Given an acyclic representation $\alpha$ of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement $T_\alpha$ of the Ray-Singer torsion associated to $\alpha$, which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by Turaev. $T_\alpha$ is equal to the graded determinant of the odd signature operator up to a correction term, the metric anomaly, needed to make it independent of the choice of the Riemannian metric.

$T_\alpha$ is a holomorphic function on the space of such representations of the fundamental group. When $\alpha$ is a unitary representation, the absolute value of $T_\alpha$ is equal to the Ray-Singer torsion and the phase of $T_\alpha$ is proportional to the $\eta$-invariant of the odd signature operator. The fact that the Ray-Singer torsion and the $\eta$-invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants. In particular, using these methods we compute the quotient of the refined analytic torsion and Turaev’s refinement of the combinatorial torsion generalizing in this way the classical Cheeger-M¨uller theorem. As an application, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the $\eta$-invariant.

As part of our construction of $T_\alpha$ we prove several new results about determinants and $\eta$-invariants of non self-adjoint elliptic operators.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.