  In Arakelov theory a completion of an arithmetic surface is achieved by
enlarging the group of divisors by formal linear combinations of the ``closed
fibers at infinity''. Manin described the dual graph of any such closed fiber
in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody
endowed with a Schottky uniformization. In this paper we consider arithmetic
surfaces over the ring of integers in a number field, with fibers of genus
$g\geq 2$. We use Connes' theory of spectral triples to relate the hyperbolic
geometry of the handlebody to Deninger's Archimedean cohomology and the
cohomology of the cone of the local monodromy $N$ at arithmetic infinity as
introduced by the first author of this paper.

Algebraic Geometry;Differential Geometry;Number Theory