The Reflective Lorentzian Lattices of Rank 3 by Daniel Allcock


035bb6bfa408dc2-261x361.jpg Author Daniel Allcock
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Number 1033 The Reflective Lorentzian Lattices of Rank 3 Daniel Allcock November 2012 • Volume 220 • Number 1033 (first of 4 numbers) • ISSN 0065-9266 Library of Congress Cataloging-in-Publication Data Allcock, Daniel, 1969-author. The reflective Lorentzian lattices of rank 3 / Daniel Allcock. p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 1033) “November 2012, volume 220, number 1033 (first of 4 numbers).” Includes bibliographical references and index. ISBN 978-0-8218-6911-6 (alk. paper) 1. Lattice theory. 2. Automorphisms. I. Title. QA171.5.A45 2012 2012025978 511.33—dc23 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on the Abstract page of each article. 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Contact the AMS for copyright status of individual articles. R , Zentralblatt MATH, Science Citation This publication is indexed in Mathematical Reviews  R , SciSearch  R , Research Alert  R , CompuMath Citation Index  R , Current Index  R /Physical, Chemical & Earth Sciences. This publication is archived in Contents  Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines  established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12 Contents Introduction vii Chapter 1. Background 1.1. Background 1.2. The Conway-Sloane genus symbol 1 1 3 Chapter 2. The Classification Theorem 2.1. The shape of a hyperbolic polygon 2.2. The shape of a 2-dimensional Weyl chamber 2.3. Corner symbols 2.4. The classification theorem 7 7 11 16 18 Chapter 3. The Reflective Lattices 3.1. How to read the table 3.2. Table of rank 3 reflective Lorentzian lattices 23 23 27 Bibliography 107 iii Abstract We classify all the symmetric integer bilinear forms of signature (2, 1) whose isometry groups are generated up to finite index by reflections. There are 8,595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin’s enumeration of the strongly square-free cases. Our technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg’s algorithm and a “method of bijections”. We also correct a minor error in Conway and Sloane’s definition of their canonical 2-adic symbol. Received by the editor August 30, 2010, and, in revised form, November 19, 2010 and February 22, 2011. Article electronically published on March 1, 2012; S 0065-9266(2012)00648-4. 2010 Mathematics Subject Classification. Primary 11H56; Secondary 20F55, 22E40. Key words and phrases. Lorentzian lattice, Weyl group, Coxeter group, Vinberg’s algorithm. Partly supported by NSF grant DMS-0600112. Affiliation at time of publication: Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, TX 78712. c 2012 American Mathematical Society v Introduction Lorentzian lattices, that is, integral symmetric bilinear forms of signature (n, 1), play a major role in K3 surface theory and the structure theory of hyperbolic KacMoody algebras. In both cases the lattices which are reflective, meaning that their isometry groups are generated by reflections up to finite index, play a special role. In the KM case they provide candidates for root lattices of KMA’s with hyperbolic Weyl groups that are large enough to be interesting. For K3’s they are important because a K3 surface has finite automorphism group if and only if its Picard group is “2-reflective”, meaning that its isometry group is generated up to finite index by reflections in classes with self-intersection −2 (see [20]). Even if the Picard group is not 2-reflective, but is still reflective, then one can often describe the automorphism group of the surface very explicitly [26][6][15]. The problem of classifying all reflective lattices of given rank is also of interest in its own right from the perspectives of Coxeter groups and arithmetic subgroups of O(n, 1). In this paper we give the classification of reflective Lorentzian lattices of rank 3, that is, of signature (2, 1). There are 8595 of them up to scale, but this large number makes the result seem more complicated than it is. For example, 176 lattices share a single Weyl group (W68 in the table). Our work is logically independent of Nikulin’s monumental paper [16], but follows it conceptually. Most of our techniques run parallel to or are refinements of Nikulin’s methods. His work had the same motivations as ours, and solves a problem both more general and more specific. He considered Lorentzian lattices that satisfy the more general condition of being “almost reflective”, meaning that the Weyl chamber is small enough that its isometry group is Zl by finite. This is a very natural class of lattices from both the K3 and KM perspectives. He also restricts attention to lattices that are strongly square-free (SSF), which in rank 3 means they have square-free determinant. The full classification is desirable from the KM perspective, because for example the root lattice of even a finite-dimensional simple Lie algebra need not be SSF. On the other hand, every lattice canonically determines a SSF lattice, so that his classification does yield all maximal-underinclusion Coxeter groups arising from lattices. The SSF condition also enabled Nikulin to use some tools from the theory of quadratic forms that we avoid. There are 1097 lattices in Nikulin’s 160 SSF lattices’ duality classes (defined in section 1.1), while our 8595 lattices fall into 1419 duality classes. In fact, as Nikulin explains, his 1097 lattices are exactly the “elementary” ones, i.e., those with discriminant group a product of cyclic groups of prime order. The reason is that each duality class containing an elementary lattice consists entirely of such lattices and contains a unique SSF lattice. We have striven to keep the table of lattices to a manageable size, while remaining complete in the sense that the full classification can be mechanically recovered vii viii INTRODUCTION from the information we give. So we have printed only 704 of them and given operations to obtain the others. Even after this reduction, the table is still large. The unabridged list of lattices is available separately as [1]. An even more complete version, including the Gram matrices, is available from the author’s web page. Anyone wanting to work seriously with the lattices will need computer-readable data, which is very easy to extract from the arxiv’s TeX source file for this paper. We have arranged for the TeX source to be simultaneously a Perl script that prints out all 8595 lattices (or just some of them) in computer-readable format. If the file is saved as file.tex then simply enter perl file.tex at the unix command line. For each reflective lattice L, we worked out the conjugacy class of its Weyl group W (L) in O(2, 1), the normalizers of W (L) in Aut L and O(2, 1), the area of its Weyl chamber, its genus as a bilinear form over Z, one construction of L for each corner of its chamber, and the relations among the lattices under p-duality and p-filling (defined in section 1.1). Considerable summary information appears in section 3.1, for example largest and smallest Weyl chambers, most and fewest lattices with given Weyl group, most edges (28), numbers of chambers with interesting properties like regularity or compactness, genera containing multiple reflective lattices, and the most-negative determinants and largest root norms. Here is a brief survey of related work. Analogues of Nikulin’s work have been done for ranks 4 and 5 by R. Scharlau [21] and C. Walhorn [29]; see also [23]. In rank ≥ 21, Esselmann [11] proved there are only 2 reflective lattices, both in rank 22—the even sublattice of Z21,1 (recognized as reflective by Borcherds [2]) and its dual lattice. Also, Nikulin [13] proved that there are only finitely many reflective lattices in each rank, so that a classification is possible. The lattices of most interest for K3 surfaces are the 2-reflective ones, and these have been completely classified by Nikulin [17][18] and Vinberg [27]. Beyond these results, the field is mostly a long list of examples found by many authors, with only glimmers of a general theory, like Borcherds’ suggested almostbijection between “good” reflection groups and “good” automorphic forms [4, §12]. To choose a few highlights, we mention Coxeter and Whitrow’s analysis [10] of Z3,1 , Vinberg’s famous paper [25] treating Zn,1 for n ≤ 17, its sequel with Kaplinskaja [28] extending this to n ≤ 19, Conway’s analysis [7] of the even unimodular Lorentzian lattice of rank 26, and Borcherds’ method of deducing reflectivity of a lattice from the existence of a suitable automorphic form [5]. Before this, Vinberg’s algorithm [25] was the only systematic way to prove a given lattice reflective. Here is an overview of the paper. We begin with background material in section 1.1 and then in section 1.2 we discuss the effects of various “moves” between lattices on the Conway-Sloane genus symbol. We also resolve a small error in their definition of the canonical 2-adic symbol. Then we begin the proof of the main theorem. First we consider the shape of a hyperbolic polygon P in section 2.1. The main result is the existence of 3, 4 or 5 consecutive edges such that the lines in the hyperbolic plane H 2 containing them are “not too far apart”. This is our version of Nikulin’s method of narrow parts of polyhedra [16, §4][13], and some of the same constants appear. Our technique is based the bisectors of angles and edges of P , while his is based on choosing a suitable point of P ’s interior and considering how the family of lines through it meet ∂P . Our method is simpler and improves certain bounds, but has no extension to higher-dimensional hyperbolic spaces, while INTRODUCTION ix his method works in arbitrary dimensions. For a more detailed comparison, please see the end of section 2.1. In section 2.2 we convert these bounds on the shape of the Weyl chamber into an enumeration of inner product matrices of 3, 4 or 5 consecutive simple roots of W (L). In section 2.3 we introduce “corner symbols”. Essentially, given L and 4,3 104 two consecutive simple roots r and s, we define a symbol like 2 ∗4 4 or 26 ∞b displaying their norms, with the super- and subscripts giving enough information to recover the inclusion (r, s) → L. So two lattices, each with a chamber and a pair of consecutive simple roots, are isometric by an isometry identifying these data if and only if the corner symbols and lattice determinants coincide. Some device of this sort is required, even for sorting the reflective lattices into isometry classes, because the genus of a lattice is not a strong enough invariant to distinguish it. In one case (see section 3.1), two lattices with the same genus even have different Weyl groups. We also use corner symbols in the proof of the main theorem (theorem 13 in section 2.4) and for tabulating our results. Besides the proof of the main theorem, section 2.4 also introduces what we call the “method of bijections”, which is a general algorithm for determining whether a finite-index subgroup of a Coxeter group is generated by reflections up to finite index. Finally, section 3.1 explains how to read the table, which occupies section 3.2. That table displays 704 lattices explicitly, together with operations to apply to obtain the full list of lattices. Computer calculations were important at every stage of this project, as explained in detail in section 2.4. We wrote our programs in C++, using the PARI software library [19], and the full classification runs in around an hour on the author’s laptop computer. The source code is available on request. Although there is no a posteriori check, like the mass formula used when enumerating unimodular lattices [8], we can offer several remarks in favor of the reliability of our results. First, in the SSF case we recovered Nikulin’s results exactly; second, many steps of the calculation provided proofs of their results (e.g., that a lattice is reflective or not); and third, our list of lattices is closed under the operations of p-duality and p-filling (see section 1.1) and “mainification” (section 1.1). This is a meaningful check because our enumeration methods have nothing to do with these relations among lattices. I am grateful to the Max Planck Institute in Bonn, where some of this work was carried out. CHAPTER 1 Background 1.1. Background For experts, the key points in this section are (i) lattices need not be integral and (ii) the definition of “reflective hull”. The notation x, . . . , z indicates the Z-span of some vectors. Lattices: A lattice means a free abelian group L equipped with a Q-valued symmetric bilinear form, called the inner product. If L has signature (n, 1) then we call it Lorentzian. If the inner product is Z-valued then L is called integral. In this case we call the gcd of L’s inner products the scale of L. We call L unscaled if it is integral and its scale is 1. The norm r 2 of a vector means its inner product with itself, and r is called primitive if L contains none of the elements r/n, n > 1. L is called even if all its vectors have even norm, and odd otherwise. If L is nondegenerate then its dual, written L∗ , means the set of vectors in L ⊗ Q having integer inner products with all of L. So L is integral just if L ⊆ L∗ . In this case L’s discriminant group is defined as Δ(L):= L∗ /L, a finite abelian group. The determinant det L means the determinant of any inner product matrix, and when L is integral we have | det L| = |Δ(L)|. Roots: A root means a primitive lattice vector r of positive norm, such that reflection in r, given by x·r x → x − 2 2 r r preserves L. Under the primitivity hypothesis, this condition on the reflection is equivalent to L · r ⊆ 12 r 2 Z. Now suppose r1 , . . . , rn are roots in a lattice L, whose span is nondegenerate. Then their reflective hull r1 , . . . , rn rh means r1 , . . . , rn rh := {v ∈ r1 , . . . , rn  ⊗ Q : v · ri ∈ 12 ri2 Z for all i}. The key property of the reflective hull is that it contains the projection of L to r1 , . . . , rn  ⊗ Q. In the special case that r1 , . . . , rn generate L up to finite index, we have r1 , . . . , rn  ⊆ L ⊆ r1 , . . . , rn rh . Then there are only finitely many possibilities for L, giving us some control over a lattice when all we know is some of its roots. As far as I know, the reflective hull was first introduced in [22], in terms of the “reduced discriminant group”. As examples and for use in the proof of theorem 10, we mention that if r and s are simple roots for an A21 , A2 , B2 or G2 root system, then r, srh /r, s is (Z/2)2 , Z/3, Z/2 or trivial, respectively. Weyl Group: We write W (L) for the Weyl group of L, meaning the subgroup of O(L) generated by reflections in roots. When L is Lorentzian, O(L) acts on hyperbolic space H n , which is defined as the image of the negative-norm vectors of L ⊗ R in P (L ⊗ R). For r a root, we refer to r ⊥ ⊆ L ⊗ R or the corresponding subset 1 2 1. BACKGROUND of H n as r’s mirror. The mirrors form a locally finite hyperplane arrangement in H n , and a Weyl chamber (or just chamber) means the closure in H n of a component of the complement of the arrangement. Often one chooses a chamber C and calls it “the” Weyl chamber. The general theory of Coxeter groups (see e.g. [24]) implies that C is a fundamental domain for W (L). Note that C may have infinitely many sides, or even be all of H n (if L has no roots). A Lorentzian lattice L is called reflective if C has finite volume. The purpose of this paper is to classify all reflective lattices of signature (2, 1). For a discrete group Γ of isometries of H n , not necessarily generated by reflections, its reflection subgroup means the group generated by its reflections, and the Weyl chamber of Γ means the Weyl chamber of this subgroup. Simple roots: The preimage of C in R2,1 − {0} has two components; choose one ˜ For every facet of C, ˜ there is a unique outward-pointing root r orand call it C. ˜ These thogonal to that facet (outward-pointing means that r⊥ separates r from C). are called the simple roots, and their pairwise inner products are ≤ 0. Changing the choice of C˜ simply negates the simple roots, so the choice of component will never matter to us. These considerations also apply to subgroups of W (L) that are generated by reflections. Vinberg’s algorithm: Given a Lorentzian lattice L, a negative-norm vector k (called the controlling vector) and a set of simple roots for the stabilizer of k in W (L), there is a unique extension of these roots to a set of simple roots for W (L). Vinberg’s algorithm [25] finds this extension, by iteratively adjoining batches of new simple roots. If L is reflective then there are only finitely many simple roots, so after some number of iterations the algorithm will have found them all. One can recognize when this happens because the polygon defined by the so-far-found simple roots has finite area. Then the algorithm terminates. On the other hand, if C has infinite area then the algorithm will still find all the simple roots, but must run forever to get them. In this case one must recognize L as non-reflective, which one does by looking for automorphisms of L preserving C. See section 2.4 for the details of the method we used. There is a variation on Vinberg’s algorithm using a null vector as controlling vector [7], but the original algorithm was sufficient for our purposes. p-duality: For p a prime, the p-dual of a nondegenerate lattice L is the lattice in L ⊗ Q characterized by  L∗ ⊗ Zq if q = p p-dual(L) ⊗ Zq = if q = p L ⊗ Zq for all primes q, where Zq is the ring of q-adic integers. Applying p-duality twice recovers L, so L and its p-dual have the same automorphism group. When L is integral one can be much more concrete: p-dual(L) is the sublattice of L∗ corresponding to the p-power part of Δ(L). In our applications L will be unscaled, so the p-power part of Δ(L) has the form ⊕ni=1 Z/pai where n = dim L and at least one of the ai ’s is 0. In this case we define the “rescaled p-dual” of L as p-dual(L) with all inner products multiplied by pa , where a = max{a1 , . . . , an }. This lattice is also unscaled, and the p-power part of its discriminant group is ⊕ni=1 Z/pa−ai . Repeating the operation recovers L. We say that unscaled lattices are in the same duality class if they are related by a chain of rescaled p-dualities. Our main interest in rescaled p-duality is that all the lattices in a duality class have the same isometry group, so we may usually 1.2. THE CONWAY-SLOANE GENUS SYMBOL 3 replace any one of them by a member of the class with smallest | det |. We will usually suppress the word “rescaled” except for emphasis, since whether we intend p-duality or rescaled p-duality will be clear from context. The effect of p-duality on the Conway-Sloane genus symbol is explained in section 1.2. p-filling: Suppose L is an integral lattice, p is a prime, and Δ(L) has some elements of order p2 . Then we define the p-filling of L as the sublattice of L∗ corresponding to pn−1 A ⊆ Δ(L), where A is the p-power part of Δ(L) and pn is the largest power of p among the orders of elements of Δ(L). One can check that p-fill(L) is integral, and obviously its determinant is smaller in absolute value. The operation is not reversible, but we do have the inclusion Aut L ⊆ Aut(p-fill(L)). As for p-duality, the effect of p-filling on the Conway-Sloane genus symbol is explained in section 1.2. Strongly square-free lattices: Given any integral lattice L, one can apply pfilling operations until no more are possible, arriving at a lattice whose discriminant group is a sum of cyclic groups of prime order. Then applying rescaled p-duality for some primes p leaves a lattice whose discriminant group has rank (i.e., cardinality of its smallest generating set) at most 12 dim L. Such a lattice is called strongly square-free (SSF). For rank 3 lattices this is equivalent to |Δ(L)| being squarefree. Nikulin’s paper [16] classified the SSF reflective Lorentzian lattices of rank 3. Watson [30][31] introduced operations leading from any lattice to a SSF one, but ours are not related to his in any simple way. Mainification: This is an operation that can be performed on an unscaled odd integral lattice L for which the 20 Jordan constituent of the 2-adic lattice L⊗Z2 has dimension 1 or 2. It means passage to L’s even sublattice, followed by multiplying inner products by 14 or 12 to obtain an unscaled lattice. See section 1.2 for the effect of this in the Conway-Sloane genus symbol. Every automorphism of L preserves L’s mainification. For rank 3 lattices, not admitting mainification is equivalent to being “main” in Nikulin’s terminology [16]. We dislike our name for the operation because it is ugly and the mainification of a lattice need not be main. 1.2. The Conway-Sloane genus symbol Genera of integer bilinear forms present some complicated phenomena, and Conway and Sloane introduced a notation for working with them that is as simple as possible [9, ch. 15]. In this section we state how to apply our operations of filling, duality and mainification in terms of their notation. We also correct a minor error in their formulation of the canonical 2-adic symbol. p-filling: This affects only the p-adic symbol. One simply takes the Jordan constituent of largest scale pa+2 and replaces a + 2 by a. (Recall that p-filling is only defined if the largest scale is ≥ p2 .) The scale of the resulting summand of p-fill(L) ⊗ Zp may be the same as that of another constituent of L. In this case one takes the direct sum of these sublattices of p-fill(L) ⊗ Z2 to obtain the pa -constituent. The direct sum is computed as follows, where we write q for pa : q ε1 n1 ⊕ q ε2 n2 = q (ε1 ε2 )(n1 +n2 ) if p is odd, while (ε ε )(n1 +n2 ) 1 2 qtε11 n1 ⊕ qtε22 n2 = qt1 +t 2 (ε ε2 )(n1 +n2 ) ε2 n2 qtε11 n1 ⊕ qII = qt1 1 4 1. BACKGROUND (ε ε2 )(n1 +n2 ) ε1 n1 ε2 n2 qII ⊕ qII = qII 1 if p = 2. Here the εi are signs, ni ≥ 0 are the ranks and ti ∈ Z/8 are the oddities. In short, signs multiply and ranks and subscripts add, subject to the special rules involving II. Here are some examples of 2-filling: 111 817 25611 ·12 91 → 111 817 6411 ·12 91 − −2 − −2 − 1−2 2 ]5 = [12 21 ]1 2 83 → 12 23 = [1 11−1 [41 81 ]2 → 11−1 [41 21 ]2 = [11 21 41 ]1 111 417 1611 → 111 417 ⊕ 411 = 111 420 12II 411 → 12II ⊕ 111 = 131 . Rescaling: Multiplying all inner products by a rational number s has the following effect on the l-adic symbols for all primes l that divide neither the numerator nor denominator of s. The signofa constituent (power of l)±n is flipped just if n is odd and the Legendre symbol sl is −1. If l = 2 we must supplement this rule by describing how to alter the subscript. A subscript II is left alone, while a subscript t ∈ Z/8 is multiplied by s. Scaling all inner products by l affects the l-adic symbol by multiplying each scale by l, leaving the rest of the l-adic symbol unchanged. Rescaled p-duality: Say q = pa is the largest scale appearing in the p-adic symbol. One obtains the rescaled p-dual by an alteration that doesn’t affect L ⊗ Zl for l = p, followed by rescaling by pa . So for l = p the effect of rescaled p-duality is the same as that of rescaling. The p-adic symbol is altered by replacing the scale pai of each constituent by pa−ai , leaving superscripts and subscripts alone. Since one usually writes the constituents in increasing order of scale, this also reverses the order of the constituents. A single example of rescaled 2-duality should illustrate the process: [11 21 ]6 811 ·32 9− ·11 7− 491 → 111 [41 81 ]6 ·32 91 ·11 7− 491 Mainification: This is an alteration of L that doesn’t affect L ⊗ Zl for odd l, followed by rescaling by 14 or 12 . So for odd l mainification has the same effect as rescaling. Recall that the mainification is only defined if L is odd with 20 constituent of rank 1 or 2. Referring to (33) and (34) of [9, ch. 15], this means that −2 −2 2 2 the constituent is one of 11±1 , 1−1 ±3 , 1±2 , 1±2 , 10 or 14 . To find the 2-adic symbol 0 of main(L), begin by altering the 2 -constituent as follows: ±1 1±1 t → 4t ±2 1±2 ±2 → 2±2 120 → 22II −2 1−2 4 → 2II . The scale of the result may be the same as that of another constituent present, in which case one takes their direct sum as above. Finally, one divides every scale by the smallest scale present (2 or 4), leaving all superscripts and subscripts alone. This scaling factor is also the one to use to work out the effect on the odd l-adic symbols. Here are some examples: 122 817 → 222 817 → 122 417 − − 1 2 − − − 1 2 − 1 1 − 126 8− 5 ·1 3 9 → 26 85 ·1 3 9 → 16 45 ·1 3 9 111 817 25611 ·12 91 → 411 817 25611 ·12 91 = [41 81 ]0 25611 ·12 91 → [11 21 ]0 6411 ·12 91 [11 2− 41 ]3 ·1−2 3− → [2− 42 ]3 ·1−2 3− → [1− 22 ]3 ·1−2 31 . 1.2. THE CONWAY-SLOANE GENUS SYMBOL 5 The canonical 2-adic symbol: There is an error in the Conway-Sloane definition of the canonical 2-adic symbol. Distinct 2-adic symbols can represent isometric 2adic lattices, and they give a calculus for reducing every 2-adic symbol to a canonical form. We have already used it in some of the examples above. First one uses “sign walking” to change some of the signs (and simultaneously some of the oddities), and then one uses “oddity fusion”. “Compartments” and “trains” refer to subchains of the 2-adic symbol that govern these operations. (The compartments are delimited by brackets [· · · ].) Conway and Sloane state that one can use sign walking to convert any 2-adic symbol into one which has plus signs everywhere except perhaps the first positive-dimensional constituent in each train. This is not true: sign walking would take the 2-dimensional lattice 111 2− 3 with total oddity 1 + 3 = 4 to one of the form 1 2 with total oddity 0. But the oddity of the first (resp. second) constituent 1− ··· ··· must be ±3 (resp. ±1) by (33) of [9, ch. 15], so the total oddity cannot be 0. So the canonical form we use is slightly more complicated: minus signs are allowed either on the first positive-dimensional constituent of a train, or on the 1 − r··· with total oddity 4 or second constituent of a compartment of the form q··· − − q··· r··· with total oddity 0. Here q and r are consecutive powers of 2. Oddity fusion remains unchanged. Note that the second form [q − r − ]0 can only occur if the q-constituent is the first positive-dimensional constituent of its train. Using [9, ch. 15, thm. 10] we checked that two 2-adic lattices are isomorphic if and only if their canonical symbols (in our sense) are identical. CHAPTER 2 The Classification Theorem 2.1. The shape of a hyperbolic polygon In this section we develop our version of Nikulin’s method of narrow parts of polyhedra [13][16]. We will prove the following theorem, asserting that any finitesided finite-area polygon P in H 2 has one of three geometric properties, and then give numerical consequences of them. These consequences have the general form: P has several consecutive edges, such that the distances between the lines containing them are less than some a priori bounds. See the remark at the end of the section for a detailed comparison of our method with Nikulin’s. Theorem (and Definition) 1. Suppose P is a finite-sided finite-area polygon in H 2 . Then one of its edges is short, meaning that the angle bisectors based at its endpoints cross. Furthermore, one of the following holds: (i) P has a short edge orthogonal to at most one of its neighbors, with its neighbors not orthogonal to each other. (ii) P has at least 5 edges and a short pair (S, T ), meaning: S is a short edge orthogonal to its neighbors, T is one of these neighbors, and the perpendicular bisector of S meets the angle bisector based at the far vertex of T . (iii) P has at least 6 edges and a close pair of short edges {S, S  }, meaning: S and S  are short edges orthogonal to their neighbors, their perpendicular bisectors cross, and some long (i.e., not short) edge is adjacent to both of them. For our purposes, the perpendicular bisector of a finite-length edge of a polygon P means the ray orthogonal to that edge, departing from the midpoint of that edge, into the interior of P . The angle bisector at a finite vertex is a ray defined in the usual way, and the angle bisector at an ideal vertex means that vertex together with the set of points equidistant from (the lines containing) the two edges meeting there. We think of it as a ray based at the ideal vertex. Lemma 2. Suppose P is a convex polygon in H 2 ∪ ∂H 2 and x1 , . . . , xn≥2 are distinct points of ∂P , numbered consecutively in the positive direction around ∂P , with subscripts read modulo n. Suppose a ray bi departs from each xi , into the interior of P . Then for some neighboring pair bi , bi+1 of these rays, either they meet or one of them meets the open arc (xi , xi+1 ) of ∂P . Remarks. When speaking of arcs of ∂P , we always mean those traversed in the positive direction. Also, the proof works for the closure of any open convex subset of H 2 ; one should interpret ∂P as the topological boundary of P in H 2 ∪ ∂H 2 . Proof. By convexity of P and the fact that bi enters the interior of P , its closure ¯bi meets ∂P at one other point yi . (We refer to ¯bi rather than bi because 7 8 2. THE CLASSIFICATION THEOREM yi might lie in ∂H 2 .) Now, yi lies in exactly one arc of ∂P of the form (xj , xj+1 ], and we define si as the number of steps from xi to xj . Formally: si is the number in {0, . . . , n − 1} such that i + si ≡ j mod n. Now choose i with si minimal. If si = 0 then bi meets (xi , xi+1 ) or contains xi+1 ∈ bi+1 and we’re done, so suppose si > 0. Now, either bi+1 meets (xi , xi+1 ), so we’re done, or it meets bi , so we’re done, or else yi+1 lies in (xi+1 , yi ] ⊆ (xi+1 , xj+1 ]. In the last case si+1 < si , so we have a contradiction of minimality.  Proof of theorem 1. To find a short edge, apply lemma 2 with x1 , . . . , xn the vertices of P and bi the angle bisector at xi . No bi can meet [xi−1 , xi ) or (xi , xi+1 ], so some neighboring pair of bisectors must cross, yielding a short edge. Suppose for the moment that some short edge is non-orthogonal to at least one of its neighbors. Then conclusion (i) applies, except if its neighbors, call them T and T  , are orthogonal to each other. This can only happen if P is a triangle. In a triangle all edges are short and at most one angle equals π/2. Therefore conclusion (i) applies with T or T  taken as the short edge. We suppose for the rest of the proof that every short edge of P is orthogonal to its neighbors, because this is the only case where anything remains to prove. Next we show that P has ≥ 5 sides; suppose otherwise. We already treated triangles, so suppose P is a quadrilateral. Because H 2 has negative curvature, P has a vertex with angle = π/2; write E and F for the edges incident there. The angle bisector based there meets one of the other two edges (perhaps at infinity), say the one next to E. Drawing a picture shows that E is short. So P has a short edge involved in an angle = π/2, a contradiction. Next we claim that if S and S  are adjacent short edges, say with S  at least as long as S, then (S, S  ) is a short pair. Write b for the perpendicular bisector of S, γ for the angle bisector at S ∩ S  , and α (resp. α ) for the angle bisector at the other vertex of S (resp. S  ). We hope the left half of figure 2.1 will help the reader; the intersection point whose existence we want to prove is marked. Now, α meets γ because S is short, and b passes though α ∩ γ because it is the perpendicular bisector of an edge whose two angles are equal. Similarly, γ meets α because S  is short. Furthermore, the intersection point α ∩ γ is at least as far away from S ∩ S  as α ∩ γ is, because S  is at least as long as S. If S  is strictly longer than S, then b enters the interior of the triangle bounded by γ, α and S  by passing through γ. Since b cannot meet S  (by S ⊥ S  ), it must meet α as it exits the triangle. So (S, S  ) is a short pair. (In the limiting case where S and S  have the same length, α, b, γ and α all meet at a single point.) Now an argument similar to the one used for quadrilaterals shows that a pentagon has a short pair. Namely, suppose S is a short edge, and write T, T  for its neighbors and U, U  for their other neighbors. The perpendicular bisector of S cannot meet T or T  (since S ⊥ T, T  ), so it meets U or U  (perhaps at infinity), say U . Then the angle bisector at T ∩ U either meets b, so (S, T ) is a short pair, or it meets S. In the latter case, T is also short, and the previous paragraph assures us that either (S, T ) or (T, S) is a short pair. This finishes the treatment of a pentagon, so we take P to have ≥ 6 edges in the rest of the proof. At this point it suffices to prove that P has a short pair, or a close pair of short edges. We apply lemma 2 again. Let x1 , . . . , xn be the midpoints of the short edges and the vertices not on short edges, in whatever order they occur around ∂P . If xi is the midpoint of an edge, let bi be the perpendicular bisector of that edge. If 2.1. THE SHAPE OF A HYPERBOLIC POLYGON 9 α S bi b γ α S  S bi+1 S T Figure 1. See the proof of theorem 1. xi is a vertex not on a short edge, let bi be the angle bisector there. Lemma 2 now gives us some i such that either bi and bi+1 meet, or one of them meets (xi , xi+1 ). We now consider the various possibilities: If both are angle bisectors then xi and xi+1 must be the endpoints of one edge, or else there would be some other xj between them. An angle bisector based at an endpoint of the edge cannot meet the interior of that edge, so bi and bi+1 must cross, so the edge must be short. But this is impossible because then bi and bi+1 would not be among b1 , . . . , bn . So this case cannot occur. If one, say bi , is the perpendicular bisector of a short edge S, and the other is an angle bisector, then we may suppose the following holds: the edge T after S is long, and bi+1 is the angle bisector based at the endpoint of T away from S. Otherwise, some other xj would come between xi and xi+1 . If bi and bi+1 meet then (S, T ) is a short pair. Also, bi cannot meet (xi , xi+1 ) since S ⊥ T . The only other possibility is that bi+1 crosses S between xi and S ∩ T . But in this case the angle bisectors based at the endpoints of T would cross, in which case T is short, a contradiction. Finally, suppose bi and bi+1 are the perpendicular bisectors of short edges S, S  . If S, S  are adjacent then one of the pairs (S, S  ), (S  , S) is a short pair; so we suppose they are not adjacent. There must be exactly one edge between them, which must be long, for otherwise some other xj would come between xi and xi+1 . If bi and bi+1 cross then S and S  forms a close pair of short edges and we are done. The only other possibility (up to exchanging bi ↔ bi+1 , S ↔ S  ) is pictured in the right half of figure 2.1. Namely, bi+1 meets S between S ∩ T and the midpoint of S. In this case, the angle bisector based at S ∩ T either meets bi+1 , so (S  , T ) is a  short pair, or it meets S  , which implies that T is short, a contradiction. In lemmas 3–5 we give numerical inequalities expressing the geometric configurations from theorem 1. We have tried to formulate the results as uniformly as possible. In each of the cases (short edge, short pair, close pair of short edges), we consider three edges R, T, R of a convex polygon in H 2 , consecutive except that there might be a short edge S (resp. S  ) between R (resp. R ) and T . We write rˆ, tˆ, rˆ for the corresponding outward-pointing unit vectors in R2,1 (and sˆ, sˆ when r · tˆ and μ = −ˆ r  · tˆ, and the hypotheses of the S, S  are present). We define μ = −ˆ

Author Daniel Allcock Isbn 9780821869116 File size 909.97KB Year 2012 Pages 108 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare The author classifies all the symmetric integer bilinear forms of signature $(2,1)$ whose isometry groups are generated up to finite index by reflections. There are 8,595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin’s enumeration of the strongly square-free cases. The author’s technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg’s algorithm and a “method of bijections”. He also corrects a minor error in Conway and Sloane’s definition of their canonical $2$-adic symbol.     Download (909.97KB) Kazhdan’s Property 3 Manifolds Which Are End 1 Movable The Congruences of a Finite Lattice: A “Proof-by-Picture” Approach The Congruences of a Finite Lattice Problems On Mapping Class Groups And Related Topics Load more posts

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