A realization of the Hecke algebra on the space of period functions for Γ0 (n)

Fraczek, Markus Szymon; Mayer, Dieter; Mühlenbruch, Tobias

doi:10.1515/crelle.2007.014

Cited by 10 publications

(13 citation statements)

References 9 publications

(25 reference statements)

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“…For these finite index subgroups Deitmar and Hilgert [DH07] provided period functions for the Maass cusp forms of these groups also using representations. For the groups Γ 0 (N ), the combination of [HMM05] and [FMM07] shows a close relation between eigenfunctions of certain transfer operators and Maass cusp forms.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

Möller

Pohl

2011

Ergod. Th. Dynam. Sys.

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We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

Möller

Pohl

2011

Ergod. Th. Dynam. Sys.

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“…In this limit the Hecke triangle group G q tends to the theta group Γ θ , generated by Sz = −1 z and T z = z + 2. This group is conjugate to the Hecke congruence subgroup Γ 0 (2), for which a transfer operator has been constructed in [7] and [5]. One should understand how these two different transfer operators are related to each other.…”

Section: The Selberg Zeta Function For Hecke Triangle Groupsmentioning

confidence: 99%

The transfer operator for the Hecke triangle groups

Mayer¹,

Mühlenbruch²,

Strömberg³

2012

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups G_q, q=3,4,..., which are non-arithmetic for q \not= 3,4,6. For this we make use of a Poincare map for the geodesic flow on the corresponding Hecke surfaces which has been constructed in arXiv:0801.3951 and which is closely related to the natural extension of the generating map for the so called Hurwitz-Nakada continued fractions. We derive simple functional equations for the eigenfunctions of the transfer operator which for eigenvalues rho =1 are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.Comment: 30 pages; revised versio

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“…For χ being the trivial character, [34] and [14] showed that the map (25) f −1 = α s (g 3,1 )ϕ, ϕ = α s (g −1 3,1 )f −1 provides an isomorphism between the eigenfunctions of L slow,± The combination of [15,16,26,18,22] shows that (25) provides also an isomorphism for certain representations χ. These studies take advantage of the special structure of L fast,±…”

Section: 1mentioning

confidence: 99%

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Adam¹,

Pohl²

2018

Ergod. Th. Dynam. Sys.

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Over the last few years Pohl (partly jointly with coauthors) developed dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces Γ\H with cusps and all finite-dimensional unitary representations χ of Γ.The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for (Γ, χ). Further, if Γ is cofinite and χ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for Γ. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances.In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface Γ\H of finite or infinite area and any finite-dimensional unitary representation χ of the Hecke triangle group Γ. In particular we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

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A realization of the Hecke algebra on the space of period functions for Γ0 (n)

Cited by 10 publications

References 9 publications

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

The transfer operator for the Hecke triangle groups

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

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