Infinitesimal Rigidity for Cubulated Manifolds

TL;DR

This paper establishes the infinitesimal rigidity of certain hyperbolic 4- and 5-manifolds constructed via cyclic coverings of right-angled polytopes, advancing understanding of their geometric properties.

Contribution

It introduces a general strategy for analyzing the infinitesimal rigidity of cyclic coverings of manifolds derived from right-angled polytopes, including new examples in higher dimensions.

Findings

Proved infinitesimal rigidity for specific hyperbolic 4- and 5-manifolds.

Developed a general method for studying rigidity of cyclic coverings.

Identified a 5-manifold diffeomorphic to a product with a non-hyperbolic aspherical 4-manifold.

Abstract

We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers arXiv:2009.04997 [math.GT] and arXiv:2105.14795 [math.GT]. The 5-dimensional example is diffeomorphic to $N \times \mathbb{R}$ for some aspherical 4-manifold $N$ which does not admit any hyperbolic structure. To this purpose we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.