Classification of stably reflective hyperbolic \(\mathbb{Z}[\sqrt 2 ]\)-lattices of rank 4. (English. Russian original) Zbl 1444.11149
Dokl. Math. 99, No. 3, 241-244 (2019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 486, No. 1, 7-11 (2019).
Summary: It is proved that the fundamental polyhedron of a \(\mathbb{Q}[\sqrt 2 ]\)-arithmetic reflection group in the three-dimensional Lobachevsky space has an edge such that the distance between its framing faces is sufficiently small. This result is used to classify the stably reflective hyperbolic \(\mathbb{Z}[\sqrt 2 ]\)- lattices of rank 4.
MSC:
| 11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |
| 11E12 | Quadratic forms over global rings and fields |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 51F15 | Reflection groups, reflection geometries |
References:
| [1] | Agol, I.; Belolipetsky, M.; Storm, P.; Whyte, K., No article title, Groups Geom. Dyn., 2, 481-498 (2008) · Zbl 1194.22011 · doi:10.4171/GGD/47 |
| [2] | Belolipetsky, M., No article title, Bull. Am. Math. Soc. New Ser., 53, 437-475 (2016) · Zbl 1342.22017 · doi:10.1090/bull/1530 |
| [3] | Bogachev, N. V., No article title, Russ. Math. Surv., 72, 179-181 (2017) · Zbl 1387.11036 · doi:10.1070/RM9756 |
| [4] | Bogachev, N. V., No article title, Izv. Math., 83, 1-19 (2019) · Zbl 1427.11064 · doi:10.1070/IM8766 |
| [5] | Bogachev, N. V.; Perepechko, A. Yu., No article title, Math. Notes, 103, 836-840 (2018) · Zbl 1418.11103 · doi:10.1134/S0001434618050164 |
| [6] | Bugaenko, V. O., No article title, Adv. Sov. Math., 8, 33-55 (1992) |
| [7] | E. B. Vinberg, Math. USSR-Sb. 1 (3), 429-444 (1967). · doi:10.1070/SM1967v001n03ABEH001992 |
| [8] | E. B. Vinberg, Math. USSR-Sb. 16 (1), 17-35 (1972). · Zbl 0252.20054 · doi:10.1070/SM1972v016n01ABEH001346 |
| [9] | E. B. Vinberg, Tr. Mosk. Mat. O-va 47, 68-102 (1984). |
| [10] | Mark, A., No article title, Math. Proc. Cambridge Phil. Soc., 12, 1-37 (2016) |
| [11] | V. V. Nikulin, Tr. Mat. Inst. im. V.A. Steklova 165, 119-142 (1984). · Zbl 0577.10019 |
| [12] | Nikulin, V. V., No article title, Proc. Steklov Inst. Math., 230, 1-241 (2000) |
| [13] | Nikulin, V. V., No article title, Izv. Math., 71, 53-56 (2007) · Zbl 1131.22009 · doi:10.1070/IM2007v071n01ABEH002349 |
| [14] | https://github.com/nvbogachev/VinAlg-Z-sqrt-2 |
| [15] | N. Bogachev and A. Perepechko, Vinberg’s Algorithm.https://doi.org/10.5281/zenodo.1098448, https://github.com/aperep/vinberg-algorithm, 2017. · Zbl 1418.11103 |
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