arXiv subjects

Louis Soares

Publications and source records attributed to Louis Soares.

9 recordsLinked to original sources

arXiv2023

Low-lying resonances for infinite-area hyperbolic surfaces with long closed geodesics

We consider sequences $(X_n)_{n\in \mathbb{N}}$ of coverings of convex cocompact hyperbolic surfaces $X$ with Euler characterictic $\chi(X_n)$ tending to $-\infty$ as $n\to \infty.$ We prove that for $n$ large enough, each $X_n$ has an abundance of "low-lying" resonances, provided the length of the shortest closed geodesic on $X_n$ grows sufficiently fast. When applied to congruence covers we obtain a bound that improves upon a result of Jakobson, Naud, and the author in \cite{JNS}. Our proof uses the wave 0-trace formula of Guillop\'{e}--Zworski \cite{GZ99} together with specifically tailored test-functions with rapidly decaying Fourier transform.

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arXiv2023

Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma\backslash \mathbb{H}^2$ be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic $L$-functions, we establish a uniform and explicit spectral gap for the Laplacian on the Hecke congruence covers $ X_0(p) = \Gamma_0(p)\backslash \mathbb{H}^2$ of $X$ for "almost" all primes $p$, provided the limit set of $\Gamma$ is thick enough.

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arXiv2023

Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions

Let $X$ be a convex cocompact hyperbolic surface, and let $\delta$ denote the Hausdorff dimension of its limit set. Let $N_X(\sigma,T)$ denote the number of resonances of $X$ inside the box $[\sigma, \delta] + i[0,T]$. We prove that for all $\sigma > \delta/2$, we have \[ N_X(\sigma,T) \ll_\epsilon T^{1 + \delta - 2(2\sigma - \delta) + \epsilon}. \] This strengthens the previously established "improved" fractal Weyl bounds due to Naud \cite{Naud14} and Dyatlov \cite{Dya19}. Moreover, this result implies that for every $\epsilon > 0$, there exist resonance-free rectangular boxes of arbitrary height within the strip \[ \left\{\, s \in \mathbb{C} : \tfrac{3}{4}\delta + \epsilon < \mathrm{Re}(s) < \delta\, \right\}. \] Our proof combines Naud's approach \cite{Naud14} with the refined transfer operator machinery developed by Dyatlov-Zworski \cite{DyZw18}, as well as a new estimate for oscillatory integrals that arise naturally in our analysis.

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arXiv2021

Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders

We prove that every family of coverings of any infinite-area, convex cocompact hyperbolic surface has uniform spectral gap, provided that the associated Schreier graphs form a family of two-sided expanders. This extends the results of Brooks, Burger, and Bourgain-Gamburd-Sarnak to a setting where the Laplacian has no $L^2$-eigenvalues. In particular, the notion of spectral gap needs to be redefined in terms of the resonances of the Laplacian. As an immediate corollary, we obtain uniform spectral gap for congruence covers of convex cocompact surfaces, a result previously established by Oh-Winter and Bourgain-Kontorovich-Magee. Moreover, given any convex cocompact hyperbolic surface $X$, we provide a new "universal" resonance-free region for $X$, by which we mean a region in the complex plane that contains no resonances for any finite cover of $X$. This enlarges the universal resonance-free region given by Magee-Naud. Our methods rely on the thermodynamic formalism for twisted Selberg zeta functions.

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arXiv2020

Density of Resonances for Covers of Schottky Surfaces

We investigate how bounds of resonance counting functions for Schottky surfaces behave under transitions to covering surfaces of finite degree. We consider the classical resonance counting function asking for the number of resonances in large (and growing) disks centered at the origin of $\mathbb{C}$, as well as the (fractal) resonance counting function asking for the number of resonances in boxes near the axis of the critical exponent. For the former counting function we provide a transfer-operator-based proof that bounding constants can be chosen such that the transformation behavior under transition to covers is as for the Weyl law in the case of surfaces of finite area. For the latter counting function we deduce a bound in terms of the covering degree and the minimal length of a periodic geodesic on the covering surface. This yields an improved fractal Weyl upper bound. In the setting of Schottky surfaces, these estimates refine previous results due to Guillopé--Zworski and Guillopé--Lin--Zworski. When applied to principal congruence covers, these results yield new estimates for the resonance counting functions in the level aspect, which have recently been investigated by Jakobson--Naud. The techniques used in this article are based on the thermodynamic formalism for $L$-functions (twisted Selberg zeta functions), and twisted transfer operators.

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arXiv2020

Hecke triangle groups, transfer operators and Hausdorff dimension

We consider the family of Hecke triangle groups $ Γ_{w} = \langle S, T_w\rangle $ generated by the Möbius transformations $ S : z\mapsto -1/z $ and $ T_{w} : z \mapsto z+w $ with $ w > 2.$ In this case the corresponding hyperbolic quotient $ Γ_{w}\backslash\mathbb{H}^2 $ is an infinite-area orbifold. Moreover, the limit set of $ Γ_w $ is a Cantor-like fractal whose Hausdorff dimension we denote by $ δ(w). $ The first result of this paper asserts that the twisted Selberg zeta function $ Z_{Γ_{ w}}(s, ρ) $, where $ ρ: Γ_{w} \to \mathrm{U}(V) $ is an arbitrary finite-dimensional unitary representation, can be realized as the Fredholm determinant of a Mayer-type transfer operator. This result has a number of applications. We study the distribution of the zeros in the half-plane $\mathrm{Re}(s) > \frac{1}{2}$ of the Selberg zeta function of a special family of subgroups $( Γ_w^n )_{n\in \mathbb{N}} $ of $Γ_w$. These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $X_w^n = Γ_w^n \backslash \mathbb{H}^2$. We show that the classical Selberg zeta function $Z_{Γ_w}(s)$ can be approximated by determinants of finite matrices whose entries are explicitly given in terms of the Riemann zeta function. Moreover, we prove an asymptotic expansion for the Hausdorff dimension $δ(w)$ as $w\to \infty$.

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arXiv2018

Fractal Weyl bounds and Hecke triangle groups

Let $Γ_{w}$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $\varrho\colonΓ_w\to U(V)$ be a finite-dimensional unitary representation of $Γ_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of $Γ_{w}$ twisted by $\varrho$. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $\exp\left( C_{\varepsilon} \vert s\vert^{δ+ \varepsilon} \right)$, where $δ= δ_{w}$ denotes the Hausdorff dimension of the limit set of $Γ_{w}$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=\widetildeΓ\backslash\mathbb{H}$ where $\widetildeΓ$ is a finite index, torsion-free subgroup of $Γ_w$.

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arXiv2017

Large covers and sharp resonances of hyperbolic surfaces

Let $Γ$ be a convex co-compact discrete group of isometries of the hyperbolic plane $\mathbb{H}^2$, and $X=Γ\backslash \mathbb{H}^2$ the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian for large degree covers of $X$ given by a finite index normal subgroup of $Γ$. Using various techniques of thermodynamical formalism and representation theory, we prove two new existence results of "sharp non-trivial resonances" close to $\Re(s)=δ_Γ$, both in the large degree limit, for abelian covers and also infinite index congruence subgroups of $SL2(\mathbb{Z})$.

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arXiv2017

Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

Let $ X = Γ\setminus \mathbb{H} $ be a non-elementary geometrically finite hyperbolic surface and let $ δ$ denote the Hausdorff dimension of the limit set $ Λ(Γ) $. We prove that for every $ \varepsilon > 0 $ the surface $ X $ admits a finite cover $ X' $ such that the Selberg zeta function associated to $ X' $ has a zero $ s\neq δ$ with $ | δ- s| < \varepsilon $. For $ δ> \frac{1}{2} $ we exploit the combinatorial interpretation of spectral gap in terms of expander graphs. For $ δ\leq \frac{1}{2} $ the proof is based on the thermodynamic formalism approach for L-functions associated to hyperbolic surfaces and an analogue of the Artin-Takagi formula for these L-functions.

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