Author | Daniel Allcock | |

Isbn | 9780821869116 | |

File size | 909.97KB | |

Year | 2012 | |

Pages | 108 | |

Language | English | |

File format | ||

Category | mathematics |

Number 1033
The Reflective
Lorentzian Lattices of Rank 3
Daniel Allcock
November 2012 • Volume 220 • Number 1033 (first of 4 numbers)
• ISSN 0065-9266
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Allcock, Daniel, 1969-author.
The reﬂective Lorentzian lattices of rank 3 / Daniel Allcock.
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“November 2012, volume 220, number 1033 (ﬁrst of 4 numbers).”
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1. Lattice theory. 2. Automorphisms. I. Title.
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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) Kazhdans 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