A Milnor-Wood inequality for complex hyperbolic lattices in quaternionic space

TL;DR

This paper establishes a Milnor-Wood inequality for complex hyperbolic lattice representations in quaternionic hyperbolic space, revealing rigidity properties and characterizing cases of equality as totally geodesic representations.

Contribution

It introduces a Milnor-Wood inequality in a new setting and proves a global rigidity theorem for totally geodesic representations of complex hyperbolic lattices.

Findings

Equality in the inequality implies totally geodesic representations.

The inequality provides a new rigidity criterion for such representations.

The work extends classical results to quaternionic hyperbolic geometry.

Abstract

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.