Functional determinants, generalized BTZ geometries and Selberg zeta function

We continue the study of a special entry in the AdS/CFT dictionary, namely a holographic formula relating the functional determinant of the scattering operator in an asymptotically locally anti-de Sitter space to a relative functional determinant of the scalar Laplacian in the bulk. A heuristic derivation of the formula involves a one-loop quantum effect in the bulk and the corresponding sub-leading correction at large N on the boundary. We presently explore the formula in the background of a higher dimensional version of the Euclidean BTZ black hole, obtained as a quotient of hyperbolic space by a discrete subgroup of isometries generated by a loxodromic (or hyperbolic) element consisting of dilation (temperature) and torsion angles (twist). The bulk computation is done using heat-kernel techniques and fractional calculus. At the boundary, we acquire a recursive scheme that allows us to successively include rotation blocks in spacelike planes in the embedding space. The determinants are compactly expressed in terms of an associated (Patterson-)Selberg zeta function and a connection to quasi-normal frequencies is discussed.