Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry

TL;DR

This paper explores the intersection of non-commutative geometry, hyperbolic handlebodies, and Arakelov geometry, relating geometric structures to cohomological theories in arithmetic surfaces over number fields.

Contribution

It introduces a novel approach linking Connes' spectral triples with Deninger's Archimedean cohomology and local monodromy in the context of arithmetic surfaces.

Findings

Relates hyperbolic geometry of handlebodies to cohomological theories

Connects non-commutative geometry with Arakelov theory

Provides a framework for understanding arithmetic infinity via spectral triples

Abstract

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.