Abstract
AbstractWe 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 (Battista in Trans Am Math Soc 375(04):2597–2625, 2022; Italiano et al. in Invent Math, 2022. https://doi.org/10.1007/s00222-022-01141-w). The 5-dimensional example is diffeomorphic to $$N\times {{\mathbb {R}}}$$
N
×
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.
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Battista, L.: Code for infinitesimal rigidity of cubulated manifolds. https://people.dm.unipi.it/battista/code/irfcm. Accessed 20 Dec 2021
2. Battista, L., Martelli, B.: Hyperbolic 4-manifolds with perfect circle-valued Morse functions. Trans. Am. Math. Soc. 375(04), 2597–2625 (2022)
3. Bergeron, N., Gelander, T.: A note on local rigidity. Geom. Dedic. 107, 111–131 (2004)
4. Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)
5. Brock, J.F., Bromberg, K.W.: On the density of geometrically finite Kleinian groups. Acta Math. 192(1), 33–93 (2004)