Infinitely many commensurability classes of compact Coxeter polyhedra in \(\mathbb{H}^4\) and \(\mathbb{H}^5\). (English) Zbl 07857679
Summary: We prove that certain families of compact Coxeter polyhedra in \(4\)- and \(5\)-dimensional hyperbolic space constructed by Makarov give rise to infinitely many commensurability classes of reflection groups in these dimensions.
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 51F15 | Reflection groups, reflection geometries |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
Software:
SageMathReferences:
| [1] | Agol, I., Systoles of hyperbolic 4-manifolds, 2006, preprint |
| [2] | Agol, I.; Belolipetsky, M.; Storm, P.; Whyte, K., Finiteness of arithmetic hyperbolic reflection groups, Groups Geom. Dyn., 2, 481-498, 2008 · Zbl 1194.22011 |
| [3] | Allcock, D., Infinitely many hyperbolic Coxeter groups through dimension 19, Geom. Topol., 10, 737-758, 2006 · Zbl 1133.20025 |
| [4] | Andreev, E. M., Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.), 81, 123, 445-478, 1970 · Zbl 0194.23202 |
| [5] | Bader, U.; Fisher, D.; Miller, N.; Stover, M., Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds, Invent. Math., 233, 169-222, 2023 · Zbl 1534.53056 |
| [6] | Baldi, G.; Ullmo, E., Special subvarieties of non-arithmetic ball quotients and Hodge theory, Ann. Math. (2), 197, 159-220, 2023 · Zbl 1519.14025 |
| [7] | Belolipetsky, M.; Bogachev, N.; Kolpakov, A.; Slavich, L., Subspace stabilisers in hyperbolic lattices, 2021 |
| [8] | Belolipetsky, M. V.; Thomson, S. A., Systoles of hyperbolic manifolds, Algebraic Geom. Topol., 11, 1455-1469, 2011 · Zbl 1248.22004 |
| [9] | Bergeron, N.; Haglund, F.; Wise, D. T., Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. (2), 83, 431-448, 2011 · Zbl 1236.57021 |
| [10] | Bogachev, N.; Guschin, D.; Vesnin, A., Arithmeticity of ideal hyperbolic right-angled polyhedra and hyperbolic link complements, 2023, preprint |
| [11] | Bogachev, N.; Kolpakov, A., On faces of quasi-arithmetic Coxeter polytopes, Int. Math. Res. Not., 3078-3096, 2021 · Zbl 1478.52008 |
| [12] | Bugaenko, V. O., Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring \(\mathbf{Z} [(\sqrt{ 5} + 1) / 2]\), Vestn. Mosk. Univ., Ser. I, Mat. Mekh., 6-12, 1984 · Zbl 0571.10024 |
| [13] | Bugaenko, V. O., Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, (Lie Groups, Their Discrete Subgroups, and Invariant Theory. Lie Groups, Their Discrete Subgroups, and Invariant Theory, Adv. Soviet Math., vol. 8, 1992, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 33-55 · Zbl 0768.20020 |
| [14] | Chen, H., Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?, (version: 2014-04-23) |
| [15] | Corlette, K., Archimedean superrigidity and hyperbolic geometry, Ann. Math. (2), 135, 165-182, 1992 · Zbl 0768.53025 |
| [16] | Dotti, E., On the commensurability of hyperbolic Coxeter groups, Manuscr. Math., 173, 203-222, 2024 · Zbl 1530.20120 |
| [17] | Dotti, E.; Kolpakov, A., Infinitely many quasi-arithmetic maximal reflection groups, Proc. Am. Math. Soc., 150, 4203-4211, 2022 · Zbl 1502.11108 |
| [18] | Esnault, H.; Groechenig, M., Cohomologically rigid local systems and integrality, Sel. Math. New Ser., 24, 4279-4292, 2018 · Zbl 1408.14037 |
| [19] | Felikson, A.; Tumarkin, P., Essential hyperbolic Coxeter polytopes, Isr. J. Math., 199, 113-161, 2014 · Zbl 1301.51013 |
| [20] | Fisher, D.; Hurtado, S., A new proof of finiteness of maximal arithmetic reflection groups, Ann. Henri Lebesgue, 6, 151-159, 2023 · Zbl 1516.20082 |
| [21] | Gromov, M.; Piatetski-Shapiro, I. I., Non-arithmetic groups in Lobachevsky spaces, Publ. Math. Inst. Hautes Études Sci., 66, 93-103, 1988 · Zbl 0649.22007 |
| [22] | Gromov, M.; Schoen, R., Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math. Inst. Hautes Études Sci., 165-246, 1992 · Zbl 0896.58024 |
| [23] | Maclachlan, C.; Reid, A. W., The Arithmetic of Hyperbolic 3-Manifolds, Grad. Texts Math., vol. 219, 2003, Springer: Springer New York, NY · Zbl 1025.57001 |
| [24] | Makarov, V., On Fedorov’s groups in four- and five-dimensional Lobachevskij spaces, Issled. po Obshch. Algebre. Kishinev, 1, 120-129, 1968 |
| [25] | Margulis, G. A., Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 17, 1991, Springer-Verlag: Springer-Verlag Berlin · Zbl 0732.22008 |
| [26] | Mila, O., The trace field of hyperbolic gluings, Int. Math. Res. Not., 2021, 4392-4412, 2021 · Zbl 1486.57031 |
| [27] | Nikulin, V. V., On the classification of arithmetic groups generated by reflections in Lobachevskiĭ spaces, Izv. Akad. Nauk SSSR, Ser. Mat., 45, 113-142, 1981, 240 · Zbl 0477.22008 |
| [28] | Nikulin, V. V., Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces, Izv. Math., 71, 53-56, 2007 · Zbl 1131.22009 |
| [29] | Potyagailo, L.; Vinberg, È. B., On right-angled reflection groups in hyperbolic spaces, Comment. Math. Helv., 80, 63-73, 2005 · Zbl 1072.20046 |
| [30] | Raimbault, J., A note on maximal lattice growth in \(\operatorname{SO}(1, n)\), Int. Math. Res. Not., 3722-3731, 2013 · Zbl 1367.22005 |
| [31] | Raimbault, J., Coxeter polytopes and Benjamini-Schramm convergence, 2022 |
| [32] | Roeder, R. K.W.; Hubbard, J. H.; Dunbar, W. D., Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble), 57, 825-882, 2007 · Zbl 1127.51012 |
| [33] | SageMath, the sage mathematics software system (version 9.5), 2022, The Sage Developers |
| [34] | Thomson, S., Quasi-arithmeticity of lattices in \(\operatorname{PO}(n, 1)\), Geom. Dedic., 180, 85-94, 2016 · Zbl 1334.22012 |
| [35] | Vinberg, È. B., Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. (N.S.). Mat. Sb. (N.S.), Mat. Sb. (N.S.), 73, 115, 303-488, 1967, correction · Zbl 0166.16303 |
| [36] | Vinberg, È. B., Rings of definition of dense subgroups of semisimple linear groups, Math. USSR, Izv., 5, 45-55, 1972 · Zbl 0252.20043 |
| [37] | Vinberg, È. B., The nonexistence of crystallographic reflection groups in Lobachevskiĭ spaces of large dimension, Funkc. Anal. Prilozh., 15, 67-68, 1981 · Zbl 0462.51013 |
| [38] | Vinberg, È. B., Absence of crystallographic groups of reflections in Lobachevskiĭ spaces of large dimension, Tr. Mosk. Mat. Obŝ., 47, 68-102, 1984, 246 · Zbl 0462.51013 |
| [39] | Vinberg, È. B., Hyperbolic reflection groups, Russ. Math. Surv., 40, 31-75, 1985 · Zbl 0579.51015 |
| [40] | Vinberg, È. B., Non-arithmetic hyperbolic reflection groups in higher dimensions, Mosc. Math. J., 15, 593-602, 2015, 606 · Zbl 1359.51004 |
| [41] | Zimmer, R. J., Ergodic Theory and Semisimple Groups, Monogr. Math., vol. 81, 1984, Birkhäuser: Birkhäuser Basel, Cham · Zbl 0571.58015 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
