The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations

AbstractWe prove the existence of a piecewise analytic expanding map associated to certain Kleinian groups without parabolics acting in the 3-dimensional hyperbolic space. These groups have a fundamental domain ℛ with the property that the geodesic planes containing each face are part of the tesselation. We use this map together with the methods of thermodynamic formalism to give another proof that the Selberg zeta function for such groups has a meromorphic extension to ℂ.

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Divisor of the Selberg zeta function for kleinian groups

Journées Équations aux dérivées partielles ◽

10.5802/jedp.463 ◽

1994 ◽

pp. 1-9

Author(s):

Peter A. Perry

Keyword(s):

Zeta Function ◽

Kleinian Groups ◽

Selberg Zeta Function

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The Selberg trace formula for modular correspondences

Nagoya Mathematical Journal ◽

10.1017/s0027763000001823 ◽

1990 ◽

Vol 117 ◽

pp. 93-123

Author(s):

Shigeki Akiyama ◽

Yoshio Tanigawa

Keyword(s):

Zeta Function ◽

Kernel Function ◽

Trace Formula ◽

Conjugacy Classes ◽

Hecke Operators ◽

Cusp Forms ◽

Selberg Trace Formula ◽

Weight One ◽

Space Of Cusp Forms

In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

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Selberg's trace formula on thek-regular tree and applications

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117120311126x ◽

2003 ◽

Vol 2003 (8) ◽

pp. 501-526 ◽

Cited By ~ 4

Author(s):

Audrey Terras ◽

Dorothy Wallace

Keyword(s):

Asymptotic Formula ◽

Zeta Function ◽

Trace Formula ◽

Geodesic Flow ◽

Prime Number Theorem ◽

Selberg Trace Formula ◽

Regular Tree ◽

Graph Theoretic ◽

Upper Half Plane ◽

The Laplace Operator

We survey graph theoretic analogues of the Selberg trace and pretrace formulas along with some applications. This paper includes a review of the basic geometry of ak-regular treeΞ(symmetry group, geodesics, horocycles, and the analogue of the Laplace operator). A detailed discussion of the spherical functions is given. The spherical and horocycle transforms are considered (along with three basic examples, which may be viewed as a short table of these transforms). Two versions of the pretrace formula for a finite connectedk-regular graphX≅Γ\Ξare given along with two applications. The first application is to obtain an asymptotic formula for the number of closed paths of lengthrinX(without backtracking but possibly with tails). The second application is to deduce the chaotic properties of the induced geodesic flow onX(which is analogous to a result of Wallace for a compact quotient of the Poincaré upper half plane). Finally, the Selberg trace formula is deduced and applied to the Ihara zeta function ofX, leading to a graph theoretic analogue of the prime number theorem.

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Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

Annales Henri Poincaré ◽

10.1007/s00023-021-01117-1 ◽

2021 ◽

Author(s):

Louis Soares

Keyword(s):

Hausdorff Dimension ◽

Zeta Function ◽

Fredholm Determinant ◽

Transfer Operator ◽

Selberg Zeta Function ◽

Triangle Groups ◽

Finite Dimensional ◽

The Family ◽

Eigenvalues Of The Laplacian ◽

Riemann Zeta

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 → ∞ .

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