Geometric and arithmetic properties of Löbell polyhedra

Bogachev, Nikolay¹ ; Douba, Sami²

¹Department of Computer and Mathematical Sciences, University of Toronto Scarborough, Toronto, ON, Canada, Institute for Information Transmission Problems, Moscow, Russia
²Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2281-2295

Résumé

The Löbell polyhedra form an infinite family of compact right-angled hyperbolic polyhedra in dimension 3. We observe, through both elementary and more conceptual means, that the “systoles” of the Löbell polyhedra approach 0, so that these polyhedra give rise to particularly straightforward examples of closed hyperbolic 3-manifolds with arbitrarily small systole, and constitute an infinite family even up to commensurability. By computing number-theoretic invariants of these polyhedra, we refine the latter result, and also determine precisely which of the Löbell polyhedra are quasi-arithmetic.

Détail

DOI : 10.2140/agt.2025.25.2281

Keywords: geometry, low-dimensional topology, hyperbolic 3-manifolds, Coxeter groups

Bibliographie

[1] I Agol, Systoles of hyperbolic 4-manifolds, preprint (2006)

[2] I Agol, M Belolipetsky, P Storm, K Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups Geom. Dyn. 2 (2008) 481 | DOI

[3] E M Andreev, On convex polyhedra in Lobachevsky spaces, Mat. Sb. 81 (1970) 445

[4] E M Andreev, On convex polyhedra of finite volume in Lobachevsky space, Mat. Sb. 83 (1970) 256 | DOI

[5] O Antolín-Camarena, G R Maloney, R K W Roeder, Computing arithmetic invariants for hyperbolic reflection groups, from: "Complex dynamics" (editor D Schleicher), A K Peters (2009) 597 | DOI

[6] A Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994) 207 | DOI

[7] M V Belolipetsky, S A Thomson, Systoles of hyperbolic manifolds, Algebr. Geom. Topol. 11 (2011) 1455 | DOI

[8] R Benedetti, C Petronio, Lectures on hyperbolic geometry, Springer (1992) | DOI

[9] N Bergeron, F Haglund, D T Wise, Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. 83 (2011) 431 | DOI

[10] N Bogachev, A Kolpakov, On faces of quasi-arithmetic Coxeter polytopes, Int. Math. Res. Not. 2021 (2021) 3078 | DOI

[11] W D Dunbar, G R Meyerhoff, Volumes of hyperbolic 3-orbifolds, Indiana Univ. Math. J. 43 (1994) 611 | DOI

[12] D Fisher, S Hurtado, A new proof of finiteness of maximal arithmetic reflection groups, Ann. H. Lebesgue 6 (2023) 151 | DOI

[13] T Gelander, Homotopy type and volume of locally symmetric manifolds, Duke Math. J. 124 (2004) 459 | DOI

[14] M Gromov, I Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. 66 (1988) 93

[15] M Gromov, W Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987) 1 | DOI

[16] R Kellerhals, A polyhedral approach to the arithmetic and geometry of hyperbolic link complements, J. Knot Theory Ramifications 32 (2023) 2350052 | DOI

[17] A Kolpakov, Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers, European J. Combin. 33 (2012) 1709 | DOI

[18] F Löbell, Beispiele geschlossener dreidimensionaler Clifford–Kleinscher Räume negativer Krümmung, Ber. Verh. Sächs Akad. Wiss. Leipzig Math.-Phys. Kl. 83 (1931) 167

[19] C Maclachlan, A W Reid, The arithmetic of hyperbolic 3-manifolds, 219, Springer (2003) | DOI

[20] G A Margulis, Discrete subgroups of semisimple Lie groups, 17, Springer (1991) | DOI

[21] A D Mednykh, A Y Vesnin, Löbell manifolds revised, Sib. Èlektron. Mat. Izv. 4 (2007) 605

[22] J S Meyer, C Millichap, R Trapp, Arithmeticity and hidden symmetries of fully augmented pretzel link complements, New York J. Math. 26 (2020) 149

[23] G D Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 53 | DOI

[24] V V Nikulin, Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces, Izv. Ross. Akad. Nauk Ser. Mat. 71 (2007) 55 | DOI

[25] M Phillips, S Levy, T Munzner, Geomview

[26] G Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math. 21 (1973) 255 | DOI

[27] R K W Roeder, J H Hubbard, W D Dunbar, Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble) 57 (2007) 825 | DOI

[28] K Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977) 91 | DOI

[29] S Thomson, Quasi-arithmeticity of lattices in PO(n,1), Geom. Dedicata 180 (2016) 85 | DOI

[30] A Y Vesnin, Three-dimensional hyperbolic manifolds of Löbell type, Sibirsk. Mat. Zh. 28 (1987) 50

[31] A Y Vesnin, Three-dimensional hyperbolic manifolds with a general fundamental polyhedron, Mat. Zametki 49 (1991) 29, 157 | DOI

[32] A Y Vesnin, Volumes of Löbell 3-manifolds, Mat. Zametki 64 (1998) 17 | DOI

[33] A Y Vesnin, Right-angled polytopes and three-dimensional hyperbolic manifolds, Uspekhi Mat. Nauk 72 (2017) 147 | DOI

[34] È B Vinberg, Discrete groups generated by reflections in Lobachevsky spaces, Mat. Sb. 72 (1967) 471

[35] È B Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 45