Modular orbits on the representation spaces of compact abelian Lie groups

TL;DR

This paper investigates the dynamics of the mapping class group acting on torus character varieties, providing conditions for orbit density and applying these results to give a new proof of Kronecker's theorem in Diophantine approximation.

Contribution

It offers a novel topological-dynamical analysis of the mapping class group's action on character varieties and connects this to classical number theory results.

Findings

Characterization of dense orbits of the mapping class group

Necessary and sufficient conditions for orbit density

A dynamical proof of Kronecker's theorem

Abstract

Let $S$ be a closed surface of genus $g$ greater than zero. In the present paper we study the topological-dynamical action of the mapping class group on the $\Bbb T^n$-character variety giving necessary and sufficient conditions for Mod$(S)$-orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's Theorem concerning inhomogeneous diophantine approximation.