Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

Fedosova, Ksenia; Pohl, Anke

doi:10.1007/s00029-019-0534-3

Cited by 19 publications

(21 citation statements)

References 57 publications

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“…By the Venkov-Zograf factorization formula [26,25] (see also Fedosova-Pohl [6] for the infinite-area case), the Selberg zeta function of any finite-index subgroup Γ of Γ w equals the Selberg zeta function of Γ w twisted with the representation of Γ w that is induced by the trivial one-dimensional representation of Γ. This allows us to deduce the following corollaries of Theorem 1.1.…”

Section: Introduction and Resultsmentioning

confidence: 88%

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Fractal Weyl bounds and Hecke triangle groups

Naud¹,

Pohl²,

Soares³

2019

Electronic Research Announcements

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Let Γw be a non-cofinite Hecke triangle group with cusp width w > 2 and let : Γw → U (V ) be a finite-dimensional unitary representation of Γw. In this note we announce a new fractal upper bound for the Selberg zeta function of Γw twisted by . In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by exp Cε|s| δ+ε , where δ = δw denotes the Hausdorff dimension of the limit set of Γw. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces X = Γ\H 2 where Γ is a finite index, torsion-free subgroup of Γw.

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Section: Introduction and Resultsmentioning

confidence: 88%

“…For example, we may consider the subgroup of index 2 which is freely generated by the elements T w and ST w S, as well as its abelian covers. It extends meromorphically to all of C. The set of poles is contained in 1 2 (1 − N 0 ), and bounds on the order of poles can be given [21,6].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Fractal Weyl bounds and Hecke triangle groups

Naud¹,

Pohl²,

Soares³

2019

Electronic Research Announcements

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“…For any Fuchsian group Γ with parabolic elements, the class of representations with non-expanding cusp monodromy is a proper superset of those which are only unitary at cusps. We refer to [7,Section 5] for examples of representations with nonexpanding cusp monodromy that are not unitary at cusps. Throughout the last few decades, representations with non-expanding cusp monodromy have already been seen to be of huge interest.…”

Section: Introductionmentioning

confidence: 99%

“…In [7], we established, for any geometrically finite Fuchsian group Γ, convergence of the Selberg zeta functions for Γ with twists of non-expanding cusp monodromy. We further showed that for a certain rather large subclass of Fuchsian groups these Selberg zeta functions have a meromorphic continuation to all of C. Moreover, we proved that for twists without non-expanding cusp monodromy, the twisted Euler products used for the definition of Selberg zeta functions do not converge.…”

Section: Introductionmentioning

confidence: 99%

“…As explained in [4,Introduction], a key ingredient for establishing convergence is to show that, for any cusp representative c, the operator χ(g −1 ) • P c can be bounded polynomially in terms of Im(σ −1 c g.z) with all bounding constants depending continuously on z ∈ H and being independent of (almost all) [g] ∈ Γ c \Γ. For providing such a result we take advantage of our detailed study in [7] on the weights that χ attributes to periodic geodesics with long cusp excursions on Γ\H. The findings allow us now to establish the following polynomial bound.…”

Section: Introductionmentioning

confidence: 99%

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Eisenstein series twisted with non-expanding cusp monodromies

Fedosova

Pohl

2020

Ramanujan J

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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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Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

Cited by 19 publications

References 57 publications

Fractal Weyl bounds and Hecke triangle groups

Fractal Weyl bounds and Hecke triangle groups

Eisenstein series twisted with non-expanding cusp monodromies

Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

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