The combinatorics of CAT(0) cubical complexes

Transcription

1 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

2 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

3 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 of the same sidelengths in the Euclidean plane R 2. Consider any two points on the sides of T at distance d. Let the corresponding points on the sides of T have distance d. Then d d. X R 2 a b d c a d c b

4 Preliminaries: cubical complexes A cubical complex is a space obtained by gluing cubes (of possibly different dimensions) along their faces. (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.

5 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.

6 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.

7 preliminaries examples SPACE characterizations OF PHYLOGENETIC TREES applications9 Example 2. Phylogenetic trees (Billera, Holmes, Vogtmann): 12 pentagons become 12 vertices of degree 5. The shaded region shows a single tile of the tiling by associahedra. Goal: Predict the evolutionary tree of n current-day species/languages/... i g j k h l Approach: Build a space of all trees T n. Navigate it. Thm. (BHV) T n is a CAT(0) cubical complex. l g b c c e a f k a j e Figure 4: Cubical tiling of M 0,5, where the arrows indicate oriented identification Cor. T n has unique geodesics. A problem with the above representation is that we are interested in the abstract combinatorial information contained in the tree, which does not depend on how Cor. the tree Average" is embedded intrees the plane. exist. The space of trees as described in this paper is in fact a quotient of M 0,n+1, but a direct f d d b h i

8 Example 3. Geometric Group Theory. A right-angled Coxeter group is a group of the Example form W (G) = v V v 2 = 1 for v V, (uv) 2 Regular = 1 for uv E CAT(0) Cube a 2 = b 2 = c 2 = d 2 = 1 Complexes Rick Scott Intro to cube Example complexes Cube complexes CAT(0) (ab) 2 = (ac) 2 = (bc) 2 = (cd) 2 = 1 Vertex-regular Regular CAT(0) Cube Complexes Thm. (Davis) Key Examples RAMRG s Rick Scott For the graph the Davis complex is c a bc Growth series Intro to cube Definition W (G) acts very nicely" onc a d complexes CAT(0) cubical complex X(G). Properties of CAT(0) Cube complexes cube complexes abc ac CAT(0) Vertex-regular Key Examples RACG s RAMRG s Growth series Definition Properties of CAT(0) cube complexes Recurrence relations Growth formula Examples Remarks Proof sketch Example abc b b bc ac c cd 1 d RACG s Recurrence relations Growth formula Examples Remarks Proof sketch Example For the graph a ab b b a c d the Dav cd 1 d ab a Use the geometry of X(G) to study the group W (G).

9 Cube complexes CAT(0) Vertex-regular RACG s RAMRG s Definition Properties of CAT(0) cube complexes Recurrence relations Growth formula Examples Remarks Proof sketch ular ) Cube lexes Scott cube xes plexes gular preliminaries examples characterizations applications amples series s of CAT(0) plexes ce relations rmula tch 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) Geometric Version Characterizations: Which cube complexes are CAT(0)? 1. Gromov s characterization. A P B Q Q C d(p,q) > d(p,q ) Vertex Links Not Thin P Regular CAT(0) Cube Complexes Rick Scott Intro to cube complexes Key Examples Growth series 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. Q P Q d(p,q) < d(p,q ) Thin P Regular CAT(0) Cube Complexes Rick Scott Intro to cube complexes Cube complexes CAT(0) Vertex-regular Key Examples RACG s RAMRG s Growth series In general, CAT(0) is a subtle condition. For cubical complexes: Theorem. (Gromov) A cubical In a cubical complex complex, is CAT(0) theif links and only ifof it vertices is simplyare connected simplicial and complexes. the link of every vertex is a flag simplicial complex. flag: Aifsimplicial the 1-skeleton complex of al simplex is a T is in, then T is in. flag complex if whenever the

10 Characterizations: Which cube complexes are CAT(0)? 2. Our characterization. CAT(0) RACG cubical vs. complexes RAMRG look like" distributive lattices. Can we make this precise? Can we describe them globally? Regular CAT(0) Cube Complexes Rick Scott RACG: Intro to cube complexes Cube complexes CAT(0) Vertex-regular Key Examples RACG s RAMRG s Growth series Definition Properties of CAT(0) cube complexes Recurrence relations Growth formula Examples Remarks Proof sketch RAMRG: Theorem. (AOS) (Pointed) CAT(0) cubical complexes are in bijection with posets with incompatible pairs PIP: A poset P and a set of incompatible pairs" {x, y}, with x, y incompatible, y < z x, z incompatible. 6 4

11 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)

12 Theorem. (AOS) (Pointed) CAT(0) cubical complexes are in bijection with posets with incompatible pairs. Bijection. : Fix a home" vertex v v 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.

13 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 v Note. When we have no incompatible pairs, this is the bijection: distributive lattices posets 246

14 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.

15 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 Z d. 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 Z d v (Proof also by Brodzki, Campbell, Guentner, Niblo, Wright (08).)

16 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 Then X is the state complex for this robot. 2

17 Application 3. Finding geodesics in CAT(0) cube complexes. Motivation: Algorithm. (Owen-Provan 09) A polynomial-time algorithm to find the geodesic between trees T 1 and T 2 in the space of trees T n. ( 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.

18 muchas gracias many thanks