The existence of metrics ofnonpositive curvature on the

Brady-Krammer complexes forfinite-type Artin groups

- Dissertation Preliminary Exam

?

A1

A2

A3

A4

A5

A6

A7

A8

A9

B2

B3

B4

B5

B6

B7

B8

B9

D4

D5

D6

D7

D8

D9

I2(m)

H3

H4F4

E6

E7

E8

Woonjung Choi

Department of Mathematics

Texas A&M University

1

OverviewMy dissertation focuses on the existence or

non-existence of metrics of a particular type

on some simplicial complexes of interest to ge-

ometric group theorists.

1) the groups of our interest,

2) the type of metrics for consideration,

3) the specific simplicial complexes to be in-

vestigated

4) previously known results,

5) the approaches to the proof of the prob-

lem.

2

Problem Statements

3

Finite-type Artin groupsCoxeter groups and Artin groups are classes

of groups defined by finite presentations of a

very special form. This general form is derived

from the presentations for all finite reflection

groups (in the case of Coxeter groups) and

from the braid groups (for Artin groups). Cox-

eter groups and Artin groups are well-behaved

enough to enable strong results, but compli-

cated enough to produce numerous counterex-

amples to long-standing conjectures. As a re-

sult, they are important testing ground for the

various general theories of nonpositive curva-

ture within geometric group theory.

4

Coxeter and Artin groups

Let Γ be a finite graph with edges labeled by

integers greater than 1, and let (a, b)n be the

first n letters of (ab)n.

Definition The Artin group AΓ is generated

by its vertices with a relation (a, b)n = (b, a)n

whenever a and b are joined by an edge labeled

n.

Definition The Coxeter group WΓ is the Artin

group AΓ modulo the relations a2 = 1 ∀a ∈

Vert(Γ).

Graph

a

b

c2

3 4

Artin presentation〈a, b, c| aba = bab, ac = ca, bcbc = cbcb〉

Coxeter presentation⟨

a, b, c|aba = bab, ac = ca, bcbc = cbcb

a2 = b2 = c2 = 1

⟩

5

Finite-type Artin groups

The finite Coxeter groups have been classified.An Artin group defined by the same labeledgraph as a finite Coxeter is called a finite-typeArtin.

An....

1 2 3 n

Bn....

1 2 3 n

Dn....

1 2 3

n − 1

n − 2n − 3

n

E81 2 3

4

5 6 7 8

E71 2 3

4

5 6 7

E61 2 3

4

5 6

F4 1 2 3 4

H4 1 2 3 4

H3 1 2 3

I2(m) 1 2

m

6

CAT(0)

Definition Let (X, d) be a metric space. A

geodesic path joining x ∈ X to y ∈ X (or more

briefly a geodesic from x to y) is a map c from a

closed interval [0, l] ⊂ R to X such that c(0) =

x, c(l) = y and d(c(t), c(t′)) =∣

∣t − t′∣

∣ for all

t, t′ ∈ [0, l]. In particular l = d (x, y).

Definition (X, d) is said to be a geodesic met-

ric space if every two points in X are joined by

a geodesic.

Definition Let G be a group acting by isome-

tries on a metric space X. The action is said

to be proper (alternatively, “G acts properly

on X”) if for each x ∈ X there exists a num-

ber r > 0 such that the set {g ∈ G|gB(x, r) ∩B(x, r) 6= ∅}. It is said to be cocompact if

there exists a compact set K ⊆ X such that

X = GK.

7

CAT(0)

Definition A geodesic metric space X is CAT(0)

if for any geodesic triangle connecting any points

x, y, z ∈ X, for any point p on the geodesic con-

necting x and z,

dX(p, z) ≤ dX ′(p′, z′)

where X ′ is the comparison triangle in E2 with

the same distances between vertices.

x′ y′

z′

p′x y

z

p

X E2

A group is CAT(0) if it acts properly cocom-

pactly by isometries on a CAT(0) metric space.

8

Brady-Krammer Complexes

(For a finite type Artin group)

(G, S)Coxeter

group

standard

generating set

(G, T)

set of

all reflections

PosetCayley graph

of (G, T) graded

by distance from 1

PGinterval [1, δ]

Kgeometric

realization

K'Brady-Krammer

Complex

δ = Coxeter element =∏

s∈S

s

9

Brady-Krammer Complexes

(For a finite type Artin group)

Example

c

1

b

aba = bab

ba ab

a

a

b

(A2, {a,b})

ab

= ca = bc

1

ba

c

ba = ac

= cb

(A2, {a,b,c})

ba

= ac = cb

1

a b c

ab = ca

= bc

Poset

ba c

ab = bc = ca

1

PA2

ba

bc

c

a

a

b

c

10

Brady-Krammer Complexes

(For a finite type Artin group)

Example

(G, S)Coxeter

group

standard

generating set

(G, T)

set of

all reflections

PosetCayley graph

of (G, T) graded

by distance from 1

PGinterval [1, δ]

Kgeometric

realization

K'Brady-Krammer

Complex

c

1

b

aba = bab

ba ab

a

a

b

(A2, {a,b})

ab

= ca = bc

1

ba

c

ba = ac

= cb

(A2, {a,b,c})

ba

= ac = cb

1

a b c

ab = ca

= bc

Poset

ba c

ab = bc = ca

1

PA2

ba

bc

c

a

a

b

c

δ = Coxeter element =∏

s∈S

s

11

Brady-Krammer Complexes

(For a finite type Artin group)

• For a finite Coxeter group (G, S) where S is a stan-dard generating set, it can be generated by using allreflections T which are the conjugates of standardgenerators.

• Create a new Cayley graph with respect to T .

• Orient the graph based on lT (length function) andget a poset where lT(x) := the least length from 1in the Cayley graph.

• Pick a coxeter element, which is a product of allstandard generators, and restrict the poset to theinterval between the coxeter element and the iden-tity. Then the interval, PG, is a poset of factoriza-tions of the coxeter element into reflections whichis self-dual and a lattice.

• Realize this poset as a simplicial complex and labelthe edges with the group element.

• Glue simplices according to their labels.

12

F4 Poset

13

Piecewise Euclidean Complexes

Definition A piecewise Euclidean (PE) complex X isa simplicial complex in which each simplex is given aEuclidean metric and the induced metrics on the inter-sections always agree.

Definition Let X be a PE complex and x be a point inX. The link of x in X is the set of unit tangent vectorsto X at x. When x lies in the interior of a cell B of X,the link of B in X is the set of unit tangent vectors tox in X which are orthogonal to B.

Theorem. A PE complex is CAT(0) iff the link of eachcell does not contain a closed geodesic loop of lengthless than 2π.

v v ’

14

CAT(0) and Artin groups

Theorem (T.Brady-J.McCammond).The finite-

type Artin groups with at most 3 generators

are CAT(0)-groups and the Artin groups A4

and B4 are CAT(0) groups.

Conjecture. The Brady-Krammer complex is

CAT(0) for all Artin groups of finite type.

15

Approach

16

CAT(0) metrics on D4 and F4

Conjecture (Counter Conjecture).The Brady-

Krammer complexes for D4 and F4 do not sup-

port PE CAT(0) metrics in which the symme-

tries of the group lead to symmetries in the

metric.

17

Proof Idea

1. Consider the group action on the univer-

sal cover of the Brady-Krammer complex.

Since the group has a subgroup which is a

product of another subgroup and a cyclic

group with finite index, by the splitting the-

orem, the complex which is the universal

cover is a product of another complex with

R. Since a product of a CAT(0) space

and R is CAT(0), we only need to check

if the cross section of the universal cover

has CAT(0) structure.

2. I will examine which Euclidean metrics on

the 3-dimensional cross-section complex have

dihedral angles which make the edge links

(which are finite graphs) large.

18

3. We hope to show that both D4 and F4 have

the system with only one solution which

assigns to each simplex a metric commonly

used in the study of Coxeter groups.

4. Since it was already shown by Jon McCam-

mond and Tom Brady that this particular

metric fails to be nonpositively curved for

these complexes, no such metric exists.

The software

The program coxeter.g is a set of GAP rou-

tines used to examine Brady-Kramer complexes.

It is initially developed to test the curvature of

the Brady-Krammer complexes using the “nat-

ural” metric and I extensively reworked the rou-

tines so that they

• find the poset

• find the 3-dimensional structure of the cross-section

• find representive vertex of the poset and edge links(up to automorphism)

• find the graphs for the edge links

• find the simple cycles in these graph

• find the linear system of inequalities which need tobe satisfied by the dihedral angles of the tetrahedra.

19

Types D4 and F4

D4 has:

• 162 simplices

• 3 types of tetrahedra in the cross section

• 4 vertex types to check

• 21 inequalities in 9 variables

• 11 simplified inequalities in 9 variables

F4 has:

• 432 simplices

• 4 types of tetrahedra in the cross section

• 7 vertex types to check

• 81 inequalities in 13 variables

• 27 simplified inequalities in 13 variables

20

Type H4

The case of H4 is hard to resolve because

the defining diagram has no symmetries, which

greatly increases the number of equations and

variables involved in the computations.

H4 has:

• 1350 simplices

• 16 types of tetrahedra in the cross section

• 10 vertex types to check

• 2986 inequalities in 96 variables

• 638 simplified inequalities in 96 variables

The F4 and D4 cases produced systems small

enough to analyze by hand. This system is not.

Therefore, the case of H4 will be a subject of

future research.

21