Hecke Triangle Groups Research Articles

Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume

We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.

Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

We consider the family of Hecke triangle groups Gamma _{w} = langle S, T_wrangle generated by the Möbius transformations S : zmapsto -1/z and T_{w} : z mapsto z+w with w > 2. In this case, the corresponding hyperbolic quotient Gamma _{w}backslash {mathbb {H}}^2 is an infinite-area orbifold. Moreover, the limit set of Gamma _w is a Cantor-like fractal whose Hausdorff dimension we denote by delta (w). The first result of this paper asserts that the twisted Selberg zeta function Z_{Gamma _{ w}}(s, rho ) , where rho : Gamma _{w} rightarrow 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 ( Gamma _w^N )_{Nin {mathbb {N}}} of Gamma _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. We show that the classical Selberg zeta function Z_{Gamma _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) as wrightarrow infty .

A higher moment formula for the Siegel–Veech transform over quotients by Hecke triangle groups

We compute higher moments of the Siegel–Veech transform over quotients of SL (2,\mathbb{R}) by the Hecke triangle groups. After fixing a normalization of the Haar measure on SL (2,\mathbb{R}) we use geometric results and linear algebra to create explicit integration formulas which give information about densities of k -tuples of vectors in discrete subsets of \mathbb{R}^2 which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over SL (2,\mathbb{R})/ SL (2,\mathbb{Z}) .

Heat kernel asymptotics on sequences of elliptically degenerating Riemann surfaces

This is the first of two articles in which we define an elliptically degenerating family of hyperbolic Riemann surfaces and study the asymptotic behavior of the associated spectral theory. Our study is motivated by a result which Hejhal attributes to Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we prove various results regarding the asymptotic behavior of heat kernels and traces of heat kernels for both real and complex time. In Garbin et al. (2018) [8], we will use the results from this article and study the asymptotic behavior of numerous spectral functions through elliptic degeneration, including spectral counting functions, Selberg's zeta function, Hurwitz-type zeta functions, determinants of the Laplacian, wave kernels, spectral projections, small eigenfunctions, and small eigenvalues. The method of proof we employ follows the template set in previous articles which study spectral theory on degenerating families of finite volume Riemann surfaces (Huntley et al. (1995) [14] and (1997) [15], Jorgenson et al. (1997) [20] and (1997) [17]) and on degenerating families of finite volume hyperbolic three manifolds (Dodziuk et al. (1998) [4].) Although the types of results developed here and in Garbin et al. (2018) [8], are similar to those in existing articles, it is necessary to thoroughly present all details in the setting of elliptic degeneration in order to uncover all nuances in this setting.

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.

Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

In this article we study the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration, in particular the behavior of several spectral invariants. Some of these invariants, such as the Selberg zeta function and the spectral counting functions associated to small eigenvalues below 1/4, converge to their respective counterparts on the limiting surface. Other spectral invariants, such as the spectral zeta function and the logarithm of the determinant of the Laplacian, diverge. In these latter cases, we identify diverging terms and remove their contributions, thus regularizing convergence of these spectral invariants. Our study is motivated by a result from [Hej3], which D. Hejhal attributes to A. Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we obtain a quantitative result to Selberg’s remark.

Fractal Weyl bounds and Hecke triangle groups

Let $ \Gamma_w $ be a non-cofinite Hecke triangle group with cusp width $ w>2 $ and let $ \varrho\colon\Gamma_w\to U(V) $ be a finite-dimensional unitary representation of $ \Gamma_w $. In this note we announce a new fractal upper bound for the Selberg zeta function of $ \Gamma_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^{\delta + \varepsilon} \right) $, where $ \delta = \delta_w $ denotes the Hausdorff dimension of the limit set of $ \Gamma_w. $ This bound implies fractal Weyl bounds on the resonances of the Laplacian for any geometrically finite surface $ X = \widetilde{\Gamma}\backslash \mathbb{H}^2 $ whose fundamental group $ \widetilde{\Gamma} $ is a finite index, torsion-free subgroup of $ \Gamma_w $.

Algebraic curves uniformized by congruence subgroups of triangle groups

We construct certain subgroups of hyperbolic triangle groups which we call \congruence subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the eld of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups.

Reciprocity of Dedekind sums and the Euler class

Dedekind sums are arithmetic sums that were first introduced by Dedekind in the context of elliptic functions and modular forms, and later recognized to be surprisingly ubiquitous. Among the variations and generalizations introduced since, there is a construction of Dedekind sums for lattices in $\mathrm{SL}_2(\mathbf{R}). Building upon work of Asai, we prove the reciprocity law for these Dedekind sums, based on a concrete realization of the Euler class. As an application, we obtain an explicit formula expressing Dedekind sums for Hecke triangle groups in terms of continued fractions.

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

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.

Templates for geodesic flows

AbstractWe construct templates for geodesic flows on an infinite family of Hecke triangle groups. Our results generalize those of E. Ghys [Knots and dynamics. Proc. Int. Congress of Mathematicians. Vol. 1. International Congress of Mathematicians, Zürich, 2007], who constructed a template for the modular flow in the complement of the trefoil knot in $S^3$. A significant difficulty that arises in any attempt to go beyond the modular flow is the fact that for other Hecke triangles the geodesic flow cannot be viewed as a flow in $S^3$, and one is led to consider embeddings into lens spaces. Our final result is an explicit description of a single ‘Hecke template’ which contains all other templates we construct, allowing a topological study of the periodic orbits of different Hecke triangle groups all at once.

The transfer operator for the Hecke triangle groups

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.

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

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.

Metric and arithmetic properties of mediant-Rosen maps

A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.

Generalized continued fractions and orbits under the action of Hecke triangle groups

Generalized continued fractions and orbits under the action of Hecke triangle groups

Multiplicities of periodic orbit lengths for non-arithmetic models

Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length. The main ingredient used is the construction of joint distribution of periodic orbits when group matrices are transformed by field isomorphisms. The method can be generalized to other groups for which traces of group matrices are integers of an algebraic field of finite degree.

On Dirichlet series and Hecke triangle groups of infinite volume

Here k is an arbitrary real number (the weight) and λ > 2, but otherwise unrestricted. The proof entails the use of the Riemann–Hecke–Bochner correspondence between modular integrals on the one side, and Dirichlet series with functional equations, on the other. It requires, as well, an application of Eichler’s generalized Poincare series. Since λ > 2 here, the underlying group of invariance, the Hecke triangle group

Length spectra of the Hecke triangle groups

Length spectra of the Hecke triangle groups

On the existence of Maass cusp forms on hyperbolic surfaces with cone points

The perturbation theory of the Laplace spectrum of hyperbolic surfaces with conical singularities belonging to a fixed conformal class is developed. As an application, it is shown that the generic such surface with cusps has no Maass cusp forms ( L 2 {L^2} eigenfunctions) under specific eigenvalue multiplicity assumptions. It is also shown that eigenvalues depend monotonically on the cone angles. From this, one obtains Neumann eigenvalue monotonicity for geodesic triangles in H 2 {{\mathbf {H}}^2} and a lower bound of 1 2 π 2 \frac {1}{2}{\pi ^2} for the eigenvalues of ‘odd’ Maass cusp forms associated to Hecke triangle groups.