Infinitesimal rigidity for cubulated manifolds
Tóm tắt
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 (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
$$N\times {{\mathbb {R}}}$$
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.
Tài liệu tham khảo
Battista, L.: Code for infinitesimal rigidity of cubulated manifolds. https://people.dm.unipi.it/battista/code/irfcm. Accessed 20 Dec 2021
Battista, L., Martelli, B.: Hyperbolic 4-manifolds with perfect circle-valued Morse functions. Trans. Am. Math. Soc. 375(04), 2597–2625 (2022)
Bergeron, N., Gelander, T.: A note on local rigidity. Geom. Dedic. 107, 111–131 (2004)
Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)
Brock, J.F., Bromberg, K.W.: On the density of geometrically finite Kleinian groups. Acta Math. 192(1), 33–93 (2004)
Bromberg, K.: Projective structures with degenerate holonomy and the bers density conjecture. Ann. Math. 166(1), 77–93 (2007)
Cooper, D., Hodgson, C.D., Kerckhoff, S.: Three-dimensional Orbifolds and Cone-manifolds. MSJ Memoirs, Vol. 5. Mathematical Society of Japan (2000)
Ferrari, L., Kolpakov, A., Slavich, L.: Cusps of hyperbolic 4-manifolds and rational homology spheres. Proc. Lond. Math. Soc. 123(6), 636–648 (2020)
Italiano, G., Martelli, B., Migliorini, M.: hyperbolic 5-manifolds that fiber over \(S^1\). Invent. Math. (2022). https://doi.org/10.1007/s00222-022-01141-w
Jankiewicz, K., Norin, S., Wise, D.T.: Virtually fibering right-angled coxeter groups. J. Inst. Math. Jussieu 20(3), 957–987 (2021)
Kapovich, M.: Kleinian groups in higher dimensions. Prog. Math. 265, 02 (2007)
Kerckhoff, S.P., Storm, P.A.: Local rigidity of hyperbolic manifolds with geodesic boundary. J. Topol. 5, 757–784 (2009)
Kolpakov, A., Slavich, L.: Hyperbolic 4-manifolds, colourings and mutations. Proc. Lond. Math. Soc. 113(2), 163–184 (2016)
Martelli, B.: An introduction to geometric topology (2016)
MathWorks. Matlab help page on SVD decomposition. https://www.mathworks.com/help/matlab/math/singular-values.html
Namazi, H., Souto, J.: Non-realizability and ending laminations: proof of the density conjecture. Acta Math. 209(2), 323–395 (2012)
Potyagailo, L., Vinberg, E.: On right-angled reflection groups in hyperbolic spaces. Comment. Math. Helv. 80, 63–73 (2005)
Weil, A.: Remarks on the cohomology of groups. Ann. Math. 80(1), 149–157 (1964)