Determinants of Laplacians and isopolar metrics on surfaces of infinite area

Borthwick, David; Judge, Chris; Perry, Peter

doi:10.1215/s0012-7094-03-11814-1

Cited by 28 publications

(40 citation statements)

References 44 publications

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“…By Lemma 3.1 this implies that Z Γ N w (s) has at least cN zeros (counted with multiplicities) in the disk |s − δ| < ε. Finally, the result of Borthwick-Judge-Perry [7] shows that all these zeros lie on the real interval (δ − ε, δ] and they correspond one to one to the eigenvalues s(1 − s) of the Laplacian on X N w . The proof of Theorem 1.4 is now complete.…”

Section: Proof Of Theorem 14mentioning

confidence: 93%

“…Part (ii) of Theorem 1.4 is now a straightforward consequence of two well-known results in the spectral theory of hyperbolic surfaces, which we recall here. First, given an arbitrary torsion-free, finitely generated Fuchsian group Γ, the result of Borthwick-Judge-Perry [7] asserts that the zeros of Z Γ (s) in Re(s) > 1 2 correspond, with multiplicities, to the L 2 -eigenvalues λ = s(1 − s) ∈ (0, 1 4 ) of the Laplacian Δ X on X = Γ\H 2 . Second, from Ballmann-Mathiesen-Mondal [3] we know that the number of eigenvalues of Δ X in (0, 1 4 ) is bounded above by −χ(X), where χ(X) denotes the Euler characteristic of the surface X.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…Part (i) of Theorem 1.4 should be compared with [34, Theorem 6.1], which states that for all w > 2 the base eigenvalue δ(w)(1 − δ(w)) is the only Laplace eigenvalue for the Hecke orbifold X w . From Borthwick-Judge-Perry [7] we know that if Γ is a finitely generated, torsion-free Fuchsian group, then the zeros of Z Γ (s) in the half-plane Re(s) > 1 2 correspond to the L 2 -eigenvalues s(1 − s) of the Laplacian on Γ\H 2 . Thus [34,Theorem 6.1] is morally equivalent to Part (i) of Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

Soares¹

2021

Ann. Henri Poincaré

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We consider the family of Hecke triangle groups $$ \Gamma _{w} = \langle S, T_w\rangle $$ Γ w = ⟨ S , T w ⟩ generated by the Möbius transformations $$ S : z\mapsto -1/z $$ S : z ↦ - 1 / z and $$ T_{w} : z \mapsto z+w $$ T w : z ↦ z + w with $$ w > 2.$$ w > 2 . In this case, the corresponding hyperbolic quotient $$ \Gamma _{w}\backslash {\mathbb {H}}^2 $$ Γ w \ H 2 is an infinite-area orbifold. Moreover, the limit set of $$ \Gamma _w $$ Γ w is a Cantor-like fractal whose Hausdorff dimension we denote by $$ \delta (w). $$ δ ( w ) . The first result of this paper asserts that the twisted Selberg zeta function $$ Z_{\Gamma _{ w}}(s, \rho ) $$ Z Γ w ( s , ρ ) , where $$ \rho : \Gamma _{w} \rightarrow \mathrm {U}(V) $$ ρ : Γ w → 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 $$\mathrm {Re}(s) > \frac{1}{2}$$ Re ( s ) > 1 2 of the Selberg zeta function of a special family of subgroups $$( \Gamma _w^N )_{N\in {\mathbb {N}}} $$ ( Γ w N ) N ∈ N of $$\Gamma _w$$ Γ w . These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $$X_w^N = \Gamma _w^N \backslash {\mathbb {H}}^2$$ X w N = Γ w N \ H 2 . We show that the classical Selberg zeta function $$Z_{\Gamma _w}(s)$$ 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 $$\delta (w)$$ δ ( w ) as $$w\rightarrow \infty $$ w → ∞ .

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Section: Proof Of Theorem 14mentioning

confidence: 93%

Section: Proof Of Theorem 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

Soares¹

2021

Ann. Henri Poincaré

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“…From [PP01] as well as from [BJP03] it follows that the multiset of zeros of Z X consists of the multiset R(X) of resonances of X and the topological zeros. The latter are well-understood: they are located at −N 0 with well-known multiplicities.…”

Section: Selberg Zeta Functionmentioning

confidence: 99%

Numerical resonances for Schottky surfaces via Lagrange-Chebyshev approximation

Bandtlow,

Pohl,

Schick

et al. 2020

Preprint

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“…By [3], the set R(X) of resonances of X is contained in the set of zeros of Z Γ , counted with multiplicities. This property of the Selberg zeta function allows us to translate upper estimates of the resonance counting functions N X and M X into counting problems of the number of zeros of Z Γ in certain domains.…”

Section: Schottky Groups and Schottky Surfacesmentioning

confidence: 99%

Density of Resonances for Covers of Schottky Surfaces

Pohl¹,

Soares²

2018

Preprint

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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 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 prove 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 implies an improved fractal Weyl upper bound. In the setting of Schottky surfaces, these estimates refine previously known results due to Guillopé-Zworski and Guillopé-Lin-Zworski. When applied to principal congruence covers, both 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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Determinants of Laplacians and isopolar metrics on surfaces of infinite area

Cited by 28 publications

References 44 publications

Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

Numerical resonances for Schottky surfaces via Lagrange-Chebyshev approximation

Density of Resonances for Covers of Schottky Surfaces

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