Резюме (Рефлективные гиперболические решётки)

Federal State Budget Educational Institution of Higher ProfessionalEducation M. V. Lomonosov Moscow State UniversityFaculty of Mechanics and MathematicsDepartment of Higher AlgebraNikolay V. BogachevReflective hyperbolic latticesSummary of the PhD thesisfor the purpose of obtainingDoctor of Philosophy in Mathematics HSEAcademic supervisor:Doctor of Sciences, Professor Ernest B.

VinbergMoscow 2019Chapter 1. IntroductionAbstractThis thesis is devoted to classification of reflective hyperbolic lattices, which is an openproblem since the 1970s. The dissertation was written under the supervision of Prof. E. B. Vinberg during my postgraduate study at the Department of Higher Algebra of the Faculty ofMechanics and Mathematics of the Moscow State University.The thesis consists of five chapters. The first chapter is an introduction to the subject. Itincludes only some basic definitions along with a number of known facts and open problemsas well as the formulations of the main results of the dissertation. Chapter 2 contains someauxiliary results, including a description of models of spaces of constant curvature, acuteangled polyhedra in them, discrete groups of reflections, and, finally, the fundamentals ofthe theory of reflective hyperbolic lattices and arithmetic hyperbolic reflection groups.The main results of this dissertation are obtained in Chapters 3, 4, and 5.

Chapter 3gives a theoretical description of Vinberg’s Algorithm and also the description of the project(our joint work with A. Yu. Perepechko) VinAl of a computer implementation of Vinberg’sAlgorithm for hyperbolic lattices over Z. Finally, Chapters 4 and 5 contain√ the results ofclassification of stably reflective hyperbolic lattices of rank 4 over Z and Z[ 2], respectively.Discrete reflection groupsLet Xn be one of the three spaces of constant curvature, that is, either the Euclidean spaceE , or the n-dimensional sphere Sn , or the n-dimensional (hyperbolic) Lobachevsky space Hn .Consider a convex polytope P in the space Xn .

If we act on P by the group Γ generatedby reflections in the hyperplanes of its faces it can occur that the images of this polyhedroncorresponding to different elements of Γ will cover the entire space Xn and will not overlapwith each other. In this case we say that Γ is a discrete reflection group, and the polytope P isthe fundamental polyhedron for Γ. If the polytope P is bounded (or, equivalently, compact),then the group Γ is called a cocompact reflection group, and if the polytope P has a finitevolume, then the group Γ is called cofinite or a discrete group of finite covolume.Which properties characterize such polyhedra P? For example, any two hyperplanes Hiand H j bounding P either do not intersect or form a dihedral angle equal to π/ni j , whereni j ∈ Z, ni j ≥ 2.Such polyhedra are called Coxeter polyhedra, since the discrete reflection groups of finitecovolume (hence their finite volume fundamental polyhedra) for Xn = En , Sn were determinedand found by H.S.M.

Coxter in 1933 (see [19]).In 1967 (see [37]), E. B. Vinberg developed his theory of discrete groups generated byreflections in the Lobachevsky spaces. He proposed new methods for studying hyperbolicreflection groups, in particular, a description of such groups in the form of the so-called Coxeter diagrams. He formulated and proved the arithmeticity criterion for hyperbolic reflectiongroups and constructed a number of various examples.n1Arithmetic reflection groups and reflective hyperbolic latticesSuppose F is a totally real number field with the ring of integers A = OF . For conveniencewe will assume that it is a principal ideal domain.Definition 1. A free finitely generated A-module L with an inner product of signature(n, 1) is said to be a hyperbolic lattice if, for each non-identity embedding σ : F → R, thequadratic space L ⊗σ(A) R is positive definite.

(The inner product in L is associated withsome admissible quadratic form.)Suppose that L is a hyperbolic lattice. Then the vector space En,1 = L ⊗id(A) R is identifiedwith the (n + 1)-dimensional real Minkowski space. The group Γ = O′ (L) of integer (thatis, with coefficients in A) linear transformations preserving the lattice L and mapping eachconnected component of the coneC = {v ∈ En,1 | (v, v) < 0} = C+ ∪ C−into itself is a discrete group of motions of the Lobachevsky space. Here we mean the vectormodel Hn given as a set of points of the hyperboloid{v ∈ En,1 | (v, v) = −1},in the convex open cone C+ . The group of motions is Isom(Hn ) = O′ (n, 1), which is the groupof pseudoorthogonal transformations of the space En,1 that leaves invariant the cone C+ .It is known from the general theory of arithmetic discrete groups (A.

Borel & HarishChandra [11] and G. Mostov & T. Tamagawa [24] in 1962) that if F = Q and the lattice Lis isotropic (that is, the quadratic form associated with it represents zero), then the quotientspace Hn /Γ (the fundamental domain of Γ) is not compact, but is of finite volume (in this casewe say that Γ is a discrete subgroup of finite covolume), and in all other cases it is compact.For F = Q, these assertions were first proved by B.A. Venkov in 1937 (see [16]).Definition 2. Two subgroups Γ1 and Γ2 of some group are said to be commensurableif the group Γ1 ∩ Γ2 is a subgroup of finite index in each of them.Definition 3.

The groups Γ obtained in the above way and the subgroups of the groupIsom(Hn ) that are commensurable with them are called arithmetic discrete groups of thesimplest type. The field F is called the definition field (or the ground field) of the group Γ(and all subgroups commensurable with it).A primitive vector e of a quadratic lattice L is called a root or, more precisely, a k-root,where k = (e, e) ∈ A>0 if 2(e, x) ∈ kA for all x ∈ L.

Every root e defines an orthogonal reflection(called a k-reflection if (e, e) = k) in the space L ⊗id(A) RRe : x 7→ x −2(e, x)e,(e, e)which preserves the lattice L. In the hyperbolic case, Re determines the reflection of the spaceHn with respect to the hyperplaneHe = {x ∈ Hn | (x, e) = 0},called the mirror of the reflection Re .We denote by Or (L) the subgroup of the group O′ (L) generated by all reflections containedin it.2Definition 4. A hyperbolic lattice L is said to be reflective if the index [O′ (L) : Or (L)] isfinite.Theorem 1. (Vinberg, 1967, see [37])A discrete reflection group of finite covolume is an arithmetic group with a ground fieldF (or an F-arithmetic reflection group) if it is a subgroup of finite index in a group of theform O′ (L), where L is some (automatically reflective) hyperbolic lattice over a totally realnumber field F.Now we formulate some fundamental theorems on the existence of arithmetic reflectiongroups and cocompact reflection groups in the Lobachevsky spaces.Theorem 2.

(Vinberg, 1984, see [41])1. Compact Coxeter polyhedra do not exist in the Lobachevsky spaces Hn for n ≥ 30.2. Arithmetic reflection groups do not exist in the Lobachevsky spaces Hn for n ≥ 30.The next important result belongs to several authors.Theorem 3. For each n ≥ 2, up to scaling, there are only finitely many reflective hyperbolic lattices. Similarly, up to conjugacy, there are only finitely many maximal arithmeticreflection groups in the spaces Hn .The proof of this theorem is divided into the following stages:• 1980, 1981 – V. V.

Nikulin proved that there are only finitely many maximal arithmeticreflection groups in the spaces Hn for n ≥ 10, see [26, 28];• 2005 – D. D. Long, C. Maclachlan and A. W. Reid proved the finiteness of maximalarithmetic reflection groups in dimension n = 2, see [21];• 2005 — I. Agol proved the finiteness in dimension n = 3, see [1];• 2007 — V. V. Nikulin proved by induction the finiteness in the remaining dimensions4 ≤ n ≤ 9, see [31];• 2008 — I. Agol, M. Belolipetsky, P.

Storm, and K. Whyte independently proved thefiniteness theorem for all dimensions by their spectral method, see [2] (see also therecent survey [7] of M. Belolipetsky).The above results give the hope that all reflective hyperbolic lattices, as well as maximalarithmetic hyperbolic reflection groups can be classified.Open problemsOur discussion above leads us to the following fundamental open problems connectedwith the theory of discrete reflection groups and Coxeter polytopes in the Lobachevsky spacesHn .3Problem 1.

Which is the maximal dimension of the Lobachevsky space in which thereexist compact Coxeter polytopes? A similar question is open for Coxeter polytopes of finitevolume.Problem 2. Classification of reflective hyperbolic lattices and maximal arithmetic hyperbolic reflection groups.Remark 1. The problem of classification of reflective hyperbolic lattices was actuallyposed in cited work of Vinberg in 1967. Further results obtained in the 1970-80s (and alsosome recent results) definitely confirm that there is a hope to solve these problems.A very efficient tool for solving problems 1 and 2 is Vinberg’s Algorithm (1972, see [38])of constructing the fundamental polyhedron for a hyperbolic reflection group.

Practically itis efficient for arithmetic reflection groups. It enables one given a lattice to determine if thislattice is reflective.The record example of a compact Coxeter polyhedron was found by V.O. Bugaenko forn = 8 (see [15]), although the maximal possible dimension is bounded by the inequalityn < 30 (see Theorem 2 above).A record example of a Coxeter polyhedron of finite volume belongs to R. Borcherds in thedimension n = 21 (see [12]). It is known that the Coxeter polytopes of finite volume can existonly for n < 996 (see papers [33] of M.

Prokhorov and [20] A. Khovanskii, 1986).Both examples came from arithmetic reflection groups. Bugaenko’s polyhedron√ is thefundamental polyhedron for some arithmetic reflection group over the field Q[ 2] in thespace H8 , the Borcherds polyhedron is the fundamental polyhedron for some arithmetic groupof reflections over a field Q in the space H21 .Moreover, D. Allcock, using an elegant and a simple doubling trick, has constructed infinite series (see [3]) of finite volume Coxeter polytopes in Lobachevsky spaces through dimension 19, and also of compact Coxeter polytopes through dimension 6. We also note thatin dimensions 7 and 8 they can be taken to be either arithmetic or nonarithmetic.As for the second problem, it is also far from being completely solved.