An Extension of the Thermodynamic Formalism Approach to Selberg’s Zeta Function for General Modular Groups

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

“…In view of the already existing strict transfer operator approaches and the different methods for their construction [42,16,1,44,29,30,3,41,4,34,17,37,22,6,28,38,32,40], it might well be that this requirement is not a severe restriction on Γ at all. Moreover, it is most likely that with the methods we propose in this article the meromorphic continuability of (1) can also be shown starting with nonstrict transfer operator approaches as provided in [41,34,31] (for certain classes of cofinite Fuchsian groups).…”

“…A natural isomorphism, on which we rely in Section 3 below, is given as follows: To a hyperbolic element h ∈ Γ we assign the geodesic γ h on H for which γ h (+∞) is the attracting fixed point of h, and γ h (−∞) is the repelling fixed point. Note that γ h n = γ h for any n ∈ N. Then we identify an equivalence class [g] ∈ [Γ] p of a primitive hyperbolic element g ∈ Γ with the periodic geodesic γ g on X for which γ g is a representing geodesic on H. In this case, the (primitive) length ℓ( γ g ) of γ g is given by (6) ℓ( γ g ) = log N (g).…”

“…As mentioned in the Introduction, the number of geometrically finite Fuchsian groups for which strict transfer operator approaches are established is increasing [16,1,44,3,4,17,37,6,28,38,32,39,40]. The hyperbolic funnel groups are (as Fuchsian Schottky groups) among these Fuchsian groups.…”

Transfer operators for $\Gamma_{0}(n)$ and the Hecke operators for the period functions of $\PSL(2,\mathbb{Z})$

“…The proof follows the same line of argument as in [2]. In fact, using the kth Taylor polynomial of f at 0 we have:…”

“…An easy calculation shows that the operators L s,+ L s,− and L 2 s can be conjugated by the matrix ρ(( 1 0 0 −1 )). On the other hand it was shown in [2] that the Selberg zeta function Z Γ0(n) (s) for the group Γ 0 (n) can be expressed in terms of the Fredholm determinant of the operator L s,+ L s,− as Z Γ0(n) (s) = det(1 − L s,+ L s,− ) and hence also as Z Γ0(n) (s) = det(1 − L 2 s ) = det(1 + L s ) det(1 − L s ). This shows that using the operator L s the Selberg zeta function for the group Γ 0 (n) factorizes as in the case of the modular group and hence this transfer operator facilitates also the discussion of the period functions for Γ 0 (n).…”