Branched Bending in Finite-Volume Hyperbolic Manifolds

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.