Let Γ ⊆ PSL2(R) be a Fuchsian subgroup of the first kind acting on the upper half-plane H. Consider the d-dimensional space of cusp forms S Γ k of weight 2k for Γ, and let {f1, . . . , f d } be an orthonormal basis of S Γ k with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity S Γ k (z) := d j=1 |fj (z)| 2 Im(z) 2k is bounded as OΓ(k) in the cocompact setting, and as OΓ(k 3/2 ) in the cofinite case, where the implied constants depend solely on Γ. We also show that the implied constants are uniform if Γ is replaced by a subgroup of finite index.

ABSTRACT. For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification point to the zeta function. In fact, if2 ] is the ring of Eisenstein integers, then the Selberg zeta-function of PSL(2, O) contains ramification points and is the sixth-root of a meromorphic function.

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ABSTRACT. For cofinite Kleinian groups (or equivalently, finite-volume three-dimensional hyperbolic orbifolds) with finite-dimensional unitary representations, we evaluate the regularized determinant of the Laplacian using W. Müller's regularization. We give an explicit formula relating the determinant to the Selberg zeta-function.

Let Γ ⊂ PSL2(R) be a Fuchsian subgroup of the first kind acting on the upper half-plane H. Consider the d 2k -dimensional space of cusp forms S Γ 2k of weight 2k for Γ, and let {f1, . . . , f d 2k } be an orthonormal basis of S Γ 2k with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity S Γ 2k (z) := d 2k j=1 |fj (z)| 2 Im(z) 2k as z ranges through H.