Резюме (Рефлективные гиперболические решётки), страница 3

The faces F3 and F4 are called the framing edges of the edge E, and thenumber |(u3 , u4 )| is its width.We associate with the edge E the set ᾱ = (α12 , α13 , α23 , α14 , α24 ), where αi j is the anglebetween the faces Fi and F j .Theorem 4. The fundamental polyhedron of every Q-arithmetic cocompact group ofreflections in H3 has an edge of width less than 4.14.In fact, a stronger result is obtained.

Namely, it is proved that there is an edge of widthtᾱ , where tᾱ ≤ 4.14 is a number depending on the set ᾱ of dihedral angles around this edge.To obtain this result, the following method is used. Let P be the fundamental polytope ofa Q-arithmetic cocompact reflection group in H3 . Following Nikulin, we consider a point Oinside the polyhedron P. Let E be the outermost2 edge from it. We denote the vertices of theedge E by V1 and V2 , and the dihedral angles between the faces Fi and F j will be denoted byαi j .Let E1 and E3 be the edges of the polytope P outgoing from the vertex V1 and let E2 andE4 be the edges outgoing from V2 such that the edges E1 and E2 lie in the face F1 . The length8Fig.

1. The outermost edgeof the edge E is denoted by a, and the plane angles between the edges E j and E are denotedby α j (see Figure 1).The following result is true for an arbitrary compact acute-angled polytope in H3 .Theorem 5. The length of the outermost edge satisfies the inequality tanh(ln(ctg( α12 )))  tanh(ln(ctg( α12 )))  . + arcsinh (α ) 4(α ) 4a < arcsinh 3tan 2tan 24Then it remains to estimate, by using a linear inequality, the width of the edge through itslength. To do this, we use the fact that we initially considered the fundamental polyhedronof the Q-arithmetic cocompact reflection group in H3 . As we see, the estimates in Theorem5 depend on the set of angles around this edge, therefore, the estimates for the width of theedge also depend on it.To formulate the results of classification of stably reflective lattices we introduce somenotation for hyperbolic lattices:• [C] is a quadratic lattice whose inner product in some basis is given by a symmetricmatrix C,• d(L) := det C is the discriminant of the lattice L = [C],• L ⊕ M is the orthogonal sum of the lattices L and M,• [k]L is the quadratic lattice obtained from L by multiplying all inner products by k ∈ A.is the adjoint lattice.2In an acute-angled polyhedron the distance from an interior point to a face (of any dimension) is equal tothe distance to the plane of this face.9Theorem 6.

Any stably reflective anisotropic hyperbolic lattice of rank 4 over Z is either isomorphic to [−7] ⊕ [1] ⊕ [1] ⊕ [1] or [−15] ⊕ [1] ⊕ [1] ⊕ [1], or to an even index 2 sublatticeof one of them.Actually, these lattices are even 2-reflective (see [42]).√Chapter 5. Stably reflective Z[ 2]-lattices of rank 4√Theorem 7. The fundamental polyhedron of any Q[ 2]-arithmetic group of reflectionsin H3 has an edge of width less than 4.14.As above, actually a stronger result is obtained. Namely, it is proved that there is an edgeof width tᾱ , where tᾱ ≤ 4.14 is a number depending on the set ᾱ of dihedral angles around thisedge.√Theorem 8.

Any maximal stably reflective hyperbolic lattice of rank 4 over Z[ 2] isisomorphic to one of the following seven lattices:№1234567L√[−1 − 2] ⊕ [1] ⊕ [1] ⊕ [1]√[−1 − 2 2] ⊕ [1] ⊕ [1] ⊕ [1]√[−5 − 4 2] ⊕ [1] ⊕ [1] ⊕ [1]√[−11 − 8 2] ⊕ [1] ⊕ [1] ⊕ [1]√[− 2] ⊕ [1] ⊕ [1] ⊕ [1]√  2−1−√ 2 2 −√1 ⊕ [1] −1√2 √− 22−1 2− 2√[−7 − 5 2] ⊕ [1] ⊕ [1] ⊕ [1]# faces565176Discriminant√−1 − 2√−1 − 2 2√−5 − 4 2√−11 − 8 2√− 26√− 25√−7 − 5 2Approbation of the workThe results of the thesis have been reported at the following meetings:• the seminar “Lie groups and invariant theory”, led by E.B.

Vinberg, D.A. Timashev andI.V. Arzhantsev, the Faculty of Mechanics and Mathematics, Moscow State University,May 2016 and October 2017;• the Sixth School-Conference “Lie Algebras, Algebraic Groups and Invariant Theory”,MSU & IUM, Moscow, Russia, January–February 2017;• S.P. Novikov’s Seminar “Geometry, topology and mathematical physics”, the Faculty ofMechanics and Mathematics, Moscow State University, March 2017;• the international conference “Geometry and Topology” in honor of C. Bavard, Instituteof Mathematics, Bordeaux, France, November 2017;• the seminar “Hyperbolic geometry and combinatorial structures”, Institute of Mathematics, Neuchatel, Switzerland, November 2017;10• the seminar “Automorphic forms and their applications”, led by V.A. Grytsenko, theFaculty of Mathematics, HSE, Moscow, Russia, February 2018;• the international conference “Automorphic forms and algebraic geometry”, PDMI SteklovInstitute of RAS, St.

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