### Arithmetic Hyperbolic Reflection Groups

#### April 27, 2016    refl

• This is an attempt to collect some results concerning classification of arithmetic hyperbolic reflection groups and reflective hyperbolic lattices.
• This page is under construction. Any suggestions and corrections are welcome.
• For more detailed discussion of the general theory of hyperbolic Coxeter polytopes see the page of Anna Felikson and Pavel Tumarkin.
• This page is constructed by Nikolay Bogachev. The idea is suggested by Anna Felikson and Ruth Kellerhals.

## Basic definitions and facts

• Suppose $\mathbb{F}$ is a totally real number field with the ring of integers $A = \mathbb{O}_{\mathbb{F}}$. Suppose $f(x)$ is a quadratic forms of signature $(n,1)$ defined over $\mathbb{F}$ such that for every non-identity embedding $\sigma \colon \mathbb{F} \to \mathbb{R}$ the form $f^{\sigma}$ is positive definite. Such forms $f(x)$ are said to be admissible.

• Suppose $O’(f, A)$ is the group of integral automorphisms (i.e. the automorphisms with coefficients from $A$) of the form $f(x)$, preserving the $n$-dimensional Lobachevsky (hyperbolic) space $\mathbb{H^n}$.

• By the result of A. Borel and Harish-Chandra (see [BHC62]) or G. Mostow and Tamagawa (see [MT62]), the group $O’(f, A)$ is the cocompact discrete group of motions of the space $\mathbb{H}^n$ with an exception of the field $\mathbb{F} = \mathbb{Q}$, when such group could be of cofinite volume, but not cocompact. The case $\mathbb{F} = \mathbb{Q}$ (when $\mathbb{O}_{\mathbb{F}} = \mathbb{Z}$) was studied by Venkov in 1937 (see [Ven37]). The groups $O’(f,\mathbb{Z})$ is cocompact only in the case, when the form $f(x)$ is anisotropic. All rational quadratic forms $f(x)$ of 5 and more variables are isotropic.

• Such groups $O’(f, \mathbb{O}_{\mathbb{F}})$ are said to be $\textbf{arithmetic lattices of simplest type}$.

• Suppose $O_r(f, A)$ is the subgroup generated by all reflection in $O’(f, A)$. If it is the finite index subgroup of $O’(f, A)$, then the form $f(x)$ is called reflective. It is equivalent to the fact that the fundamental (Coxeter) polyhedron of the group $O_r(f, A)$ has a finite volume in the space $\mathbb{H}^n$.

• In other words, a free finitely-generated $A$-module $L$ with an inner product of signature $(n,1)$ is said to be $\textbf{a hyperbolic lattice}$, if for each non-identity embedding $\sigma \colon \mathbb{F} \to \mathbb{R}$ the quadratic space $L \otimes_{\sigma(A)} \mathbb{R}$ is positive definite.

• Suppose $O’(L)$ is the group of integral automorphisms of a lattice $L$, preserving the $n$-dimensional Lobachevsky (hyperbolic) space $\mathbb{H^n}$.

• If the subgroup $O_r(L)$ of $O’(L)$ generated by all reflections is of finite index, then the lattice $L$ is said to be $\textbf{reflective}$. It is equivalent to the fact that the fundamental (Coxeter) polyhedron of the group $O_r(L)$ has a finite volume in the space $\mathbb{H}^n$.

• All the groups of type $O_r(f, A)$ as well as all the groups, commensurable with them, are called $\textbf{arithmetic hyperbolic reflection groups}$ with the $\textbf{field of definitions}$ (or the $\textbf{ground field}$) $\mathbb{F}$.

## Existence and Finiteness Theorems

1. (Nikulin, 2007) There are only finitely many of arithmetic maximal hyperbolic reflection groups. See [Nik07].
2. (Vinberg, 1984) In Lobachevsky spaces $\mathbb{H}^n$ of dimension $n \ge 30$ there are no arithmetic hyperbolic reflection groups. See [Vin84].
3. (Vinberg, 1984) In Lobachevsky spaces $\mathbb{H}^n$ of dimension $n \ge 22$ there are no arithmetic hyperbolic reflection groups with a ground field other than $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\cos(2\pi/7))$. See [Vin84].
4. (Vinberg, 1984) In Lobachevsky spaces $\mathbb{H}^n$ of dimension $n \ge 14$ there are no arithmetic hyperbolic reflection groups with a ground field other than $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$, $\mathbb{Q}(\sqrt{5})$, $\mathbb{Q}(\sqrt{6})$, $\mathbb{Q}(\sqrt{2}, \sqrt{3})$, $\mathbb{Q}(\sqrt{2}, \sqrt{5})$ and $\mathbb{Q}(\cos(2\pi/m))$, where $m=7,9,11,15,16,$ or $20$. See [Vin84].
5. (Nikulin, 2011) In the Lobachevsky spaces $\mathbb{H}^n$ of dimension $4 \leq n \leq 13$ the degree $d = [\mathbb{F} : \mathbb{Q}]$ is at most $25$. See [Nik11].
6. (Belolipetsky, 2011) In the Lobachevsky space $\mathbb{H}^3$ the degree $d = [\mathbb{F} : \mathbb{Q}]$ is at most $9$. See [Bel09] and [Bel11].
7. (Linowitz, 2017) In the Lobachevsky plane $\mathbb{H}^2$ the degree $d = [\mathbb{F} : \mathbb{Q}]$ is at most $7$. See [Lin17].

## Vinberg’s Algorithm

In 1972 (see [Vin72] and [Vin73]), Vinberg proposed an effective algorithm of constructing the fundamental polyhedron $P$ for hyperbolic reflection group. It works for each reflection group, but is efficient only for groups of the form $O_r (L)$.

• See here the brief description.
• R. Guglielmetti’s implementation for hyperbolic lattices (over a series of ground fields) with an orthogonal basis (the program with the documentation is available here, and also some information you can find in [Gugl17])
• Sowtware implementation by Bogachev and Perepechko for arbitrary $\textbf{integral}$ hyperbolic lattices (the program, some brief description). See also the paper [BP18].

## Classification results

1. $\mathbb{F} = \mathbb{Q}$

• V.V. Nikulin, 1979,1981,1984. $2$-reflective hyperbolic lattices (for all $1 < n < 21$ and $n \ne 4$). See [Nik79], [Nik81] and [Nik84].
• E.Vinberg, 1972. Unimodular reflective hyperbolic lattices (reflective for all $n \le 19$). See [Vin72] and [VK78].

• $n=2$.

• V. Nikulin, 2000. Classification of reflective lattices of rank $3$ with square free discriminants. See [Nik00]
• D. Allcock, 2012. Full classification of Reflective Lorentzian Lattices of Rank $3$. See [All12]
• $n=3$.

• E.B. Vinberg, 2007. $2$-reflective hyperbolic lattices of rank $4$. See [Vin07].
• R. Scharlau, 1989. Reflective isotropic hyperbolic lattices. See [Sch89]
• N.V. Bogachev, 2016-2017. $(1.2)$-reflective anisotropic hyperbolic lattices of rank $4$. See [B17] and [B18].
• Anisotropic case: $\textbf{Open Problem}$
• $n=4$. C. Walhorn. See [SW92] and [Wal93].

• $n=5$. I. Turkalj. Classification of Relective Lorentzian Lattices of Signature $(5,1)$. See [Tur17].

• $n \ge 6$. $\quad \textbf{Open Problem}$

2. $\mathbb{F} = \mathbb{Q}[\sqrt{2}]$.

• $n=2$. A. Mark. Classification of Reflective Hyperbolic Lattices of rank $3$. See [Mar17].
• $n \ge 3$. $\quad \textbf{Open Problem}$
3. $\mathbb{F} = \mathbb{Q}[\sqrt{5}]$

• V.O. Bugaenko, 1992. Classification of Unimodular Reflective Hyperbolic Lattices (reflective for all $n \le 7$). See [Bug92].
4. $\mathbb{F} = \mathbb{Q}[\cos(2\pi/7)]$

• V.O. Bugaenko, 1992. Classification of Unimodular Reflective Hyperbolic Lattices (reflective for all $n \le 4$). See [Bug92].
5. Other Ground Fields. $\textbf{Open Problem}$.

## Methods of Classification

(Under construction…)

• Nikulin’s ideas. See [Nik00].
• Allcock’s methods. See [All12] and [Mar17].
• Scharlau’s approach for isotropic lattices. See [Sch89], [SW92], [W93] and [Tur89].
• Vinberg’s Method for Lattices of Rank $4$. See [Vin07]
• Spectral Method. See [Bel16].
• Method of the outermost edge. See [B17] and [B18].

## Open Problems

1. Classification of reflective hyperbolic lattices with ground fields other than $\mathbb{Q}$.

2. Find the list of all possible ground fields of arithmetic hyperbolic reflection groups.

3. Improve the upper bounds for degrees of ground fields of arithmetic hyperbolic reflection groups.

4. Classification of reflective anisotropic hyperbolic lattices over $\mathbb{Q}$ of rank 4.

5. Classification of reflective anisotropic hyperbolic lattices over $\mathbb{Q}$ of ranks more than 6.

6. Efficient software implementation of Vinberg’s Algorithm.

## References

[Agol06] Ian Agol. Finiteness of arithmetic Kleinian reflection groups. In International Congress of Math- ematicians. Vol. II, pages 951–960. Eur. Math. Soc., Zu ̈rich, 2006

[ABSW08] Ian Agol, Mikhail Belolipetsky, Peter Storm, and Kevin Whyte. Finiteness of arithmetic hyper- bolic reflection groups. Groups Geom. Dyn., 2(4):481–498, 2008.

[All12] D. Allcock, The Reflective Lorentzian Lattices of Rank $3$, in Mem.Amer.Math.Soc. (Amer.Math.Soc., Providence, RI, 2012), Vol. 220, No. 1033.

[Bel09] Mikhail Belolipetsky. On fields of definition of arithmetic Kleinian reflection groups. Proc. Amer. Math. Soc., 137(3):1035–1038, 2009.

[Bel11] Mikhail Belolipetsky. Finiteness theorems for congruence reflection groups. Transform. Groups, 16(4):939–954, 2011.

[Bel16] M. Belolipetsky — Arithmetic hyperbolic reflection groups

[BHC62] Armand Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann. of Math. (2), 75:485–535, 1962.

[B17] N.V. Bogachev — Reflective anisotropic hyperbolic lattices of rank 4, Russian Mathematical Surveys, 2017, vol. 1 (433), p. 179 - 181.

[B18] N.V. Bogachev — The classification of $(1.2)$-reflective anisotropic hyperbolic lattices of rank $4$. Izv.Math., 2018.

[BP18] N.V. Bogachev, A.Ju. Perepechko — Vinberg’s Algorithm for Hyperbolic Lattices.

[Bug92] V.O. Bugaenko. Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices.~— Advances in Soviet Mathematics, 1992, Volume 8, p. 33–55.

[Gugl17]

[LMR06] Darren Long, Colin Maclachlan, and Alan Reid. Arithmetic Fuchsian groups of genus zero. Pure Appl. Math. Q., 2(2, part 2):569–599, 2006.

[Lin17] B. Linowitz, Bounds for arithmetic hyperbolic reflection groups in dimension $2$.

[Mac11] Colin Maclachlan. Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2. Bull. Lond. Math. Soc., 43(1):111–123, 2011.

[Mar17] Alice Mark. The classification of rank 3 reflective hyperbolic lattices over Z(sqrt(2)). PhD thesis, University of Texas at Austin, 2015. in preparation.

[MT62] G.D. Mostow and T. Tamagawa. On the compactness of arithmetically defined homogeneous spaces. Ann. of Math, 1962, Vol.76, No. 3, pp. 446–463.

[Nik79] V. V. Nikulin. Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by 2-reflections. Dokl. Akad. Nauk SSSR, 248(6):1307–1309, 1979.

[Nik80] V. V. Nikulin. On the arithmetic groups generated by reflections in Lobacˇevski ̆ı spaces. Izv. Akad. Nauk SSSR Ser. Mat., 44(3):637–669, 719–720, 1980.

[Nik81a] V. V. Nikulin. On the classification of arithmetic groups generated by reflections in Lobachevski ̆ı spaces. Izv. Akad. Nauk SSSR Ser. Mat., 45(1):113–142, 240, 1981.

[Nik81b] V. V. Nikulin. Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. Algebro-geometric applications. In Current problems in mathematics, Vol. 18, pages 3–114. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981. English transl., J. Soviet Math. 22:4 (1983), 1401–1475.

[Nik84] V. V. Nikulin. K3 surfaces with a finite group of automorphisms and a Picard group of rank three. Trudy Mat. Inst. Steklov., 165:119–142, 1984. Algebraic geometry and its applications.

[Nik00] V. V. Nikulin. On the classification of hyperbolic root systems of rank three. Tr. Mat. Inst. Steklova, 230:256, 2000.

[Nik07] V. V. Nikulin. Finiteness of the number of arithmetic groups generated by reflections in Lobachevski ̆ı spaces. Izv. Ross. Akad. Nauk Ser. Mat., 71(1):55–60, 2007.

[Nik09] V. V. Nikulin. On ground fields of arithmetic hyperbolic reflection groups. In Groups and sym- metries, volume 47 of CRM Proc. Lecture Notes, pages 299–326. Amer. Math. Soc., Providence, RI, 2009.

[Nik11] V. V. Nikulin. The transition constant for arithmetic hyperbolic reflection groups. Izv. Ross. Akad. Nauk Ser. Mat., 75(5):103–138, 2011.

[Sch89] Rudolf Scharlau. On the classification of arithmetic reflection groups on hyperbolic 3-space. Preprint, Bielefeld, 1989.

[SW92] R. Scharlau, C. Walhorn. Integral lattices and hyperbolic reflection groups.~— Asterisque, 1992, V. 209. p. 279~—291.

[Tur17] Ivica Turkalj. Reflective Lorentzian Lattices of Signature $(5,1)$. PhD-Thesis, TU Dortmund, Germany, 2017.

[Wal93] Claudia Walhorn. Arithmetische Spiegelungsgruppen auf dem 4-dimensionalen hyperbolischen Raum. PhD thesis, Univ. Bielefeld, 1993.

[Ven37] B.A. Venkov. Izv. Akad. NaukSSSRSer.Mat.1(2),139(1937).

[Vin67] E.B. Vinberg. Discrete groups generated by reflections in Lobacˇevski ̆ı spaces. Mat. Sb. (N.S.), 72 (114):471–488; correction, ibid. 73 (115) (1967), 303, 1967.

[Vin72] E.B. Vinberg.The groups of units of certain quadratic forms. Mat.Sb.(N.S.), 87 (129): 18–36, 1972.

[Vin73] E.B. Vinberg. Some arithmetical discrete groups in Lobachevsky spaces. — In: Proc. Int. Coll. on Discrete Subgroups of Lie Groups and Appl. to Moduli (Bombay, January 1973).~— Oxford: University Press, 1975, p. 323 — 348.

[VK78] E.B. Vinberg and I.M. Kaplinskaja. The groups $O{18,1}(\mathbb{Z})$ and $O{19,1}(\mathbb{Z})$. Dokl. Akad. Nauk SSSR, 238(6): 1273–1275, 1978.

[Vin84] E.B.Vinberg. Absence of crystallographic groups of reflections in Lobachevskii spaces of large dimension. Trudy Moskov. Mat. Obshch., 47:68–102, 246, 1984.

[Vin85] E.B. Vinberg — Hyperbolic reflection groups, Russian Mathematical Surveys, 1985, vol. 40, p. 31 - 75.

[Vin93] E.B. Vinberg(ed) — Spaces of constant curvature, Geom. II, vol. 29, Encyclopaedia of Math. Sciences, Springer-Verlag, Berlin, 1993.

[Vin07] E.B. Vinberg. Classification of $2$-reflective hyperbolic lattices of rank $4$.~—Tr. Mosk. Mat. Obs., 2007: 68, p. 44–76. [Trans. Moscow Math. Soc. 2007, 39].

[Vin14] E.B. Vinberg.Non-arithmetichyperbolicreflectiongroupsinhigherdimensions.Univ.Bielefeld Preprint 14047, 2014.