scholarly journals Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow

2014 ◽  
Vol 36 (1) ◽  
pp. 142-172 ◽  
Author(s):  
ANKE D. POHL

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$In this article we show that the operator families${\mathcal{L}}_{s}^{\pm }$arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for$s\in \mathbb{C}$,$\text{Re}s={\textstyle \frac{1}{2}}$, the operator${\mathcal{L}}_{s}^{+}$(respectively${\mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue$s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

2011 ◽  
Vol 33 (1) ◽  
pp. 247-283 ◽  
Author(s):  
M. MÖLLER ◽  
A. D. POHL

AbstractWe 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.


Author(s):  
Louis Soares

AbstractWe 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 → ∞ .


2018 ◽  
Vol 40 (3) ◽  
pp. 612-662
Author(s):  
ALEXANDER ADAM ◽  
ANKE POHL

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. 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 $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. 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.


1996 ◽  
Vol 16 (4) ◽  
pp. 805-819 ◽  
Author(s):  
Hans Henrik Rugh

AbstractWe consider a generalized Fredholm determinant d(z) and a generalized Selberg zeta function ζ(ω)−1 for Axiom A diffeomorphisms of a surface and Axiom A flows on three-dimensional manifolds, respectively. We show that d(z) and ζ(ω)−1 extend to entire functions in the complex plane. That the functions are entire and not only meromorphic is proved by a new method, identifying removable singularities by a change of Markov partitions.


1932 ◽  
Vol 28 (3) ◽  
pp. 273-274 ◽  
Author(s):  
E. C. Titchmarsh

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfyingwhere A is an absolute constant.


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