Branched Bending in Finite-Volume Hyperbolic Manifolds

TL;DR

This paper introduces branched bending deformations in hyperbolic manifolds, providing bounds on their deformation space, equations in higher geometries, and a specific example involving the Borromean Rings link.

Contribution

It generalizes existing bending deformation concepts to branched complexes, establishes lower bounds on deformation dimensions, and constructs explicit examples in hyperbolic geometry.

Findings

Lower bound on the dimension of deformation space

Equations describing deformations in higher hyperbolic and projective geometries

Constructed infinitesimal deformations on Borromean Rings link complement

Abstract

We define branched bending deformations as deformations supported on a piecewise totally geodesic complex of $(n-1)$-dimensional faces meeting along $(n-2)$-dimensional branching loci. These are a generalization of bending deformations, as introduced by Johnson and Millson. We give a lower bound on the dimension of the (infinitesimal) deformation space supported on a branched bending complex, and in doing so generalize a result of Bart and Scannell. We give equations describing these deformations in the setting of deforming to higher hyperbolic geometry and real projective geometry. As a special example of branched bending, we construct infinitesimal deformations supported on the link complement of the Borromean Rings (also known as the link $6^3_2$), recovering a special case of a theorem due to Menasco and Reid.