preliminaries
examples
characterizations
The combinatorics of
CAT(0) cubical complexes
Federico Ardila
San Francisco State University
Universidad de Los Andes, Bogotá, Colombia.
AMS/SMM Joint Meeting
Berkeley, CA, USA, June 3, 2010
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Outline
1. Preliminaries: CAT(0) spaces and cubical complexes.
2. Some sources of CAT(0) cubical complexes.
• Phylogenetics
• Moving robots (and other reconfiguration systems)
• Geometric group theory and Coxeter groups
3. Characterizing CAT(0) cubical complexes:
• Gromov’s criterion.
• A combinatorial description
4. Applications.
• Embeddability conjecture
• All CAT(0) cube complexes are robotic
• An algorithm for geodesics
Work in progress with: Tia Baker, Megan Owen, Seth Sullivant
Pictures:Baker,Billera-Holmes-Vogtmann,Ghrist-Peterson,Scott
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Preliminaries: CAT(0) spaces
A metric space X is CAT(0) if it has non-positive curvature
everywhere, in the sense that triangles in X are “thinner" than
flat triangles. More precisely, we require:
• There is a unique geodesic path between any two points of X .
• (CAT(0) ineq.) Consider any triangle T in X and a comparison
triangle T 0 of the same sidelengths in the Euclidean plane R2 .
Consider any two points on the sides of T at distance d. Let the
corresponding points on the sides of T 0 have distance d 0 . Then
d ≤ d 0.
RECONFIGURATION
X
b
a
d
c
R2
13
a
b
!
d
c
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F IGURE 4. A positive
articulated
robot arm example [left] with fixed
Preliminaries:
cubical
complexes
endpoint. One generator [center] flips corners and has as its trace
the central
four edges.
other
generatorby
[right]
rotates
the end
A cubical
complex
is a The
space
obtained
gluing
cubes
(of of
the arm, and has trace equal to the two activated edges.
possibly
different dimensions) along their faces.
F IGURE 5. The state complex of a 5-link positive arm has one cell of
dimension
three,complex,
along withbut
several
of lower
dimension.
(Like
a simplicial
the cells
building
blocks
are cubes.)
We are interested in cubical complexes which are CAT(0).
systems is a discrete type of configuration space for these systems. Such spaces
• They are abundant in many contexts.
were considered independently by Abrams [1] and also by Swiatkowski [38].
• They have a very nice combinatorial structure to work with.
For example, if the graph is K5 (the complete graph on five vertices), N = 2, and
A = {0, 1, 2}, it is straightforward to show that each vertex has a neighborhood
with six edges incident and six 2-cells patched cyclically about the vertex. There-
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Example 1. Robot motion planning (Ghrist-Peterson 07)
A robot moves around, using certain discrete local moves.
Transition graph: vertices = positions. edges = moves.
Theorem (GP) This is the skeleton of a CAT(0) cube complex.
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Example 1. Robot motion planning
State complex. vertices = positions. edges = moves.
cubes = “physically independent" moves.
This works very generally for reconfiguration systems, when
we change vertex labels on a graph according to local moves.
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SPACE
OF PHYLOGENETIC TREES
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12 pentagons
12 vertices
of degree 5.
The shaded region shows a
Example 2. Phylogenetic
trees become
(Billera,
Holmes,
Vogtmann):
single tile of the tiling by associahedra.
h
g
j
1
k
a
3 2 4
4
3
2
1
d
4
4
e
3 2
c
1
1
e
g
l
3 2 4
4 3
2
i
1
Goal: Predict the evolutionary tree of
n current-day species/languages/....
1 3 2
4
b
3
4
2
3
1
4
2
4
4
3
1
2
1 3
2
3
2 4 3 1
h
3
b
3
i
4
4
c
1
1
l
2
1
3 2 1 4
1
2
d
2
Approach:
• Build a space of all trees Tn .
• Navigate it.
Figure 4: Cubical tiling of M 0,5 , where the arrows indicate oriented identification
f
f
a
k
j
A problem with
the above
is that
we are interested in
Thm. (BHV)
Cor.
Tn representation
has unique
geodesics.
the abstract combinatorial information contained in the tree, which does
not depend on how
the tree
is embedded intrees
the plane.exist.
The space of trees
Tn is a CAT(0) cubical complex.
Cor.
“Average"
as described in this paper is in fact a quotient of M 0,n+1 , but a direct
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Example 3. Geometric Group Theory.
Example
A right-angled Coxeter group is a group of the
form
Regular
W (G) = hv ∈ V | v 2 = 1 for v ∈ VCAT(0)
, (uv
)2 = 1 for uv ∈ Ei
Cube
Complexes
Rick Scott
a2
b2
c2
d2
Example
For the graph
= Example
=
=
=1
(ab)2 = (ac)2 = (bc)2 = (cd)2 = 1
the Dav
a
Intro to cube
complexes
c
d
Cube complexes
CAT(0)
Vertex-regular
Regular
CAT(0) Cube
Complexes
b
Key Examples
Example
Thm. (Davis) For the graph a
c
the Davis complex is
bc
Growth series
W (G) acts “very nicely" onc a dCAT(0) cubical complex X (G).
RACG’s
RAMRG’s
Rick Scott
Intro to cube
complexes
Properties of CAT(0)
cube complexes
Cube complexes
CAT(0)
Vertex-regular
abc
ac
Recurrence relations
b
Growth formula
Key Examples
Examples
RACG’s
Remarks
RAMRG’s
bc
Growth series
c
Proof sketch
cd
b
1
Definition
Properties of CAT(0)
cube complexes
cd
Definition
abc
ac
ab
Recurrence relations
Growth formula
Examples
Remarks
Proof sketch
b
ab
1
d
a
→ Use the geometry of X (G) to study the group W (G).
a
d
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gular
0) Cube
plexes
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Definition
Characterizations: Which cube complexes are CAT(0)?
A cube complex is called CAT(0) if every geodesic triangle
is at least as “thin” as a Euclidean triangle with the same
CAT(0) – Geometric Version
side lengths.
Scott
o cube
exes
1. Gromov’s characterization.
mplexes
Regular
CAT(0) Cube
Complexes
egular
xamples
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’s
Intro to cube
complexes
h series
P
n
P’
B
A
es of CAT(0)
mplexes
Cube complexes
Definition
A cube complex is called CAT(0) if every geodesic triangle
is at least as “thin” as a Euclidean triangle with the same
side lengths.
CAT(0)
Vertex-regular
Key Examples
RACG’s
nce relations
RAMRG’s
ormula
Growth series
s
P
Definition
s
Q’
Q
etch
Recurrence relations
Vertex Links
C
d(P,Q) > d(P’,Q’)
Not Thin
P’
Properties of CAT(0)
cube complexes
Growth formula
Q
Q’
Examples
Remarks
Proof sketch
d(P,Q) < d(P’,Q’)
Thin
In general, CAT(0) is a subtle condition. For cubical complexes:
Regular
CAT(0) Cube
Complexes
Theorem. (Gromov)
In a cubical
complex,
the iflinks
A cubical
complex
is CAT(0)
and
Intro to cube
complexes
of
vertices
are
simplicial
complexes.
only if it is simply connected and
the
link of every vertex is a flag simplicial complex.
Key Examples
complex
is a
∆ flag:Aifsimplicial
the 1-skeleton
of aL simplex
T is in ∆, then T is in ∆.
Growth series
flag complex if whenever the
Rick Scott
Cube complexes
CAT(0)
Vertex-regular
RACG’s
RAMRG’s
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Characterizations: Which cube complexes are CAT(0)?
2. Our characterization.
RACG
RAMRG“look like" distributive lattices.
CAT(0)
cubicalvs.
complexes
Can we make this precise? Can we describe them globally?
Regular
CAT(0) Cube
Complexes
RACG:
Rick Scott
Intro to cube
complexes
Cube complexes
CAT(0)
Vertex-regular
Key Examples
RACG’s
RAMRG:
RAMRG’s
Theorem. (AOS)
(Pointed) CAT(0) cubical complexes are in
bijection with posets with incompatible pairs.
Growth series
Definition
5
3
6
1
Properties of CAT(0)
cube complexes
Recurrence relations
Growth formula
Examples
Remarks
2
PIP: A poset P and a set of “incompatible pairs" {x, y }, with
x, y incompatible, y < z → x, z incompatible.
Proof sketch
4
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Theorem. (AOS)
(Pointed) CAT(0) cubical complexes are in
bijection with posets with incompatible pairs.
Sketch of proof.
(Imitate Birkhoff’s bijection: distributive lattices ↔ posets)
→: X has hyperplanes which split cubes in half. (Sageev)
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Theorem. (AOS)
(Pointed) CAT(0) cubical complexes are in
bijection with posets with incompatible pairs.
Bijection. →: Fix a “home" vertex v .
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12 5
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1
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v
2
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2
If i, j are hyperplanes, declare:
i < j if one needs to cross i before crossing j
i, j incompatible if it is impossible to cross them both.
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Theorem. (AOS)
(Pointed) CAT(0) cubical complexes are in
bijection with posets with incompatible pairs.
Bijection. ←: Given a poset with incompatible pairs P:
o S ⊆ P is an order ideal if : (i < j , j ∈ S) → i ∈ S
o S is compatible if it contains no incompatible pair.
Form a cube complex with
• vertices: compatible order ideals
• edges: ideals differing by one element
• cubes: “fill in" when you can.
1
5
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Note. When we have no incompatible pairs, this is the bijection:
distributive lattices ↔ posets
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Remark.
Sageev (95) obtained a different combinatorial description, and
his proof takes takes care of the technical details we run into.
Which description is more useful depends on the context.
Ours is particularly useful when you have a “special vertex", or
have no harm introducing one; e.g.:
• tree space: the origin
• geometric group theory: the identity
• robotics: a “home" position?
Now we discuss some applications.
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Application 1. Embeddability conjecture.
Conjecture. (Niblo, Sageev, Wise) Let u, v be vertices of a
CAT(0) cube complex X . If the interval [u, v ] only has cubes of
dimension ≤ d then it can be embedded in the cubing Zd .
Proof. (AOS 08) Root X at v → poset with incompat. pairs P.
o vertex u ↔ compatible order ideal Q ⊆ P.
o cube complex [u, v ] ↔ subposet with(out) incompat. pairs Q
So [u, v ] is basically the distributive lattice J(Q), and Dilworth
already showed (in 1950!) how to embed it in Zd .
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(Proof also by Brodzki, Campbell, Guentner, Niblo, Wright (08).)
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Application 2. All CAT(0) cube complexes are “robotic".
Theorem. (Ghrist-Peterson 07) Every CAT(0) cube complex X
can be realized as a state complex.
Their proof is somewhat indirect.
Alternative proof. (AOS 10)
Root X , let it correspond to the poset with incompatible pairs P.
A “cancer robot" takes over the poset P.
It can take over a new cell q if and only if:
o it already took over all elements p < q, and
o it hasn’t taken over any elements incompatible with q.
5
3
6
1
4
2
Then X is the state complex for this robot.
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Application 3. Finding geodesics in CAT(0) cube complexes.
Motivation:
Algorithm. (Owen-Provan 09) A polynomial-time algorithm to find
the geodesic between trees T1 and T2 in the space of trees Tn .
√
( 2-approx.: Amenta 07, exp.: GeoMeTree 08, GeodeMaps 09)
This allows us to
• find distances between trees, and
• “average" trees.
We use the combinatorial description of X to generalize this:
Algorithm. (AOS, 10) An algorithm to find the geodesic between
points p and q in a CAT(0) cube complex X .
(Polynomial time?)
This allows us to
• find the optimal robot motion between two positions, and
• navigate the state complex of any reconfiguration system.
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gracias
many thanks
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