Braid groups and buildings

Transcription

1 Braid groups and buildings z 1 z 2 z 3 z 4 PSfrag replacements z 1 z 2 z 3 z 4 Jon McCammond (U.C. Santa Barbara) 1

2 Ten years ago... Tom Brady showed me a new Eilenberg-MacLane space for the braid groups and I suggested a nice piecewise Euclidean metric. Conjecture: The resulting PE complex is nonpositively curved and consequently braid groups are CAT(0) groups. This conjecture launched a series of papers (with a computational focus) but remains open. My goal today is to explain the space, the metric and the buildings lurking in the background. Outline I. Intervals and groups II. Orthoschemes and complexes III. Buildings and continuous braids 2

3 A secondary goal: defining continuous braids The buildings in the background are related to the continuous groups that complete the following diagram: Braid? Sym Orth They are one answer the question, symmetric groups are to braid groups as orthogonal groups are to. The embedding Sym n+1 O(n) comes from the natural action of Sym n+1 on R n by permuting coordinates in R n+1 and looking perpendicular to the fixed line. One thing to notice is that the mystery groups must have a continuum of generators. 3

4 I. Intervals and groups Def: In any metric space we say that z is between x and y when d(x, z) + d(z, y) = d(x, y). The collection of all points between x and y is the interval [x, y]. Intervals are bounded partially ordered sets with an ordering z w iff w is between z and y iff z is between x and w. 4

5 Intervals in directed graphs The natural measure of length in a directed graph Γ is the length of the shortest directed combinatorial path from x to y. We use this natural (partial asymmetric) metric to define lengths of elements (l S (g) := d(1, g)) and intervals in Cayley graphs. Note that an interval in a Cayley graph naturally corresponds to a group element g because of the homogeneous nature of the graph. Ex: If G = a a 5 and S = {a} then d(1, a 3 ) = 3 and d(a 3, 1) = 2. a is between 1 and a 3 but that a is not between a 3 and 1. 5

6 From intervals to groups and spaces Let G be a marked group generated by the set S. For each interval in its Cayley graph (or equivalently each element g in G) we construct a poset P g, a group G g, and a complex K g. P g is the interval in Cayley graph of G between v 1 and v g. This has a poset structure but it includes additional information such as edge labels. The group G g and the complex K g are defined from P g so that the fundamental group of K g is G g. a b c 6

7 The group G g The group G g is the largest marked group containing P g as part of its own Cayley graph. In other words, G g is defined by only adding those relations that are visible inside the interval P g. One presentation for G g can be obtained by focusing on the directed geodesics path from v 1 to v g. Def: Let M = M g contain the words corresponding to the directed geodesic paths from v 1 to v g in the Cay(G, S), i.e. M is the set of minimal length positive factorizations of g over S. Ex 1: If G = Sym 3, S = {a = (1, 2), b = (2, 3)} and g = (1, 3), then M = {aba, bab}. Ex 2: If G = Sym 3, S = {a = (1, 2), b = (2, 3), c = (1, 3)} and g is (1, 2, 3), then M = {ab, bc, ca}. 7

8 A presentation for G g Def: The group G g := S u = v, for all u, v M. We say G g is an interval group obtained by pulling G apart at g. It is an exercise that all relations that are visible in the interval P g are consecuences of this ones listed. Note that there are natural maps G g G and G g Z so that G g is always an infinite group and a preimage of G. Ex 1: If G = Sym 3, S = {a = (1, 2), b = (2, 3)} and g = (1, 3), then G g = a, b aba = bab = Braid 3. Ex 2: If G = Sym 3, S = {a = (1, 2), b = (2, 3), c = (1, 3)} and g is (1, 2, 3), then G g = a, b, c ab = bc = ca = Braid 3. 8

9 The complex K g The complex K = K g is a quotient of the geometric realization of the poset P. The edges in the geometric realization of P have orientations from the poset order and G-labels from the fact that P is a portion of the Cayley graph of G with respect to S. The quotient we want is the one in which simplices are identified whenever we can do so respecting edge orientations and edge labels. The result is a one-vertex complex K g with π 1 (K g ) = G g. 9

10 Pulling Apart is Functorial The procedure described is functorial in the following sense. Let g be an element in an S-group G and let h be an element in a T -group H. If φ : G H is a group homomorphism such that φ(s) T, φ(g) = h, and l S (g) = l T (h), then there is an induced group homomorphism from φ : G g H h. 10

11 Example 1 generalized aba PSfrag replacements c ab a 1 b When G is a finite Coxeter group, S is a standard Coxeter generating set, and g is the longest element in G, then G g is the corresponding Artin group of finite type. 11

12 Example 2 generalized ab PSfrag replacements c a 1 b When G is a (finite) Coxeter group, S is the set of all reflections, and g is a Coxeter element, then G g is the corresponding Artin group. Conjecturally this holds for all Artin groups. 12

13 Tom s complex Tom s complex for the braid groups is the complex K that results when the symmetric group Sym n (generated by all transpositions) is pulled apart at an n-cycle. The minimal length factorizations of an n-cycle into transpositions form a well-known poset NC n called the lattice of noncrossing partitions

14 Garside structures Cayley graph intervals are graded but need not be lattices. Garside structures are lattices but need not be graded. Although neither class contains the other, the most prominent examples are in the overlap: braid groups (Garside, Dehornoy) Artin groups of finite type (Bessis, T.Brady) finite-rank free groups (Bessis, Crisp) annular braid groups (Digne) When P g is a lattice, G g is a Garside group, the elements of G g have normal forms, the word problem is decidable, K is contractible and K is a finite-dimensional K(G g, 1). (Dehornoy, Bestvina, Charney-Meier-Whittlesey) 14

15 II. Orthoschemes and complexes Def: An orthoscheme O(v 0, v 1,..., v n ) is the convex hull of a piecewise linear path that proceeds along mutually orthogonal directions u i = v i v i 1. v 3 v 2 v 0 v 1 Coxeter was interested in orthoschemes because they arise when regular polytopes are metrically barycentrically subdivided. 15

16 Unit orthoschemes Def: A unit orthoscheme is one where the vectors u i are orthonormal. These metric simplices arise in the barycentric subdivision of the n-cube of side length 2. 16

17 Orthoscheme complexes Def: If every maximal chain from the bottom to the top of a bounded poset has the same length, then this common number is called its rank. If every interval has a rank then P is graded. Def: The order complex of a graded poset can be turned into a piecewise Euclidean complex by turning each simplex into an orthoscheme. In particular, we make the edge corresponding to x < y an edge of length k where k is the rank of the poset interval P (x, y). The result is the orthoscheme complex P. Rem: Poset products lead to orthoscheme complexes that are metric products: P Q = P Q. 17

18 Example: Boolean lattices Def: A rank n boolean lattice is poset of subsets of [n] under inclusion. abc abc ab ac bc ab ac bc a b c a b c The orthoscheme complex of a rank n boolean lattice is a subdivided n-cube. This is a consequence of poset products leading to metric products. 18

19 Example: Cube complexes Every regular cell complex has a face poset and every poset has an order complex. These operations are almost inverses of each other in that the order complex of the face poset of a regular cell complex is homeomorphic to the original complex but it has the cell structure of its barycentric subdivision. Observation: If X is a cube complex where the cubes have side length 2, then the orthoscheme complex of its face poset is isometric to the original complex but with the cell structure of its barycentric subdivision. Since we know which cube complexes are CAT(0), we can reformulate these conditions as conditions on their face posets. 19

20 Example: Linear subspace posets Def: Let L n (F) be the poset of linear subspaces of the vector space F n under inclusion. Chains in L n (F) are flags, the diagonal link of the orthoscheme complex of L n (F) is a spherical building of type A n 1 (and thus CAT(1)) and the orthoscheme metric is the one that produces the correct metric on this diagonal link. The poset L 3 (F 2 ) and its diagonal link are shown. 20

21 PE complexes and PS complexes Def: Roughly speaking a piecewise Euclidean complex K is a quotient of a disjoint union of Euclidean polytopes by isometric face identification and a piecewise spherical complex is similarly defined using spherical polytopes. Def: The link of a vertex in a Euclidean polytope is the collection of unit vectors that point into the polytope. The link of a vertex in a PE complex is the natural PS complex obtained by gluing together the vertex links in each of the individual polytopes. Faces also have links. Rem: Every PS complex is the link of a vertex in a PE complex. 21

22 Curvature conditions In more general spaces, the notion of CAT(0) and CAT(1) is defined via comparison triangles but in PE and PS complexes, those definitions are equivalent to the following. Def: A PE complex (or PS complex) is locally CAT(0) (locally CAT(1)) if the link of every cell has no short local geodesic loops. A PE complex is CAT(0) if it is locally CAT(0) and simply connected. A PS complex is CAT(1) if it is locally CAT(1) and it itself has no short local geodesic loops. 22

23 Endpoints and diagonals The vertices v 0 and v n are the endpoints of the orthoscheme O(v 0,..., v n ) and the edge connecting them is its diagonal. The link of the endpoint of the unit n-orthoscheme is a Coxeter simplex of type B n. The link of its diagonal is a Coxeter simplex of type A n 1. 23

24 Links and joins The orthogonality embedded in the definition of an orthoscheme means that the links of its faces decompose into spherical joins. Def: If K and L are spherical polytopes that are vertex links of Euclidean polytopes P and Q then the spherical join K L is defined to be the corresponding vertex link in P Q. Lem: The links of simplices in a unit orthoscheme are spherical joins of spherical polytopes of type A and B. Thm(T.Brady-M) The orthoscheme complex of a bounded graded poset is CAT(0) iff its local diagonal links have no short geodesic loops. 24

25 Spindles Def: Two elements in a bounded poset are complements if they have no nontrivial upper or lower bounds. A spindle is a zig-zag path in a poset where x i 1 and x i+1 are complements in the subinterval [0, x i ] or [x i, 1]. Here are two views of a spindle of girth

26 Spindles and local geodesics Lem: Every locally geodesic path that remains in the 1-skeleton of the diagonal link of P is described by a spindle in P. Local geodesics in a local diagonal link Local geodesics that remain in its 1-skeleton Loops that correspond to spindles The three types of loops and their relations. 26

27 Poset curvature conjecture Conjecture: The orthoscheme complex of a bounded graded poset is CAT(0) iff it has no short spindles. Thm(T.Brady-M): True for posets of rank at most 4. The proof uses earlier work with Murray Elder where we classify exact what needs to be checked in this dimension. Cor: Tom s complex for Braid 5 is nonpositively curved under the orthoscheme metric and thus Braid 5 is a CAT(0) group. [Bridson, Krammer] 27

28 Modular lattices 1 x 2 k i x 4 x 1 j i k i j x 3 0 Prop: If P is a modular lattice then P has no short spindles. Conj: If P is a modular lattice then its orthoscheme complex P is CAT(0). 28

29 Partitions and Buildings One reason for believing that Tom s complex is NPC in general (i.e. the orthoscheme complex for NC n is CAT(0)) is that for every n and for every field F we have the following inclusions: NC n Π n L n (F) The middle poset is the full partition lattice and for the second inclusion we use the blocks to indicate which coordinates must be equal. The diagonal link of L n (F) is a spherical building. Because every chain in NC n and Π n is part of a Boolean lattice subposet, their diagonal links can be viewed as unions of apartments from this building (and which apartments can be explicitly described). 29

30 III. Buildings and continuous braids Suppose we consider continuous groups as GGT objects The Lie group O(2) as a space is the union of two circles. The reflections are shown on the right, the rotations are on the left and the identity is marked. We use the reflections as our generating set and the colors are used to distinguish them. What does its Cayley graph look like? 30

31 Cayley graph of O(2) The Cayley graph of O(2) with respect to the set R of all reflections has: Vertices correspond to S 1 S 1 with the discrete topology. Edges create a complete bipartite graph: there is a unique edge (with a unique color) connecting each vertex on one circle is to each vertex on the other circle. Cayley(O(2), R) thus looks like S 1 S 1 = S 3 but with a strange metric turning it into a simplicial graph. 31

32 Presentation complex for O(2) A presentation 2-complex for O(2) has one vertex, one edge of each color, and some 2-cells. One sufficient set of relations records all products of at most four reflections that are trivial. More concretely, we add: relators that make every reflection order 2, and relations that equates the various ways that a rotation can be factored into two reflections. The orthogonal groups can be viewed as continuous Coxeter groups. What I m defining is the continuous Artin group analog. 32

33 The Paris/SNCF Metric The Cayley graph of O(2) reminds one of the Paris/SNCF metric on the plane and the longitude metric on the 2-sphere. 33

34 The Longitude Metric on S 2 This is a simplicial graph, its fundamental group is free and it has a very pretty universal cover. 34

35 Its universal cover Call this circle-branching simplicial tree T S 1. 35

36 Pulling apart O(2) Apply the pulling apart construction where G = O(2), S is the set of all reflections, and g is a non-trivial rotation. Its factorization poset P and a portion of K are shown below. P = K 36

37 Structure of pulled apart O(2) Prop: When O = O(2) is pulled apart at a rotation g, the complex K is a metric product T S 1 R. R The group structure depends on the choice of g. When the rotation is rational, G g has a non-trivial center. When it is irrational, G g is centerless. 37

38 Factorizations in O(n) Thm(T.Brady-Watt): If g is fixed-point free isometry of S n 1 then its poset P of minimal length factorizations is isomorphic to the dual linear subspace poset L n (R). Moreover, the isomorphism is defined by sending h P to Fix(h). Notice that the poset structure is independent of g. What changes is the way edges in P are labeled by elements of O(n). 38

39 Structure of pulled apart O(n) Thm (M): For every fixed-point free isometry g in O(n), the group G g obtained by pulling G = O(n) apart at g (with respect to all reflections) has a finite-dimensional Eilenberg-Maclane space whose universal cover is isometric to an Ãn 1-building cross the reals (hence CAT(0)). In addition, it has a continuous Garside structure and thus the word problem is decidable. The proof is essentially an application of traditional Garside constructions to the orthogonal groups, plus the factorization structure found by Brady and Watt. 39

40 Continuous braid groups Braid n+1 G g Sym n+1 O(n) Cor (M): If g is chosen to be the image of an (n+1)-cycle under the embedding Sym n+1 O(n), then the pulled apart group G g contains the group Braid n+1 as a subgroup. Proof: The functorial nature of the construction shows that there is a group homomorphism Braid n+1 G g. Garside normal forms are used to show that this is an injection. 40

41 Artin group program Braid Artin? Sym Coxeter Lie One can try to understand an arbitrary Artin group this way. A braid group inside a pulled apart orthogonal group is the special case of a more general construction. The embedding Sym n+1 O(n) generalizes to the Tits representation of W into a generalized orthogonal group O(V, B) preserving a symmetric bilinear form. We then look to embed A inside the pulled apart version of O(V, B). This project is ongoing and has met with some intial success. 41