The combinatorics of CAT(0) cubical complexesmath.sfsu.edu/federico/Talks/ams.smm.pdf · 2.Some...

preliminaries examples characterizations applications

The combinatorics ofCAT(0) cubical complexes

Federico Ardila

San Francisco State UniversityUniversidad de Los Andes, Bogotá, Colombia.

AMS/SMM Joint MeetingBerkeley, CA, USA, June 3, 2010


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 SullivantPictures:Baker,Billera-Holmes-Vogtmann,Ghrist-Peterson,Scott


Preliminaries: CAT(0) spaces

A metric space X is CAT(0) if it has non-positive curvatureeverywhere, in the sense that triangles in X are “thinner" thanflat 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 comparisontriangle T ′ of the same sidelengths in the Euclidean plane R2.Consider any two points on the sides of T at distance d . Let thecorresponding points on the sides of T ′ have distance d ′. Then

d ≤ d ′.RECONFIGURATION 13

a b

c

d

a b

c

d!

X R2

FIGURE 9. Comparison triangles measure curvature bounds.

4.2. The link condition. There is a well-known combinatorial approach to deter-mining when a cubical complex is nonpositively curved due to Gromov.

Definition 4.3. Let X denote a cell complex and let v denote a vertex of X . The linkof v, !k[v], is defined to be the abstract simplicial complex whose k-dimensionalsimplices are the (k + 1)-dimensional cells incident to v with the natural boundaryrelationships.

Certain global topological features of a metric cubical complex are completely de-termined by the local structure of the vertex links: a theorem of Gromov [26] assertsthat a finite dimensional Euclidean cubical complex is NPC if and only if the linkof every vertex is a flag complex without digons. Recall: a digon is a pair of ver-tices connected by two edges, and a flag complex is a simplicial complex whichis maximal among all simplicial complexes with the same 1-dimensional skeleton.Gromov’s theorem permits us an elementary proof of the following general result.

Theorem 4.4. The state complex of any locally finite reconfigurable system is NPC.

PROOF: Gromov’s theorem is stated for finite dimensional Euclidean cubical com-plexes with unit length cubes. It holds, however, for non-unit length cubes whenthere are a finite number of isometry classes of cubes (the finite shapes condition) [6].Locally finite reconfigurable systems possess locally finite and finite dimensionalstate complexes, which automatically satisfy the finite shapes condition (locally).

Let u denote a vertex of S. Consider the link !k[u]. The 0-cells of the !k[u] corre-spond to all edges in S(1) incident to u; that is, actions of generators based at u. Ak-cell of !k[u] is thus a commuting set of k + 1 of these generators based at u.

We argue first that there are no digons in !k[u] for any u " S. Assume that "1 and "2

are admissible generators for the state u, and that these two generators correspondto the vertices of a digon in !k[u]. Each edge of the digon in !k[u] corresponds toa distinct 2-cell in S having a corner at u and edges at u corresponding to "1 and"2. By Definition 2.7, each such 2-cell is the equivalence class [u; ("1, "2)]: the two2-cells are therefore equivalent and not distinct.

To complete the proof, we must show that the link is a flag complex. The interpre-tation of the flag condition for a state complex is as follows: if at u " S, one hasa set of k generators "!i , of which each pair of generators commutes, then the full


Preliminaries: cubical complexes

A cubical complex is a space obtained by gluing cubes (ofpossibly different dimensions) along their faces.

8 R. GHRIST & V. PETERSON

FIGURE 4. A positive articulated robot arm example [left] with fixedendpoint. One generator [center] flips corners and has as its tracethe central four edges. The other generator [right] rotates the end ofthe arm, and has trace equal to the two activated edges.

FIGURE 5. The state complex of a 5-link positive arm has one cell ofdimension three, along with several cells of lower dimension.

systems is a discrete type of configuration space for these systems. Such spaceswere considered independently by Abrams [1] and also by Swiatkowski [38].

For example, if the graph is K5 (the complete graph on five vertices), N = 2, andA = {0, 1, 2}, it is straightforward to show that each vertex has a neighborhoodwith six edges incident and six 2-cells patched cyclically about the vertex. There-fore, S is a closed surface. One can (as in [2]) count that there are 20 vertices, 60edges, and 30 faces in the state complex. The Euler characteristic of this surface istherefore !10. This surface can be given an orientation; thus, the state complex hasgenus six.

Example 3.4 (digital microfluidics). An even better physical instantiation of the pre-vious system arises in digital microfluidics [17, 18]. In this setting, small (e.g., 1mmdiameter) droplets of fluid can be quickly and accurately manipulated on a platecovering a network of current-controlled wires by an electrowetting process thatexploits surface tension effects to propel a droplet. Applying a current drives thedroplet a discrete distance along the wire. In this setting, one desires a “laboratoryon a chip” in which droplets of various chemicals can be positioned, mixed, andthen directed to the appropriate outputs.

Representing system states as marked vertices on a graph is appropriate given thediscrete nature of the motion by electrophoresis on a graph of wires. This adds a

(Like a simplicial complex, but the building blocks are cubes.)

We are interested in cubical complexes which are CAT(0).• They are abundant in many contexts.• They have a very nice combinatorial structure to work with.


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.


Example 1. Robot motion planning

State complex. vertices = positions. edges = moves.cubes = “physically independent" moves.

This works very generally for reconfiguration systems, whenwe change vertex labels on a graph according to local moves.


Example 2. Phylogenetic trees (Billera, Holmes, Vogtmann):

SPACE OF PHYLOGENETIC TREES 9

12 pentagons become 12 vertices of degree 5. The shaded region shows asingle tile of the tiling by associahedra.

1 2 3 4 1 2 4 31 2 3 4

4 3 2 1

g

a

a

bb

c

c

d

d

e e

f f

13

24

4 3 1 2

4 2 1 3

14

32

3214

4 1 3 2

2431

32

41 1

32

44

13

2

1324

g h

h

i

j k

l

k

l i

j

Figure 4: Cubical tiling of M0,5, where the arrows indicate oriented identifications.

A problem with the above representation is that we are interested inthe abstract combinatorial information contained in the tree, which doesnot depend on how the tree is embedded in the plane. The space of treesas described in this paper is in fact a quotient of M0,n+1, but a directconstruction seems easier to visualize. One should be able to view thisspace as the subset of the cone of all metrics on a fixed finite set consistingof those metrics that are derived from trees. See, for example, Bocker andDress (1998) for the relation between trees and metrics.

2. CONSTRUCTION OF THE SPACE OF TREES

In this section, we describe a geometric model for tree space, in whicheach point represents a rooted semi-labeled tree with n leaves and positivebranch lengths on all interior edges. In general one moves around in thespace by varying the branch lengths of the trees, but when a branch length

Goal: Predict the evolutionary tree ofn current-day species/languages/....

Approach:• Build a space of all trees Tn.• Navigate it.

Thm. (BHV) Cor. Tn has unique geodesics.Tn is a CAT(0) cubical complex. Cor. “Average" trees exist.


Example 3. Geometric Group Theory.

A right-angled Coxeter group is a group of the form

W (G) = 〈v ∈ V | v2 = 1 for v ∈ V , (uv)2 = 1 for uv ∈ E〉RegularCAT(0) CubeComplexes

Rick Scott

Intro to cubecomplexesCube complexes

CAT(0)

Vertex-regular

Key ExamplesRACG’s

RAMRG’s

Growth seriesDefinition

Properties of CAT(0)cube complexes

Recurrence relations

Growth formula

Examples

Remarks

Proof sketch

Example

ExampleFor the graph

c

b

ad

the Davis complex is

abc

1 d

c

a

b

ab

cd

ac

bc

a2 = b2 = c2 = d2 = 1(ab)2 = (ac)2 = (bc)2 = (cd)2 = 1

Thm. (Davis)W (G) acts “very nicely" on a CAT(0) cubical complex X (G).

RegularCAT(0) CubeComplexes

Rick Scott


CAT(0)

Vertex-regular


RAMRG’s




Growth formula

Examples

Remarks

Proof sketch

Example

ExampleFor the graph

c

b

ad

the Davis complex is

abc

1 d

c

a

b

ab

cd

ac

bc

→ Use the geometry of X (G) to study the group W (G).


Characterizations: Which cube complexes are CAT(0)?

1. Gromov’s characterization.


Rick Scott


CAT(0)

Vertex-regular


RAMRG’s




Growth formula

Examples

Remarks

Proof sketch

CAT(0) – Geometric Version

DefinitionA cube complex is called CAT(0) if every geodesic triangleis at least as “thin” as a Euclidean triangle with the sameside lengths.

Not Thin

P’

Q’

d(P,Q) > d(P’,Q’)

BA

C

P

Q


Rick Scott


CAT(0)

Vertex-regular


RAMRG’s




Growth formula

Examples

Remarks

Proof sketch

CAT(0) – Geometric Version

DefinitionA cube complex is called CAT(0) if every geodesic triangleis at least as “thin” as a Euclidean triangle with the sameside lengths.

QQ’

P’

d(P,Q) < d(P’,Q’)

Thin

P

In general, CAT(0) is a subtle condition. For cubical complexes:

Theorem. (Gromov)A cubical complex is CAT(0) if andonly if it is simply connected and thelink of every vertex is a flag simpli-cial complex.

∆ flag: if the 1-skeleton of a simplex T is in ∆, then T is in ∆.


Rick Scott


CAT(0)

Vertex-regular


RAMRG’s




Growth formula

Examples

Remarks

Proof sketch

Vertex Links

In a cubical complex, the linksof vertices are simplicial complexes.

A simplicial complex L is aflag complex if whenever the1-skeleton of a simplex occursin L, so does the entire simplex.


Characterizations: Which cube complexes are CAT(0)?

2. Our characterization.

CAT(0) cubical complexes “look like" distributive lattices.Can we make this precise? Can we describe them globally?


Rick Scott


CAT(0)

Vertex-regular


RAMRG’s




Growth formula

Examples

Remarks

Proof sketch

RACG vs. RAMRG

RACG:

RAMRG:Theorem. (AOS)(Pointed) CAT(0) cubical complexes are inbijection with posets with incompatible pairs.

1

2

4

65

3

PIP: A poset P and a set of “incompatible pairs" {x , y}, withx , y incompatible, y < z → x , z incompatible.


Theorem. (AOS)(Pointed) CAT(0) cubical complexes are inbijection with posets with incompatible pairs.

Sketch of proof.(Imitate Birkhoff’s bijection: distributive lattices↔ posets)

→: X has hyperplanes which split cubes in half. (Sageev)



Bijection.→: Fix a “home" vertex v .

v

1

2 4

6

3

5 12345

12

123

124

234 1246

24

1234

2

2

12

1

2

4

65

3

If i , j are hyperplanes, declare:

i < j if one needs to cross i before crossing ji , j incompatible if it is impossible to cross them both.



Bijection.←: Given a poset with incompatible pairs P:

o S ⊆ P is an order ideal if : (i < j , j ∈ S) → i ∈ So S is compatible if it contains no incompatible pair.

1

2

4

65

3v

1

2 4

6

3

5 12345

12

123

124

234 1246

24

1234

2

2

12

Form a cube complex with• vertices: compatible order ideals• edges: ideals differing by one element• cubes: “fill in" when you can.

Note. When we have no incompatible pairs, this is the bijection:distributive lattices↔ posets


Remark.

Sageev (95) obtained a different combinatorial description, andhis 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", orhave no harm introducing one; e.g.:

• tree space: the origin• geometric group theory: the identity• robotics: a “home" position?

Now we discuss some applications.


Application 1. Embeddability conjecture.

Conjecture. (Niblo, Sageev, Wise) Let u, v be vertices of aCAT(0) cube complex X . If the interval [u, v ] only has cubes ofdimension ≤ 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 Dilworthalready showed (in 1950!) how to embed it in Zd .

v

1

2 4

6

3

5 12345

12

123

124

234 1246

24

1234

2

2

12

1

2

4

65

3

(Proof also by Brodzki, Campbell, Guentner, Niblo, Wright (08).)


Application 2. All CAT(0) cube complexes are “robotic".

Theorem. (Ghrist-Peterson 07) Every CAT(0) cube complex Xcan 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, ando it hasn’t taken over any elements incompatible with q.

1

2

4

65

3

Then X is the state complex for this robot.


Application 3. Finding geodesics in CAT(0) cube complexes.

Motivation:Algorithm. (Owen-Provan 09) A polynomial-time algorithm to findthe 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 betweenpoints 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.


muchas gracias

many thanks

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