Arithmetic hyperbolic reflection groups

Belolipetsky, Mikhail

doi:10.1090/bull/1530

Arithmetic hyperbolic reflection groups
HTML articles powered by AMS MathViewer

by Mikhail Belolipetsky PDF

Bull. Amer. Math. Soc. 53 (2016), 437-475 Request permission

Abstract:

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.

References

Ian Agol, Mikhail Belolipetsky, Peter Storm, and Kevin Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups Geom. Dyn. 2 (2008), no. 4, 481–498. MR 2442945, DOI 10.4171/GGD/47
Omar Antolín-Camarena, Gregory R. Maloney, and Roland K. W. Roeder, Computing arithmetic invariants for hyperbolic reflection groups, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 597–631. MR 2508271, DOI 10.1201/b10617-22
Ian Agol, Finiteness of arithmetic Kleinian reflection groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 951–960. MR 2275630
Daniel Allcock, Reflection groups on the octave hyperbolic plane, J. Algebra 213 (1999), no. 2, 467–498. MR 1673465, DOI 10.1006/jabr.1998.7671
Daniel Allcock, New complex- and quaternion-hyperbolic reflection groups, Duke Math. J. 103 (2000), no. 2, 303–333. MR 1760630, DOI 10.1215/S0012-7094-00-10326-2
Daniel Allcock, The Leech lattice and complex hyperbolic reflections, Invent. Math. 140 (2000), no. 2, 283–301. MR 1756997, DOI 10.1007/s002220050363
Daniel Allcock, Infinitely many hyperbolic Coxeter groups through dimension 19, Geom. Topol. 10 (2006), 737–758. MR 2240904, DOI 10.2140/gt.2006.10.737
Daniel Allcock, The reflective Lorentzian lattices of rank 3, Mem. Amer. Math. Soc. 220 (2012), no. 1033, x+108. MR 3025355, DOI 10.1090/S0065-9266-2012-00648-4
Daniel Allcock, Reflection centralizers in Coxeter groups, Transform. Groups 18 (2013), no. 3, 599–613. MR 3084328, DOI 10.1007/s00031-013-9236-7
Daniel Allcock, Root systems for Lorentzian Kac-Moody algebras in rank 3, Bull. Lond. Math. Soc. 47 (2015), no. 2, 325–342. MR 3335126, DOI 10.1112/blms/bdv007
E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 81 (123) (1970), 445–478 (Russian). MR 0259734
E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
D. V. Alekseevskij, È. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 1–138. MR 1254932, DOI 10.1007/978-3-662-02901-5_{1}
Alexander Graham Barnard, The singular theta correspondence, Lorentzian lattices and Borcherds-Kac-Moody algebras, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–University of California, Berkeley. MR 2705173
Mikhail Belolipetsky and Vincent Emery, On volumes of arithmetic quotients of $\textrm {PO}(n,1)^\circ$, $n$ odd, Proc. Lond. Math. Soc. (3) 105 (2012), no. 3, 541–570. MR 2974199, DOI 10.1112/plms/pds009
Mikhail Belolipetsky, On volumes of arithmetic quotients of $\textrm {SO}(1,n)$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 749–770. MR 2124587
Mikhail Belolipetsky, On volumes of arithmetic quotients of $\textrm {SO}(1,n)$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 749–770. MR 2124587
Mikhail Belolipetsky, On fields of definition of arithmetic Kleinian reflection groups, Proc. Amer. Math. Soc. 137 (2009), no. 3, 1035–1038. MR 2457444, DOI 10.1090/S0002-9939-08-09590-7
Mikhail Belolipetsky, Finiteness theorems for congruence reflection groups, Transform. Groups 16 (2011), no. 4, 939–954. MR 2852486, DOI 10.1007/s00031-011-9156-3
Mikhail Belolipetsky, “Hyperbolic orbifolds of small volume”. In International Congress of Mathematicians. Vol. II (Seoul 2014), pages 837–851. Kyung Moon SA, 2014.
Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
Luigi Bianchi, Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie, Math. Ann. 38 (1891), no. 3, 313–333 (German). MR 1510677, DOI 10.1007/BF01199425
M. Bożejko, T. Januszkiewicz, and R. J. Spatzier, Infinite Coxeter groups do not have Kazhdan’s property, J. Operator Theory 19 (1988), no. 1, 63–67. MR 950825
Mikhail Belolipetsky and Benjamin Linowitz, On fields of definition of arithmetic Kleinian reflection groups II, Int. Math. Res. Not. IMRN 9 (2014), 2559–2571. MR 3207375, DOI 10.1093/imrn/rns292
M. Burger, J.-S. Li, and P. Sarnak, Ramanujan duals and automorphic spectrum, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 253–257. MR 1118700, DOI 10.1090/S0273-0979-1992-00267-7
Mikhail Belolipetsky and John Mcleod, Reflective and quasi-reflective Bianchi groups, Transform. Groups 18 (2013), no. 4, 971–994. MR 3127984, DOI 10.1007/s00031-013-9245-6
A. Borel, Commensurability classes and volumes of hyperbolic $3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1–33. MR 616899
Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200, DOI 10.1016/0021-8693(87)90245-6
Richard E. Borcherds, Lattices like the Leech lattice, J. Algebra 130 (1990), no. 1, 219–234. MR 1045746, DOI 10.1016/0021-8693(90)90110-A
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
Richard E. Borcherds, Reflection groups of Lorentzian lattices, Duke Math. J. 104 (2000), no. 2, 319–366. MR 1773561, DOI 10.1215/S0012-7094-00-10424-3
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math. 106 (1991), no. 1, 1–11. MR 1123369, DOI 10.1007/BF01243900
V. O. Bugaenko, Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring $\textbf {Z}[(\sqrt {5}+1)/2]$, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1984), 6–12 (Russian). MR 764026
V. O. Bugaenko, On reflective unimodular hyperbolic quadratic forms, Selecta Math. Soviet. 9 (1990), no. 3, 263–271. Selected translations. MR 1074386
V. O. Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, Lie groups, their discrete subgroups, and invariant theory, Adv. Soviet Math., vol. 8, Amer. Math. Soc., Providence, RI, 1992, pp. 33–55. MR 1155663, DOI 10.1007/bf00883263
Ted Chinburg and Eduardo Friedman, The smallest arithmetic hyperbolic three-orbifold, Invent. Math. 86 (1986), no. 3, 507–527. MR 860679, DOI 10.1007/BF01389265
J. H. Conway, The automorphism group of the $26$-dimensional even unimodular Lorentzian lattice, J. Algebra 80 (1983), no. 1, 159–163. MR 690711, DOI 10.1016/0021-8693(83)90025-X
H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588–621. MR 1503182, DOI 10.2307/1968753
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
H. S. M. Coxeter and G. J. Whitrow, World-structure and non-Euclidean honeycombs, Proc. Roy. Soc. London Ser. A 201 (1950), 417–437. MR 41576, DOI 10.1098/rspa.1950.0070
Martin Deraux, Deforming the $\Bbb R$-Fuchsian $(4,4,4)$-triangle group into a lattice, Topology 45 (2006), no. 6, 989–1020. MR 2263221, DOI 10.1016/j.top.2006.06.005
I. Dolgachev and S. Kond\B{o}, A supersingular $K3$ surface in characteristic 2 and the Leech lattice, Int. Math. Res. Not. 1 (2003), 1–23. MR 1935564, DOI 10.1155/S1073792803202038
P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651
Igor V. Dolgachev, Reflection groups in algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 1–60. MR 2358376, DOI 10.1090/S0273-0979-07-01190-1
I. Dolgachev, Hyperbolic geometry and algebraic geometry. Seoul-Austin Lecture Notes, available at http://www.math.lsa.umich.edu/ idolga/lecturenotes.html, 2015.
Martin Deraux, John R. Parker, and Julien Paupert, New non-arithmetic complex hyperbolic lattices. 2015. to appear.
Guillaume Dufour, Notes on right-angled Coxeter polyhedra in hyperbolic spaces, Geom. Dedicata 147 (2010), 277–282. MR 2660580, DOI 10.1007/s10711-009-9454-2
Walther Dyck, Gruppentheoretische Studien, Math. Ann. 20 (1882), no. 1, 1–44 (German). MR 1510147, DOI 10.1007/BF01443322
Frank Esselmann, Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe, J. Number Theory 61 (1996), no. 1, 103–144 (German, with English summary). MR 1418323, DOI 10.1006/jnth.1996.0141
Elena Fuchs, Chen Meiri, and Peter Sarnak, Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 8, 1617–1671. MR 3262453, DOI 10.4171/JEMS/471
Robert Fricke, Ueber eine besondere Classe discontinuirlicher Gruppen reeller linearer Substitutionen, Math. Ann. 38 (1891), no. 1, 50–81 (German). MR 1510663, DOI 10.1007/BF01212693
Anna Felikson and Pavel Tumarkin, Essential hyperbolic Coxeter polytopes, Israel J. Math. 199 (2014), no. 1, 113–161. MR 3219531, DOI 10.1007/s11856-013-0046-3
The Geometry Center, University of Minnesota. Current homepage address http://www.geom.uiuc.edu.
Frederick W. Gehring and Gaven J. Martin, Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group, Ann. of Math. (2) 170 (2009), no. 1, 123–161. MR 2521113, DOI 10.4007/annals.2009.170.123
V. A. Gritsenko and V. V. Nikulin, On the classification of Lorentzian Kac-Moody algebras, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 79–138 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 5, 921–979. MR 1992083, DOI 10.1070/RM2002v057n05ABEH000553
Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
Jeffrey A. Harvey and Gregory Moore, Algebras, BPS states, and strings, Nuclear Phys. B 463 (1996), no. 2-3, 315–368. MR 1393643, DOI 10.1016/0550-3213(95)00605-2
Jun-ichi Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175–200. MR 141643, DOI 10.2307/2372812
E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva, and A. C. Souza Filho, From the Poincaré theorem to generators of the unit group of integral group rings of finite groups, Math. Comp. 84 (2015), no. 293, 1489–1520. MR 3315518, DOI 10.1090/S0025-5718-2014-02865-2
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, and S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, Linear Algebra Appl. 345 (2002), 119–147. MR 1883270, DOI 10.1016/S0024-3795(01)00477-3
D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390
Felix Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1878), no. 3, 428–471 (German). MR 1509988, DOI 10.1007/BF01677143
D. A. Každan and G. A. Margulis, A proof of Selberg’s hypothesis, Mat. Sb. (N.S.) 75 (117) (1968), 163–168 (Russian). MR 0223487
Grant Lakeland, Arithmetic reflection groups and congruence subgroups. PhD thesis, University of Texas at Austin, 2012.
Grant S. Lakeland, Dirichlet-Ford domains and arithmetic reflection groups, Pacific J. Math. 255 (2012), no. 2, 417–437. MR 2928559, DOI 10.2140/pjm.2012.255.417
Folke Lannér, On complexes with transitive groups of automorphisms, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 11 (1950), 71. MR 42129
Silvio Levy, editor, The eightfold way: The beauty of Klein’s quartic curve, volume 35 of Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge, 1999.
D. D. Long, C. Maclachlan, and A. W. Reid, Arithmetic Fuchsian groups of genus zero, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 569–599. MR 2251482, DOI 10.4310/PAMQ.2006.v2.n2.a9
Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, “On the generalized Ramanujan conjecture for $\textrm {GL}(n)$”. In Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), volume 66 of Proc. Sympos. Pure Math., pages 301–310. Amer. Math. Soc., Providence, RI, 1999.
Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
Shouhei Ma, Finiteness of $2$-reflective lattices of signature $(2,n)$. Preprint arXiv:1409.2969 [math.AG], 2015.
C. Maclachlan, Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2, Bull. Lond. Math. Soc. 43 (2011), no. 1, 111–123. MR 2765555, DOI 10.1112/blms/bdq085
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
Alice Mark, Reflection groups of the quadratic form $-px_0^2+x_1^2+\dots +x_n^2$ with $p$ prime, Publ. Mat. 59 (2015), no. 2, 353–372. MR 3374612
Alice Mark, The classification of rank $3$ reflective hyperbolic lattices over $\mathbb {Z}(\sqrt {2})$. University of Texas at Austin, PhD thesis, May 2015. http://hdl.handle.net/2152/31507
John Mcleod, Hyperbolic reflection groups associated to the quadratic forms $-3x^2_0+x^2_1+\dots +x^2_n$, Geom. Dedicata 152 (2011), 1–16. MR 2795233, DOI 10.1007/s10711-010-9543-2
John Angus McLeod, Arithmetic Hyperbolic Reflection Groups, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–University of Durham (United Kingdom). MR 3389315
Chen Meiri, On the non-amenability of the reflective quotient I: The rational case. Preprint arXiv:1401.7708 [math.GR], 2014.
Toshitsune Miyake, Modular forms, Reprint of the first 1989 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda. MR 2194815
T. H. Marshall and G. J. Martin, Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group, Ann. of Math. (2) 176 (2012), no. 1, 261–301. MR 2925384, DOI 10.4007/annals.2012.176.1.4
G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876
G. D. Mostow, Actions on $\mathbb {H}^n$ generated by torsion elements. MathOverflow Question http://mathoverflow.net/q/102644, 2012.
Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957, DOI 10.1007/978-1-4757-6720-9
V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by $2$-reflections, Dokl. Akad. Nauk SSSR 248 (1979), no. 6, 1307–1309 (Russian). MR 556762
V. V. Nikulin, On the arithmetic groups generated by reflections in Lobačevskiĭ spaces, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 637–669, 719–720 (Russian). MR 582161
V. V. Nikulin, On the classification of arithmetic groups generated by reflections in Lobachevskiĭ spaces, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 113–142, 240 (Russian). MR 607579
V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by $2$-reflections. Algebro-geometric applications, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 3–114 (Russian). MR 633160
V. V. Nikulin, $K3$ surfaces with a finite group of automorphisms and a Picard group of rank three, Trudy Mat. Inst. Steklov. 165 (1984), 119–142 (Russian). Algebraic geometry and its applications. MR 752938
V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 654–671. MR 934268
V. V. Nikulin, Reflection groups in Lobachevskiĭ spaces and an identity for the denominator of Lorentzian Kac-Moody algebras, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), no. 2, 73–106 (Russian, with Russian summary); English transl., Izv. Math. 60 (1996), no. 2, 305–334. MR 1399419, DOI 10.1070/IM1996v060n02ABEH000072
V. V. Nikulin, On the classification of hyperbolic root systems of rank three, Tr. Mat. Inst. Steklova 230 (2000), 256 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. Math. 3(230) (2000), 1–241. MR 1802343
V. V. Nikulin, Finiteness of the number of arithmetic groups generated by reflections in Lobachevskiĭ spaces, Izv. Ross. Akad. Nauk Ser. Mat. 71 (2007), no. 1, 55–60 (Russian, with Russian summary); English transl., Izv. Math. 71 (2007), no. 1, 53–56. MR 2477273, DOI 10.1070/IM2007v071n01ABEH002349
Viacheslav V. Nikulin, On ground fields of arithmetic hyperbolic reflection groups, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 299–326. MR 2500568, DOI 10.1090/crmp/047/19
V. V. Nikulin, The transition constant for arithmetic hyperbolic reflection groups, Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), no. 5, 103–138 (Russian, with Russian summary); English transl., Izv. Math. 75 (2011), no. 5, 971–1005. MR 2884665, DOI 10.1070/IM2011v075n05ABEH002561
A. L. Onishchik, editor, Voprosy Teorii Grupp i Gomologicheskoi Algebry. Yaroslavl, 1987. English translation: Selecta Math. Soviet., Vol. 9, No. 4 (1990).
John R. Parker, Complex hyperbolic lattices, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 1–42. MR 2581913, DOI 10.1090/conm/501/09838
Emile Picard, Sur des fonctions de deux variables indépendantes analogues aux fonctions modulaires, Acta Math. 2 (1883), no. 1, 114–135 (French). MR 1554595, DOI 10.1007/BF02415213
H. Poincaré, Théorie des groupes fuchsiens, Acta Math. 1 (1882), no. 1, 1–76 (French). MR 1554574, DOI 10.1007/BF02391835
Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
Gopal Prasad, Volumes of $S$-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117. With an appendix by Moshe Jarden and the author. MR 1019962
M. N. Prokhorov, Absence of discrete groups of reflections with a noncompact fundamental polyhedron of finite volume in a Lobachevskiĭ space of high dimension, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 2, 413–424 (Russian). MR 842588
Leonid Potyagailo and Ernest Vinberg, On right-angled reflection groups in hyperbolic spaces, Comment. Math. Helv. 80 (2005), no. 1, 63–73. MR 2130566, DOI 10.4171/CMH/4
Julien Paupert and Pierre Will, Real reflections, commutators and cross-ratios in complex hyperbolic space. Preprint arXiv:1312.3173 [math.GT], 2013.
John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR 2249478
Roland K. W. Roeder, Constructing hyperbolic polyhedra using Newton’s method, Experiment. Math. 16 (2007), no. 4, 463–492. MR 2378487
O. P. Ruzmanov, Examples of nonarithmetic crystallographic Coxeter groups in $n$-dimensional Lobachevskiĭ space when $6\leq n\leq 10$, Problems in group theory and in homological algebra (Russian), Yaroslav. Gos. Univ., Yaroslavl′1989, pp. 138–142 (Russian). MR 1068774
O. P. Ruzmanov, Subgroups of reflections in Bianchi groups, Uspekhi Mat. Nauk 45 (1990), no. 1(271), 189–190 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 1, 227–228. MR 1050942, DOI 10.1070/RM1990v045n01ABEH002323
Peter Sarnak, Notes on thin matrix groups, Thin groups and superstrong approximation, Math. Sci. Res. Inst. Publ., vol. 61, Cambridge Univ. Press, Cambridge, 2014, pp. 343–362. MR 3220897
Rudolf Scharlau and Britta Blaschke, Reflective integral lattices, J. Algebra 181 (1996), no. 3, 934–961. MR 1386588, DOI 10.1006/jabr.1996.0155
Rudolf Scharlau, On the classification of arithmetic reflection groups on hyperbolic $3$-space. Preprint, Bielefeld, 1989.
M. K. Shaiheev, Reflective subgroups in Bianchi groups, Selecta Math. Soviet. 9 (1990), no. 4, 315–322. Selected translations. MR 1078260
O. V. Shvartsman, Reflective subgroups of Bianchi groups, Selecta Math. Soviet. 9 (1990), no. 4, 323–329. Selected translations. MR 1078261
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
Matthew Stover, Hurwitz ball quotients, Math. Z. 278 (2014), no. 1-2, 75–91. MR 3267570, DOI 10.1007/s00209-014-1306-6
Rudolf Scharlau and Claudia Walhorn, Integral lattices and hyperbolic reflection groups, Astérisque 209 (1992), 15–16, 279–291. Journées Arithmétiques, 1991 (Geneva). MR 1211022
G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356. MR 61749, DOI 10.1512/iumj.1954.3.53017
William P. Thurston, Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.
J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62. MR 0224710
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0
Ivica Turkalj, Totally-reflective genera of integral lattices. Preprint arXiv:1503.04428v2 [math.NT].
È. B. Vinberg, V. V. Gorbatsevich, and O. V. Shvartsman, Discrete subgroups of Lie groups, Current problems in mathematics. Fundamental directions, Vol. 21 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 5–120, 215 (Russian). MR 968445
Marie-France Vignéras, Quelques remarques sur la conjecture $\lambda _{1}\geq {1\over 4}$, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982) Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 321–343 (French). MR 729180
È. B. Vinberg, Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 72 (114) (1967), 471–488; correction, ibid. 73 (115) (1967), 303 (Russian). MR 0207853
È. B. Vinberg, Rings of definition of dense subgroups of semisimple linear groups. , Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 45–55 (Russian). MR 0279206
È. B. Vinberg, The groups of units of certain quadratic forms, Mat. Sb. (N.S.) 87(129) (1972), 18–36 (Russian). MR 0295193
È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505
È. B. Vinberg, The nonexistence of crystallographic reflection groups in Lobachevskiĭ spaces of large dimension, Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 67–68 (Russian). MR 617472
È. B. Vinberg, Absence of crystallographic groups of reflections in Lobachevskiĭ spaces of large dimension, Trudy Moskov. Mat. Obshch. 47 (1984), 68–102, 246 (Russian). MR 774946
È. B. Vinberg, Discrete reflection groups in Lobachevsky spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 593–601. MR 804716
È. B. Vinberg, Hyperbolic groups of reflections, Uspekhi Mat. Nauk 40 (1985), no. 1(241), 29–66, 255 (Russian). MR 783604
È. B. Vinberg, Reflective subgroups in Bianchi groups, Selecta Math. Soviet. 9 (1990), no. 4, 309–314. Selected translations. MR 1078259
È. B. Vinberg, Classification of 2-reflective hyperbolic lattices of rank 4, Tr. Mosk. Mat. Obs. 68 (2007), 44–76 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2007), 39–66. MR 2429266, DOI 10.1090/s0077-1554-07-00160-4
E. B. Vinberg, Some free algebras of automorphic forms on symmetric domains of type IV, Transform. Groups 15 (2010), no. 3, 701–741. MR 2718942, DOI 10.1007/s00031-010-9107-4
È. B. Vinberg, Non-arithmetic hyperbolic reflection groups in higher dimensions. Univ. Bielefeld Preprint 14047, 2014.
È. B. Vinberg and I. M. Kaplinskaja, The groups $O_{18,1}(Z)$ and $O_{19,1}(Z)$, Dokl. Akad. Nauk SSSR 238 (1978), no. 6, 1273–1275 (Russian). MR 0476640
Claudia Walhorn, Arithmetische Spiegelungsgruppen auf dem $4$-dimensionalen hyperbolischen Raum. PhD thesis, Univ. Bielefeld, 1993.
Hsien Chung Wang, Topics on totally discontinuous groups, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 459–487. MR 0414787
Dave Witte Morris, Introduction to arithmetic groups, Deductive Press, [place of publication not identified], 2015. MR 3307755
Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325
P. Zograf, A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group, J. Reine Angew. Math. 414 (1991), 113–116. MR 1092625, DOI 10.1515/crll.1991.414.113