Volume 10, issue 2 (2006)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Infinitely many hyperbolic Coxeter groups through dimension 19

Daniel Allcock

Geometry & Topology 10 (2006) 737–758
Bibliography
1 R Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987) 133 MR913200
2 R E Borcherds, Coxeter groups, Lorentzian lattices, and $K3$ surfaces, Internat. Math. Res. Notices (1998) 1011 MR1654763
3 V O Bugaenko, Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring $\mathbf{Z}[(\sqrt{5}+1)/2]$, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1984) 6 MR764026
4 V O Bugaenko, On reflective unimodular hyperbolic quadratic forms, Selecta Math. Soviet. 9 (1990) 263 MR1074386
5 V O Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, from: "Lie groups, their discrete subgroups, and invariant theory", Adv. Soviet Math. 8, Amer. Math. Soc. (1992) 33 MR1155663
6 J H Conway, N A Sloane, Leech roots and Vinberg groups, Proc. Roy. Soc. London Ser. A 384 (1982) 233 MR684311
7 J H Conway, N A Sloane, Sphere packings, lattices and groups, Grundlehren series 290, Springer (1993) MR1194619
8 F Esselmann, The classification of compact hyperbolic Coxeter $d$–polytopes with $d+2$ facets, Comment. Math. Helv. 71 (1996) 229 MR1396674
9 I M Kaplinskaja, The discrete groups that are generated by reflections in the faces of simplicial prisms in Lobačevskiĭspaces, Mat. Zametki 15 (1974) 159 MR0360858
10 V S Makarov, The Fedorov groups of four-dimensional and five-dimensional Lobačevskiĭspace, from: "Studies in General Algebra, No. 1 (Russian)", Kišinev. Gos. Univ., Kishinev (1968) 120 MR0259735
11 PARI/GP, version 2.1.5
12 L Potyagailo, E Vinberg, On right-angled reflection groups in hyperbolic spaces, Comment. Math. Helv. 80 (2005) 63 MR2130566
13 M N Prokhorov, Absence of discrete groups of reflections with a noncompact fundamental polyhedron of finite volume in a Lobachevskiĭspace of high dimension, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 413 MR842588
14 O P Ruzmanov, Examples of nonarithmetic crystallographic Coxeter groups in $n$–dimensional Lobachevskiĭspace when $6\leq n\leq 10$, from: "Problems in group theory and in homological algebra (Russian)", Yaroslav. Gos. Univ. (1989) 138 MR1068774
15 P V Tumarkin, Compact hyperbolic Coxeter $n$–polytopes with $n+3$ facets arXiv:math.MG/0406226
16 P V Tumarkin, Hyperbolic Coxeter polytopes in $\mathbb H^m$ with $n+2$ hyperfacets, Mat. Zametki 75 (2004) 909 MR2086616
17 P V Tumarkin, Hyperbolic $n$–dimensional Coxeter polytopes with $n+3$ facets, Tr. Mosk. Mat. Obs. 65 (2004) 253 MR2193442
18 È B Vinberg, The groups of units of certain quadratic forms, Mat. Sb. $($N.S.$)$ 87(129) (1972) 18 MR0295193
19 È B Vinberg, The nonexistence of crystallographic reflection groups in Lobachevskiĭspaces of large dimension, Funktsional. Anal. i Prilozhen. 15 (1981) 67 MR617472
20 È B Vinberg, The two most algebraic $K3$ surfaces, Math. Ann. 265 (1983) 1 MR719348
21 È B Vinberg, I M Kaplinskaja, The groups $O_{18,1}(Z)$ and $O_{19,1}(Z)$, Dokl. Akad. Nauk SSSR 238 (1978) 1273 MR0476640