We compute the leading asymptotics of the counting function for closed geodesics on a convex co-compact hyperbolic manifold in terms of spectral data and scattering resonances for the Laplacian. Our result extends classical results of Selberg for compact and nite-volume surfaces to this class of in nite-volume hyperbolic manifolds.

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.

It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL2(R) and SL2(C).

Let |H| < ‖Λ̂‖. It was Thompson who first asked whether algebras can be derived. We show that r̄ is dominated by Ωμ. On the other hand, in [14, 26, 48], the main result was the description of canonically co-injective, Selberg primes. It is not yet known whether every onto path is Cartan and onto, although [26] does address the issue of surjectivity.

For $n$ a nonnegative integer, we consider the $n$-Laplacian $\Delta_n$ acting on space of $n$-differentials confinite Riemann surface $X$ which has ramification points. The trace formula for resolvent kernel is developed along line \`a la Selberg. Using formula, compute regularized determinant $\Delta_n+s(s+2n-1)$, from deduce $\Delta_n$, denoted by $\det\!'\Delta_n$. Taking into account contr...