×

Virtually special embeddings of integral Lorentzian lattices. (English) Zbl 1468.11155

Collin, Olivier (ed.) et al., Characters in low-dimensional topology. A conference celebrating the work of Steven Boyer, Université du Québec à Montréal, Montréal, Québec, Canada, June 2–6, 2018. Providence, RI: American Mathematical Society (AMS); Montreal: Centre de Recherches Mathématiques (CRM). Contemp. Math. 760, 35-44 (2020).
The theory of hyperbolic reflection groups provides many examples of finite volume hyperbolic orbifolds. However in higher dimensions these do not exist. Finite volume hyperbolic orbifolds in any dimension can also be constructed as quotient of hyperbolic space by the automorphism groups of Lorentzian lattices. In this paper, embedding of lattices into unimodular lattices of higher dimension have been constructed. The Lorentzian unimodular lattices \(I_{n,1} \) are reflective for \(2\leq n \leq 19\) ([E. B. Vinberg, Math. USSR, Sb. 16, 17–35 (1972; Zbl 0252.20054)] and [È. B. Vinberg and I. M. Kaplinskaya, Sov. Math., Dokl. 19, 194–197 (1978; Zbl 0402.20034); translation from Dokl. Akad. Nauk SSSR 238, 1273–1275 (1978)]). Moreover for \(2 \leq n \leq 8\), these are associated to reflection groups of hyperbolic right-angled polyhedra, which are geometric right angled Coxeter groups [L. Potyagailo and E. Vinberg, Comment. Math. Helv. 80, No. 1, 63–73 (2005; Zbl 1072.20046)]. Here, the lattice embedding together with explicit relationship between the unimodular lattices \(I_{n,1}\) and RACGs given by B. Everitt et al. [Math. Ann. 354, No. 3, 871–905 (2012; Zbl 1260.57028)] have been applied to construct many examples of C-special hyperbolic manifold groups in dimension 3 and 4. The results in this paper extend and improve the results of the author [Quantifying virtual properties of bianchi groups. Austin, TX: University of Texas (PhD Thesis) (2018); Groups Geom. Dyn. 14, No. 1, 41–59 (2020; Zbl 1453.20067)] and J. Deblois et al. [Trans. Am. Math. Soc. 373, No. 11, 8219–8257 (2020; Zbl 07269836)].
For the entire collection see [Zbl 1458.57001].

MSC:

11H56 Automorphism groups of lattices
20F55 Reflection and Coxeter groups (group-theoretic aspects)
22E40 Discrete subgroups of Lie groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agol, Ian, The virtual Haken conjecture, Doc. Math., 18, 1045-1087 (2013) · Zbl 1286.57019 · doi:10.1016/j.procs.2013.05.269
[2] Allcock, Daniel, The reflective Lorentzian lattices of rank 3, Mem. Amer. Math. Soc., 220, 1033, x+108 pp. (2012) · Zbl 1302.11046 · doi:10.1090/S0065-9266-2012-00648-4
[3] Allcock, Daniel, Prenilpotent pairs in the \(E_{10}\) root lattice, Math. Proc. Cambridge Philos. Soc., 164, 3, 473-483 (2018) · Zbl 1498.20128 · doi:10.1017/S0305004117000287
[4] Bergeron, Nicolas; Haglund, Fr\'{e}d\'{e}ric; Wise, Daniel T., Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. (2), 83, 2, 431-448 (2011) · Zbl 1236.57021 · doi:10.1112/jlms/jdq082
[5] Michelle Chu, Quantifying virtual properties of bianchi groups, Phd dissertation, The University of Texas at Austin, 2018.
[6] Chu, Michelle, Special subgroups of Bianchi groups, Groups Geom. Dyn., 14, 1, 41-59 (2020) · Zbl 1453.20067 · doi:10.4171/ggd/533
[7] Conway, J. H.; Sloane, N. J. A., Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 290, lxxiv+703 pp. (1999), Springer-Verlag, New York · Zbl 0915.52003 · doi:10.1007/978-1-4757-6568-7
[8] Jason DeBlois Miller Patel DeBlois, Nicholas Miller, and Priyam Patel, Effective virtual and residual properties of some arithmetic hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 373 (2020), 8219-8257. DOI: https://doi.org/10.1090/tran/8190 · Zbl 07269836
[9] St., Literatur-Berichte: Niedere Zahlentheorie, Monatsh. Math. Phys., 14, 1, A75-A77 (1903) · doi:10.1007/BF01707031
[10] Everitt, Brent; Ratcliffe, John G.; Tschantz, Steven T., Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel, Math. Ann., 354, 3, 871-905 (2012) · Zbl 1260.57028 · doi:10.1007/s00208-011-0744-2
[11] Haglund, Fr\'{e}d\'{e}ric; Wise, Daniel T., Special cube complexes, Geom. Funct. Anal., 17, 5, 1551-1620 (2008) · Zbl 1155.53025 · doi:10.1007/s00039-007-0629-4
[12] James, D. G.; Maclachlan, C., Fuchsian subgroups of Bianchi groups, Trans. Amer. Math. Soc., 348, 5, 1989-2002 (1996) · Zbl 0874.11037 · doi:10.1090/S0002-9947-96-01606-6
[13] Kaplinskaja, I. M.; Vinberg, \`E. B., The groups \(O_{18,1}(Z)\) and \(O_{19,1}(Z)\), Dokl. Akad. Nauk SSSR, 238, 6, 1273-1275 (1978) · Zbl 0402.20034
[14] Khovanski\u{\i}, A. G., Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevski\u{\i} space, Funktsional. Anal. i Prilozhen., 20, 1, 50-61, 96 (1986) · Zbl 0597.51014
[15] Potyagailo, Leonid; Vinberg, Ernest, On right-angled reflection groups in hyperbolic spaces, Comment. Math. Helv., 80, 1, 63-73 (2005) · Zbl 1072.20046 · doi:10.4171/CMH/4
[16] Prokhorov, M. N., Absence of discrete groups of reflections with a noncompact fundamental polyhedron of finite volume in a Lobachevski\u{\i} space of high dimension, Izv. Akad. Nauk SSSR Ser. Mat., 50, 2, 413-424 (1986) · Zbl 0604.51007
[17] Vinberg, \`E. B., The groups of units of certain quadratic forms, Mat. Sb. (N.S.), 87(129), 18-36 (1972) · Zbl 0244.20058
[18] Vinberg, \`E. B., Absence of crystallographic groups of reflections in Lobachevski\u{\i} spaces of large dimension, Trudy Moskov. Mat. Obshch., 47, 68-102, 246 (1984) · Zbl 0462.51013
[19] Watson, G. L., Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. (3), 12, 577-587 (1962) · Zbl 0107.26901 · doi:10.1112/plms/s3-12.1.577
[20] Watson, G. L., Transformations of a quadratic form which do not increase the class-number. II, Acta Arith., 27, 171-189 (1975) · Zbl 0257.10011 · doi:10.4064/aa-27-1-171-189
[21] Daniel T. Wise, The structure of groups with quasiconvex hierarchy, 187 pages. · Zbl 1183.20043
[22] Daniel T. Wise, The structure of groups with quasiconvex hierarchy, 187 pages. · Zbl 1183.20043
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.