Extended Affine
Complex Euclidean
The mysterious geometry of Artin groups
Talk 3: Over the horizon
Jon McCammond
UC Santa Barbara
Caen
2 Mar 2017
Lorentzian
Extended Affine
Complex Euclidean
Lorentzian
Extended Affine
Complex Euclidean
Lorentzian
Overview
The goal in this final talk is to make a few comments about
three classes of groups.
1
Extended Affine Artin groups
2
Complex Euclidean Braid Groups
3
Lorentzian Artin Groups
The first two classes are merely relatives of Artin groups, but
the similarities and differences are interesting. The third class
is the obvious next step when trying to understand Artin groups
using the signature of their associated bilinear form.
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Coxeter groups vs. Reflection groups
Remark (Coxeter groups and Reflection groups)
For every Coxeter group we can find a set of discrete vectors in
a metric vector space that are a root system for the Coxeter
group. In particular, the quadratic form preserving reflections
determined by these vectors generate the Coxeter group and
they are precisely the conjugacy class of reflections in this
Coxeter group.
It is tempting to think that all arrangements of vectors that look
like root systems generate Coxeter groups, but this is only true
with additional contraints. I want to describe an example where
this intuition fails. It starts with the D4 root system.
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The D4 root system
Definition (The D4 root system)
The ordinary D4 root system consists of the 24 vectors
ΦD4 = {±ei ± ej ∣ i, j ∈ {1, 2, 3, 4}} where the vectors ei are the
standard unit basis vectors.
Definition (Partitioning ΦD4 )
There are 3 ways to partition {1, 2, 3, 4} into two sets of two
numbers: {{1, 2}, {3, 4}}, {{1, 3}, {2, 4}} and {{1, 4}, {2, 3}}.
We can partition ΦD4 based on these three partitions.
Let Φkl = {±ei ± ej } with i ≠ j ∈ {k , l} and define
ΦA = Φ12 ∪ Φ34 , ΦB = Φ13 ∪ Φ24 , and ΦC = Φ14 ∪ Φ23 .
Then ΦD4 = ΦA ∪ ΦB ∪ ΦC . Each subset is an orthogonal frame.
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Eisenstein integers
The “root system” I want to consider exists in V = R4,2,0 , a
6-dimensional space whose positive semidefinite form has 4
positive eigenvalues and 2 zeros.
Definition (Eisenstein)
Let R be the 2-dimensional radical of the form. Let a, b and c
be three pairwise linearly independent vectors in R subject to
the constraint that a + b + c = 0. We can identify R with C so that
the Z-span of a, b and c is identified with the Eisenstein
integers E = Z[ω] where ω is a cube root of unity.
Definition
The quotient E/2E ≅ F4 is the field with 4 elements and the
nonzero elements are represented by a, b and c.
Let Φa = a + 2E, Φb = b + 2E and Φc = c + 2E.
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̃ (1,1) root system
The D
4
Definition
Let ΦAa = {u + v ∣ u ∈ ΦA , v ∈ Φa }, ΦBb = {u + v ∣ u ∈ ΦB , v ∈ Φb }
and ΦCc = {u + v ∣ u ∈ ΦC , v ∈ Φc }. And let Φ = ΦAa ∪ ΦBb ∪ ΦCc .
This is known in the literature as the extended affine root
̃ (1,1) . Under the quotient map from V to V /R,
system of type D
4
the roots in Φ are sent to the D4 root system ΦD4
Remark
It is easy to check that the reflections in these roots (preserving
the form) preserves the root system as a set. The discrete
group they generate is called an extended affine Coxeter group
and it is not a Coxeter group in the usual sense.
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Extended affine Coxeter groups
There is a definition of an extended affine root system and
corresponding notions of extended affine Coxeter groups,
extended affine Artin groups and extended affine Lie algebras.
Remark
The study of these objects was initiated by Saito in the 80s and
90s in a series of long papers and more recently there is an
entire community dedicated to their study.
Remark
These root systems have been classified and presentations are
known for the extended affine Coxeter groups and extended
affine Artin groups.
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Questions
I would propose that we investigate the similarities and
differences between these extended affine Coxeter and Artin
groups and the ordinary Coxeter and Artin groups.
Question (Good spaces)
These forms are singular. Does Tits’ contragradient trick work
in the same way? Is there a natural Davis complex for the
extended affine Coxeter groups? Is there a natural Salvetti
complex for the extended affine Artin groups?
Question (4 key questions)
Are these extended affine Artin groups torsion-free? Do they
have a decidable word problem? Do they have a non-trivial
center? Is the natural space used to define them a K (π, 1)?
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Extended Affine
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Complex Spherical Reflection Groups
Remark (Complex spherical space)
A complex spherical space is a complex vector space with
endowed with a positive definite hermitian form.
Remark (Complex spherical reflections)
A complex spherical reflection is an operator on a complex
spherical space that fixes a codimensional one subspace and
multiplying some vector by a root of unity.
Remark (Shephard-Todd)
The finite groups generated by complex spherical reflections
were classified by Shephard and Todd in 1954. There is a
single triply-indexed infinite family together with 34 exceptions.
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R EFL(G4 ) and the 24-cell
Remark (R EFL(G4 ))
The simplest exceptional example of a complex spherical
reflection is the 24 element group of type G4 which has 4
complex reflections of order 3 and it acts on the 4-dimensional
regular polytope called the 24-cell.
Remark (The 24-cell)
John Meier and I developed a technique for directly visualizing
the 24-cell that involves breaking the 3-sphere into 6 lens.
In the picture I identify C2 with the quaternions and use
quaternion labels for the 24 points. The points used are
{±1, ±i, ±j, ±k } ∪ { ±1±i±j±k
}. The element ζ = 1+i+j+k
.
2
2
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Globe gores and the 2-sphere
These are Martin Waldseemüller’s globe gores from 1507.
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6 Lenses and the 3-sphere
ζ2
ζ
i
−1
ζ2
jζ 2
i
iζ
ζ2
kζ 4
jζ
−1
−k
ζ5
ζ2
ζ4
iζ 2
k−j
2
ζ
ζ
j
−1
1
jζ 4
−j
4
ζ5
ζ2
ζ
k
−1
1
j−k
2
−j
ζ4
iζ 5
ζ5
1
i−j
2
ζ
j
−k
kζ 5
ζ4
−1
1
i−k
2
ζ
ζ2
kζ 2
k
jζ 5
kζ
ζ
j−i
2
ζ
−i
5
ζ4
−1
1
k−i
2
−i
ζ4
ζ5
1
iζ 4
ζ5
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6 Lenses and the 3-sphere (enlarged)
ζ2
ζ
i
−1
ζ2
jζ 2
i
iζ
1
i−j
2
ζ2
kζ 4
−k
kζ 5
ζ4
−1
1
i−k
2
ζ
jζ
−1
ζ
k
1
j−k
2
−k
ζ5
ζ2
iζ 5
ζ5
−j
ζ4
ζ
j
ζ4
ζ5
ζ2
ζ
j
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Hyperplane complement and the G4 braid group
Remark (Hyperplane complement)
The G4 hyperplane complement is homotopy equivalent to the
3-sphere with 4 circles removed. From the lens picture we can
see how this deformation retracts to a 2-complex with triangular
faces on which R EFL(G4 ) acts freely.
Remark (Braid Presentation)
The presentation for B RAID(G4 ) derived from this one vertex
quotient is B RAID(G4 ) = ⟨a, b, c, d ∣ abd, bcd, cad⟩ which is the
dual Garside presentation of the 3-string braid group.
This result is not new. The deformation retraction to this
2-complex is new as is the method of visualizing the process.
Extended Affine
Complex Euclidean
Complex Euclidean Reflection Groups
Remark (Complex euclidean space)
A complex euclidean space is a complex spherical space
where the location of the origin has been forgotten.
Remark (Complex euclidean reflections)
A complex euclidean reflection is an operator on a complex
euclidean space that becomes a complex spherical reflection
with an appropriate choice of origin.
Remark (Popov)
The discrete groups generated by complex euclidean
reflections were (essentially) classified by Popov in the 1982.
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Extended Affine
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History of the classification
Remark (Popov)
The inequivalent irreducible complex euclidean reflection
groups were essentially classified by Popov in 1982. He proved
structural results, gave algorithms in various subcases and
produced a complete list.
Remark (One more)
The details of the computations themselves were not included
and in 2006 Goryunov and Man found an isolated example in
dimension 2 that was not in Popov’s list.
Someone should redo the explicit computations to verify
completeness of the current list.
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Popov’s classification
There 30 infinite families and 22 isolated examples.
Remark (Infinite families)
There are 17 infinite families with a continuous parameter and
these are closely connected to the real euclidean reflection
groups. Of these there are 7 that also have a discrete
̃n ,
parameter. These correspond to the Cartan-Killing types A
̃
̃
̃
Bn , Cn and Dn . The remaining 13 infinite families have only a
discrete parameter indicating dimension.
Remark (Isolated examples)
The 22 isolated examples mostly occur in low dimensions.
̃ 4 ) acting on C2 .
We focus on the isolated example R EFL(G
Extended Affine
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A 1-dimensional complex reflection group
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A 1-dimensional example
Remark
A 2π
3 rotation fixing some point in C is an example of a complex
euclidean reflection, and the group generated by the 2π
3
rotations about each point in the Eisenstein integers Z[ω] is an
example of a 1-dimensional complex euclidean reflection.
Remark
The braid group of this example is the fundamental group of a
triangular pillow with the corners removed. Alternatively, one
can form the Vornoi cells around the set of the fixed points and
then deform way the interiors of the hexagons. The reflection
group acts freely on this “chicken wire” graph. The quotient
graph Γ has 2 vertices, 3 edges and π1 (Γ) = F2 .
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Its Voronoi cell decomposition
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Extended Affine
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A complex euclidean reflection group
In his dissertation my student Ben Coté proved the following.
Theorem (Complement complex)
̃ 4 ) deformation retracts
The hyperplane complement of R EFL(G
onto a non-positively curved piecewise euclidean 2-complex K
in which every 2-cell is an equilateral triangle and every vertex
link is a Möbius-Kantor graph.
Definition (Möbius-Kantor graph)
The Möbius-Kantor graph as a subgraph of the 1-skeleton of a
4-cube with 8 edges removed.
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Möbius-Kantor graph
C
b
D
c
d
B
A
a
D
c
b
C
A
a
B
d
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The idea between the proof
Remark (Step 1)
The key idea is to start with the discrete set of points that arise
as intersections of the hyperplanes and to deform away the
̃ 4 ) this
interiors of the corresponding Voronoi cells. For R EFL(G
Voronoi tiling is a tiling by 24-cells and the new 3-complex is
built out of octahedra.
Remark (Step 2)
The second step is to see where the hyperplanes intersect this
3-complex and it is in the interiors of the octahedra as before.
Thus we can retract to a union of local 2-complexes that look
like the R EFL(G4 ) case.
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A complex euclidean braid group
The corresponding braid group has surprising properties.
Theorem (Isolated fixed points)
The space of regular points for the complex euclidean reflection
̃ 4 ) acting on C2 is properly contained in its
group R EFL(G
hyperplane complement because of the existence of isolated
fixed points.
Theorem (Braid group)
̃ 4 ) is a CAT(0) group and it contains
The group B RAID(G
elements of order 2.
So complex euclidean braid groups sometimes have torsion!
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Complex Euclidean
Questions
There are some special aspects of this example that make it
hard to extend to other examples. Almost all questions about
complex euclidean braid groups remain unexamined.
Question
I would conjecture that most of the complex euclidean braid
groups have 2-torsion. Is there a systematic way to define a
kind of Salvetti complex for these groups? Give (conjectural)
presentations for the corresponding braid groups.
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Extended Affine
Complex Euclidean
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Types of Coxeter Groups
Remark (Signatures)
We divide Coxeter groups and Artin groups into 3 types based
on the signs of the eigenvalues of its Coxeter matrix M. When
M has no negative eigenvalues it is spherical. When M has one
negative eigenvalue it is Lorentzian. When M has more than
one negative eigenvalue it is higher-rank. Finally, when there
are 0 eigenvalues we add the adjective weakly.
In this language euclidean groups are called weakly spherical.
We label graphs by the signature of small-type Coxeter / Artin
group it defines.
If we want to understand all Artin groups, the next step is to
understand those defined by (weakly) Lorentzian graphs.
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Weakly Lorentzian
The trouble is that we understand essentially none of them.
Remark
Recall from the survey of known results that the only small-type
Lorentzian Artin groups where we know how to decide the word
problem are the large-type ones based on complete graphs.
Question
Find one small-type Lorentzian Artin group (not defined by a
complete graph) where we know a solution to the word problem.
With the remainder of the time I want to describe a variety of
(weakly) Lorentzian graphs.
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Complex Euclidean
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Lorentzian Trees
There is a very nice characterisation of the trees which define
weakly Lorentzian Artin groups.
Theorem (Maxwell, Neumaier)
A tree defines a (weakly) Lorentzian Artin group iff there is a
vertex or a closed edge whose removal leaves a graph that
defines a (weakly) spherical Artin group.
This result (and many others that I am about to state) began as
statements about the spectrum of the adjacency matrix of a
graph. I have merely reintepreted them as statements about
the type of the corresponding Coxeter / Artin group.
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Lorentzian
Lorentzian Tripod graphs
Definition (Tp,q,r )
Let Tp,q,r be the tree with only one branch point of degree 3 and
arms that contain p, q and r vertices with the branch vertex
counting as a vertex in each arm. This tree defines a spherical
Artin group iff p1 + q1 + 1r > 1, it is weakly spherical when this sum
is = 1, and it is Lorentzian when this sum is < 1.
None of these Lorentzian Artin groups are understood, but we
do have nice descriptions of the root systems for their Coxeter
groups so their study might be tractable.
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Small Lorentzian graphs
A few years ago my student Ryan Ottman proved the following.
Theorem (Blair-Ottman)
All 996 connected graphs with fewer than 8 vertices define Artin
groups that are not higher rank.
Of these graphs 13 are spherical, 9 are weakly spherical and 4
are Lorentzian groups defined by complete graphs. The
remaining 970 small connected graphs define Lorentzian Artin
groups that we do not understand.
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Kneser Coxeter groups
Definition (Kneser graphs)
The Kneser graph of order n has vertices indexed by subsets of
{1, 2, . . . , n} of size 2 and two vertices are connected iff the
associated subsets are disjoint.
For n ≤ 4 these graphs are (weakly) spherical. For n > 4 they
are Lorentzian. The smallest interesting one (with n = 5) is the
Petersen graph. This is part of a broader family called line
graph complements. Steiner triple block graph complements
are also Lorentzian.
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If there is still time
If I still have time at this point then I will switch to the blackboard
and continue listing interesting graphs that define Lorentzian
Artin groups. For example
1
the 50 vertex Hoffman-Singleton graph (H21 )
2
the 275 vertex McLaughlin graph (H22 )
3
the 26 vertex Inc(F3 P 2 ) (H25 )
These specific graphs are closed connected with sporadic finite
simple groups and the connection is not accidental. The last
one is related to the Monster finite simple group.
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Acknowledgements
Thank you for your attention
Thank you to the organizers!
And Happy Birthday Patrick!!
Lorentzian