Резюме (1137299), страница 2

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 2)

An effective description of all discrete reflection groups in the spaces Hn is obtained only for n = 2 (H. Poincaré,1882, see [32]) and for n = 3 (the famous theorems of E. M. Andreev, 1970, see [5] and [6]).In the classification of arithmetic hyperbolic reflection groups a more significant successhas been achieved. Over the definition field Q, the reflective hyperbolic lattices are classifiedfor n = 2 (V.V. Nikulin, 2000, see [30], and D. Allcock, 2011, see [4]), n = 4 (R. Sharlau andC. Walhorn, 1989–1993, see [35, 43]), n = 5 (I.

Turkalj, 2017, see [36]) and in the noncompact(isotropic) case for n = 3 (R. Sharlau and C. Walhorn, 1989–1993, see [34, 35]).√A classification of reflective hyperbolic lattices of signature (2, 1) with the definition fieldQ[ 2] was obtained by A. Mark in 2015, see [22, 23].In all other cases, Problem 2 remains open.Chapter 2. Discrete reflection groupsChapter 2 contains some auxiliary results, including a description of models of spacesof constant curvature, acute-angled polyhedra in them, discrete groups of reflections, and,4finally, the fundamentals of the theory of reflective hyperbolic lattices and arithmetic reflection groups.

Here we define the Coxeter diagrams and we also give a list of connected ellipticand parabolic Coxeter diagrams.Chapter 3. Vinberg’s AlgorithmProject VinAl: for hyperbolic lattices over ZThis chapter is devoted to Vinberg’s algorithm and the creation of a tool for solving problem 1 and 2.

With the help of different computer implementations of Vinberg’s√ algorithm, thereflectivity was investigated for dozens of hyperbolic lattices over Z and Z[ 2]. In this way,was obtained a large number of previously unknown arithmetic compact Coxeter polytopesin Lobachevsky spaces.As mentioned above, Vinberg’s algorithm is an efficient tool for constructing the fundamental polyhedra for arithmetic reflection groups. Some efforts to implement Vinberg’salgorithm by using a computer have been made since the 1980s, but all of them dealt with particular lattices, usually with an orthogonal basis.

Such programs are mentioned, e.g., in thepapers of Bugaenko (1992, see [15]), Scharlau and Walhorn (1989–1993, see [35]), Nikulin(2000, see [30]), and Allcock (2011, see [4]). But the programs themselves have not beenpublished; the only exception is Nikulin’s paper, which contains a program code for latticesof several different special forms. The only known implementation published together with adetailed documentation is Guglielmetti’s 2016 program1 , processing hyperbolic lattices withan orthogonal basis with square-free invariant factors over several ground fields.

Guglielmetti used this program in his thesis to classify reflective hyperbolic lattices with an orthogonal basis with small inner squares of its elements. His program works fairly efficiently in alldimensions in which reflective lattices exist.In this paper, we present our own implementation of Vinberg’s algorithm for arbitraryintegral (with the ground field Q) hyperbolic lattices subject to no constraints.

The project iswritten jointly with A.Yu. Perepechko in the Sage computer algebra system. It is available inthe Internet (see [9]), and it was published with a detailed description (see [45]).The program was tested on a large number of known examples of reflective hyperboliclattices. We have also found a series of new reflective lattices.Some results yielded by the program are presented in Table 1. In the table, U = [ 01 10 ]denotes the standard two-dimensional hyperbolic lattice and An denotes the Euclidean rootlattice of type An . All lattices in this table, excepting [−1] ⊕ A3 and [−4] ⊕ A3 , are new.Moreover, we have proved the reflectivity of the lattices[−2] ⊕ A2 ⊕ [1] ⊕ .

. . ⊕ [1]| {z }n−1for n ≤ 6.At present, the program works effectively for 2 ≤ n ≤ 5. Thus, it turns out to be useful,e.g., for solving the open problem of classifying reflective lattices in the dimension n = 3;it has already been successfully applied by the author to obtain partial classification results.We plan to optimize the program so as to make it efficient for n ≤ 10.1see project AlVin https://rgugliel.github.io/AlVin5Table 1: Lattices of type [−k] ⊕ A3 , [−k] ⊕ [1] ⊕ A2 for some k ≤ 15, and U ⊕ [36] ⊕ [6].L# faces t (sec)L# faces t (sec)[−1] ⊕ A340,7[−1] ⊕ [1] ⊕ A240,6[−2] ⊕ A351,9[−2] ⊕ [1] ⊕ A260,8[−3] ⊕ A351,0[−3] ⊕ [1] ⊕ A250,6[−4] ⊕ A340,66[−4] ⊕ [1] ⊕ A251,02[−5] ⊕ A361,56[−5] ⊕ [1] ⊕ A271,9[−6] ⊕ A361,5[−6] ⊕ [1] ⊕ A281,2[−8] ⊕ A371,72[−7] ⊕ [1] ⊕ A21119,2[−9] ⊕ A3979,5[−8] ⊕ [1] ⊕ A261,02[−10] ⊕ A3121,72[−9] ⊕ [1] ⊕ A250,9[−12] ⊕ A351.02[−10] ⊕ [1] ⊕ A21111[−15] ⊕ A31228,7[−15] ⊕ [1] ⊕ A21544U ⊕ [36] ⊕ [6]1556,6[−30] ⊕ [1] ⊕ A22036,6√√Table 2: Unimodular lattices over Q[ 13] и Q[ 17].Ln # facesLn # faces√√3+ 13[− 2√ ] ⊕ [1] ⊕ .

. . ⊕ [1] 24[−4 − 17] ⊕ [1] ⊕ . . . ⊕ [1] 24√3+ 13[−4 − 17] ⊕ [1] ⊕ . . . ⊕ [1] 36[− 2√ ] ⊕ [1] ⊕ . . . ⊕ [1] 39√3+ 13[−4 − 17] ⊕ [1] ⊕ . . . ⊕ [1] 420[− 2 ] ⊕ [1] ⊕ . . . ⊕ [1] 440√Vinberg’s Algorithm for hyperbolic lattices over Z[ d]√Since we also investigate the reflectivity of lattices over Z[ 2], the author decided to writea program for Vinberg’s Algorithmover quadratic fields. At the moment the author has a√program for lattices over Z[ 2], which requires some minor editing for each new lattice.

Thisprogram enables one to investigate a lattice without orthogonal bases.For lattices with an orthogonal basis was used the program of Guglielmetti mentionedabove. The work of author’s program was partly verified on the lattices from √Table 5. Inthe nearest future we plan to merge the author’s programs for lattices over Z[ 2] with theVinAl project.

Further work on the √project that implements Vinberg’s algorithm for arbitrarylattices over the quadratic fields Z[ d] is being carried out jointly with A. Yu. Perepechko.As the result of experiments with different programs were obtained some new series ofreflective hyperbolic lattices of different ranks over different quadratic fields. Some of theseresults were obtained jointly with A. A. Kolpakov in 2017–2018.The results obtained are presented in Tables 2–5.

In these tables we indicate first theform of the lattice of signature (n, 1), then we specify the dimension n of the correspondingLobachevsky space, and then the number of faces for the fundamental Coxeter polytope ofthe corresponding reflection group.Chapter 4. Stably reflective hyperbolic Z-lattices of rank 4Definition 5. A number k ∈ A, k > 0 is said to be stable if k | 2 in the ring A.6[−1 −[−1 −[−1 −[−1 −[−1 −L√5] ⊕ [1] ⊕ . . . ⊕ [1]√5] ⊕ [1] ⊕ . . . ⊕ [1]√5] ⊕ [1] ⊕ . . . ⊕ [1]√5] ⊕ [1] ⊕ . .

. ⊕ [1]√5] ⊕ [1] ⊕ . . . ⊕ [1]√Table 3: Some lattices over Q[ 5].n # facesLn√24[−1 − 5] ⊕ [2] ⊕ . . . ⊕ [2] ⊕ [1] 2√35[−1 − 5] ⊕ [2] ⊕ . . . ⊕ [2] ⊕ [1] 3√47[−1 − 5] ⊕ [2] ⊕ . . . ⊕ [2] ⊕ [1] 4√513[−1 − 5] ⊕ [2] ⊕ . . . ⊕ [2] ⊕ [1] 5√618[−1 − 5] ⊕ [2] ⊕ . . . ⊕ [2] ⊕ [1] 6# faces4571318√Table 4: Some lattices over Q[ 2].Ln√√[− 2] ⊕ [2 + 2] ⊕ [1] 2√√[− 2] ⊕ [2 + 2] ⊕ [1] ⊕ . . . ⊕ [1] 3√√[− 2] ⊕ [2 + 2] ⊕ [1] ⊕ . . . ⊕ [1] 4√√[− 2] ⊕ [2 + 2] ⊕ [1] ⊕ . . . ⊕ [1] 5Ln[− 2] ⊕ [1] ⊕ [1] 2√[− 2] ⊕ [1] ⊕ . . . ⊕ [1] 3√[− 2] ⊕ [1] ⊕ . .

. ⊕ [1] 4√[− 2] ⊕ [1] ⊕ . . . ⊕ [1] 5√Ln # faces√[−7 − 5 2] ⊕ [1] ⊕ . . . ⊕ [1] 23√[−7 − 5 2] ⊕ [1] ⊕ . . . ⊕ [1] 35√[−7 − 5 2] ⊕ [1] ⊕ . . . ⊕ [1] 47√[−7 − 5 2] ⊕ [1] ⊕ . . . ⊕ [1] 511√[−7 − 5 2] ⊕ [1] ⊕ . . . ⊕ [1] 645# faces461031√Table 5: Some lattices over Q[ 2].# facesLn # faces√√4[−1 − 2] ⊕ [2 + 2] ⊕ [1] 24√√6[−1 − 2] ⊕ [2 + 2] ⊕ [1] ⊕ .

. . ⊕ [1] 36√√8[−1 − 2] ⊕ [2 + 2] ⊕ [1] ⊕ . . . ⊕ [1] 48√√27[−1 − 2] ⊕ [2 + 2] ⊕ [1] ⊕ . . . ⊕ [1] 5277√For example, if F = Q, A = Z, then√the definition holds for numbers k ≤ 2. For A = Z[ 2]the stable numbers are 1, 2, and 2 + 2.Definition 6. A reflection Re is called stable if the number (e, e) is stable.Let L be a hyperbolic lattice over a ring of integers A. We denote by S (L) the subgroup ofO (L) generated by all stable reflections.′Definition 7. A hyperbolic lattice L is said to be stably reflective if the index [O(L) : S (L)]is finite.Remark 2. In the articles [8], [44], and [10], stably reflective Z-lattices are called (1,2)reflective, since for A = Z only the numbers 1 and 2 are stable.Definition 8. A hyperbolic Z-lattice L is called 2-reflective if the group O′ (L) is up tofinite index generated by 2-reflections.Remark 3.

All 2-reflective hyperbolic Z-lattices are already classified: for rank , 4 thiswas done by V. V. Nikulin in 1979, 1981 and 1984, see [25, 27, 29], and for the rank 4 thiswas done by E. B. Vinberg in 1981–2007 (see [42]). Presumably, stably reflective latticesshould form wider class of reflective lattices then 2-reflective.The main task in this chapter is a classification of stably reflective hyperbolic Z-latticesof rank 4.

The author hopes that the method of the outermost edge (which is a modificationof the method of narrow parts of polyhedra, applied by V. V. Nikulin) will be applicable forclassifying all reflective anisotropic hyperbolic lattices of rank 4.Let P be an acute-angled compact polytope in H3 and let E be some edge of it. We denoteby F1 and F2 the faces of the polytope P, containing the edge E. Let u3 and u4 be the unitexternal normals to the faces F3 and F4 containing the vertices of the edge E, but not theedge itself.Definition 9.

Характеристики

Список файлов диссертации