in an effective strip depending on the dimension of the limit set $δ$. Applications to lower bounds for the hyperbolic lattice point counting problem are derived.

Motivation & Objective

• To determine the maximal real part of resonances for convex co-compact hyperbolic surfaces arising from arithmetic Fuchsian groups.
• To establish effective lower bounds on the spectral gap $G(\Lambda)$, defined as the infimum of $\sigma < \delta$ such that only finitely many resonances lie in $\{\operatorname{Re}(s) \geq \sigma\}$.
• To derive quantitative improvements in the error term for the hyperbolic lattice point counting problem using resonance distribution.
• To confirm that the spectral gap $G(\Lambda)$ is strictly positive for non-elementary groups with $\delta \neq \frac{1}{2}$, supporting the conjecture $G(\Lambda) = \frac{\delta}{2}$.

Proposed method

• Uses the meromorphic continuation of the resolvent kernel $R_\Lambda(s;z,z')$ to analyze poles (resonances) via the Guillopé-Zworski parametrix construction.
• Relies on the relation between the Poincaré series $P_\Lambda(s;z,z')$ and the counting function $N(T;z,z')$ through summation by parts.
• Applies the Gauss hypergeometric function representation to relate $R_\Lambda(s;z,z')$ to $P_\Lambda(s;z,z')$ via the identity $R_\Lambda(s;z,z') = \frac{4^{s-1}}{\pi} \frac{(\Gamma(s))^2}{\Gamma(2s)} P_\Lambda(s;z,z') + H(s;z,z')$.
• Analyzes the structure of poles in the resolvent kernel, showing that the residue $A_1(\zeta;z,z')$ is a non-trivial real-analytic function away from the diagonal.
• Constructs a null set $\mathcal{N}$ of exceptional $(z,z')$ pairs where the kernel has finitely many poles in a given half-plane, using the zero set of analytic functions.
• Derives a contradiction when assuming too small a $\rho$ in the error term $O(e^{\rho T})$, proving that $\rho$ must exceed $\frac{\delta}{2} - \frac{1}{4} - \epsilon_0$.

Experimental results

Research questions

• RQ1 What is the maximal real part of resonances for convex co-compact Fuchsian groups that are subgroups of arithmetic groups, and how does it depend on the critical exponent $\delta$?
• RQ2 Can effective lower bounds be established for the spectral gap $G(\Lambda)$, defined as the infimum of $\sigma < \delta$ such that only finitely many resonances lie in $\{\operatorname{Re}(s) \geq \sigma\}$?
• RQ3 How do the distribution of resonances influence the error term in the hyperbolic lattice point counting problem?
• RQ4 Does the conjecture $G(\Lambda) = \frac{\delta}{2}$ hold, and is it supported by effective bounds in the arithmetic subgroup setting?

Key findings

• For $0 < \delta \leq \frac{1}{2}$, the spectral gap satisfies $G(\Lambda) \geq \frac{\delta(1 - 2\delta)}{2}$, which is positive for $\delta < \frac{1}{2}$.
• For $\delta > \frac{1}{2}$ and $\Lambda$ a convex co-compact subgroup of an arithmetic group, $G(\Lambda) \geq \frac{\delta}{2} - \frac{1}{4}$, which is positive for $\delta > \frac{1}{2}$.
• The lower bound $\frac{\delta}{2} - \frac{1}{4}$ is effective and improves upon previous non-effective or weaker bounds for general convex co-compact manifolds.
• The results imply that the resolvent kernel $R_\Lambda(s;z,z')$ has infinitely many poles in the half-plane $\{\operatorname{Re}(s) \geq \frac{\delta}{2} - \frac{1}{4} - \epsilon_0\}$ for almost every pair $(z,z')$ away from the diagonal.
• The lattice point counting function $N(T;z,z')$ satisfies $N(T;z,z') = \sum_j Q_j(T;z,z') e^{\delta_j T} + O(e^{\rho T})$ with $\rho \geq \frac{\delta}{2} - \frac{1}{4} - \epsilon_0$, yielding an improved error term.
• The analysis confirms that the conjectured spectral gap $G(\Lambda) = \frac{\delta}{2}$ is asymptotically consistent, as the bound behaves like $\frac{\delta}{2} + O(\delta^2)$ as $\delta \to 0$.