The hyperbolic meaning of the Milnor-Wood inequality

TL;DR

This paper introduces a new twist function for hyperbolic plane isometries, relating it to Milnor and Wood's functions, and uses it to provide geometric proofs of the Milnor-Wood inequality through hyperbolic polygon areas.

Contribution

It defines a novel twist function for hyperbolic isometries, connecting algebraic properties to geometric areas, and offers new hyperbolic-geometric proofs of classical inequalities.

Findings

The twist function relates to Milnor and Wood's functions.

Inequalities are expressed as equalities involving hyperbolic areas.

Certain products' twists equal areas of hyperbolic polygons.

Abstract

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor--Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor's function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.