The Relative Topological Complexity of Pairs of Right-Angled Artin Groups

Transcription

1 The Relative Topological Complexity of Pairs of Right-Angled Artin Groups Robert Short Lehigh University February 28, 2018 Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

2 Intuition Motion Planning Algorithm Figure 1: Let X be a topological space. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

3 Intuition Motion Planning Algorithm Figure 2: A motion planning algorithm takes a pair of points in X as its input... Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

4 Intuition Motion Planning Algorithm Figure 3:... and outputs a path between the two points. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

5 Intuition continuous Motion Planning Algorithm Figure 4: For a cmpa, if our starting and ending points vary slightly... Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

6 Intuition continuous Motion Planning Algorithm Figure 5:... then the paths selected only vary slightly as well. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

7 Topological Complexity There is a fibration π : PX X X where π(σ) = (σ(0), σ(1)). Definition A continuous motion planning algorithm of size n on X X is an partition {E i } n i=1 of X X where each E i is a Euclidean Neighborhood Retract, and there exists continuous sections of π, s i : E i PX, for each E i. Definition The topological complexity of X, denoted TC(X ), is the smallest n for which there is a cmpa of size n on X X. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

8 Lower Bound for TC(X ) Proposition (Farber, 2003) If there are elements v 1,..., v p ker[h X H X H X ] such that v 1 v p 0, then TC(X ) > p. Proposition ker[h X H X H X ] is generated as an ideal by elements of the form v = v 1 1 v where v H X. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

9 Variants of TC(X ) Equivariant Topological Complexity Symmetric Topological Complexity Efficient Topological Complexity Higher Topological Complexity Relative Topological Complexity This is what I have studied. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

10 Intuition Motion Planning from X to Y Figure 6: Let Y X, then we can consider motion planners from X to Y. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

11 Intuition Motion Planning from X to Y Figure 7: For this variant on motion planning algorithms, the input is a pair of points in X Y and the output is a path between them. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

12 Relative Topological Complexity of a Pair Let Y X. Let P X Y be the space of paths in X with endpoints in X Y. There is a fibration π : P X Y X Y where π (σ) = (σ(0), σ(1)). Definition A continuous motion planning algorithm of size n on X Y is an partition {E i } n i=1 of X Y where each E i is a Euclidean Neighborhood Retract, and there exists continuous sections of π, s i : E i P X Y, for each E i. Definition The relative topological complexity of the pair (X, Y ), denoted TC(X, Y ), is the smallest n for which there is a cmpa of size n on X Y. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

13 Lower Bound for TC(X, Y ) Proposition (S., 2017) If there are elements v 1,..., v p ker[h X H Y H Y ] such that v 1 v p 0, then TC(X, Y )> p. Proposition (S., 2017) Let ι : Y X be the inclusion map. Then, ker[h X H Y H Y ] is generated as an ideal by elements of the form v = v 1 1 ι (v) where v H X. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

14 Relation to Other Invariants Note: cat(x ) denotes the Lusternik-Schnirelmann category of X. Proposition (S., 2017) cat(x ) TC(X, Y ) TC(X ) Proposition (S., 2017) TC(X, {x 0 }) = cat(x ) for any x 0 X. TC(X, X ) = TC(X ). Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

15 Right-Angled Artin Groups Artin Groups Artin s Braid Groups Right-Angled Artin Groups RAAGs were introduced in the 1970s/80s by Baudisch. RAAGs have found applications in combinatorics, geometric group theory, and robotics. Conjecture (Ghrist, 2001) π 1 (F N (T )) is a RAAG when T is a tree. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

16 Right-Angled Artin Groups Definition A group G is a right-angled Artin group (RAAG) if it has a presentation with generators s 1,... s n and where all relations are of the form s i s j = s j s i for select index pairs i < j. Examples: The free group on n generators, F(n), is a RAAG. The free abelian group on n generators, Z(n), is a RAAG. A = a, b, c, d ab = ba, ad = da, bd = db, bc = cb is a RAAG. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

17 The Corresponding Graph For a RAAG, G, we can construct a finite simple graph Γ G where: V(Γ G ) = {s 1,..., s n }, and {s i, s j } E(Γ G ) s i s j = s j s i is a relation of G. Examples: Γ F(n) is a graph with n vertices and no edges. Γ Z(n) is the complete graph of n vertices, K n. Γ A is depicted below. b a c d Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

18 The Salvetti Complex For a RAAG, G, we can construct its Salvetti Complex, denoted S G in the following way: S 1 G = n i=1 S 1 where each circle is labeled by a generator of G. S 2 G is formed by attaching a T 2 to any pair of circles labeled by commuting generators in G. S k G is formed by attaching a T k to any k circles where any pair of labeled generators commutes in G. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

19 Salvetti Complex Examples S F(n) = n i=1 S 1. S Z(n) = T n. The cells of S A are listed below. Dimension k Set of k-cells 3 {abd} 2 {ab, ad, bd, bc} 1 {a, b, c, d} 0 {x 0 } Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

20 The Full Picture for RAAGs Group G Graph Γ G Salvetti Complex S G {generators} {vertices} {1-cells} {relations} {edges} {2-cells} {k-cliques} {k-cells} Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

21 The Topology of S G S G is a cellular subcomplex of T n when G has n generators. The cells correspond to generators of H (S G ). S G is a K(G, 1). Some Notation For any group G with subgroup H, H (G) := H (K(G, 1)) cat(g) := cat(k(g, 1)) TC(G) := TC(K(G, 1)) TC(G, H) := TC(K(G, 1), K(H, 1)) Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

22 The Topology of S G S G is a cellular subcomplex of T n when G has n generators. The cells correspond to generators of H (S G ). S G is a K(G, 1). Some Notation For any RAAG G with subgroup H an RAAG, H (G) := H (S G ) cat(g) := cat(s G ) TC(G) := TC(S G ) TC(G, H) := TC(S G, S H ) Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

23 LS-Category for RAAGs Theorem (Eilenberg-Ganea, 1957) For any group G, cat(g) = 1 + the cohomological dimension of G. Corollary If G is a RAAG, then cat(g) = 1 + max{k there is a k-clique in Γ G }. Examples: cat(f(n)) = cat ( n i=1 S 1) = 2. cat(z(n)) = cat(t n ) = n + 1. cat(a) = 4. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

24 Topological Complexity for RAAGs Theorem (Cohen-Pruidze, 2008) If G is a RAAG, then TC(G) = 1 + max{ V(K 1 ) V(K 2 ) } where the max runs over cliques K 1, K 2 Γ G. Examples: TC(F(n)) = TC ( n i=1 S 1) = 3 (if n 2). TC(Z(n)) = TC(T n ) = n + 1. TC(A) = 5. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

25 Relative Topological Complexity for RAAGs Definition Let G be a RAAG. We call H a special subgroup of G if V(Γ H ) V(Γ G ) and E(Γ H ) = E(Γ G ) V(ΓH ). In other words, we call H a special subgroup of G if Γ H is an induced subgraph of Γ G. Theorem (S., 2018) If H is a special subgroup of a RAAG G, then TC(G, H) = 1 + max{ V(K G ) V(K H ) } where the max runs over pairs of cliques {K G, K H } with K G Γ G and K H Γ H. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

26 Example 1: A with special subgroup B Let B = a, d ad = da. Then B is a special subgroup of A with Γ B depicted below. a d The generators of H (A) and H (B) are listed below. Dimension H (A) Gens H (B) Gens 3 abd 2 ab, ad, bd, bc ad 1 a, b, c, d a, d Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

27 Example 1: A with special subgroup B Let B = a, d ad = da. Then B is a special subgroup of A with Γ B depicted below. a d The generators of H (A) and H (B) are listed below. Dimension H (A) Gens H (B) Gens 3 abd 2 ab, ad, bd, bc ad 1 a, b, c, d a, d Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

28 Example 1: A with special subgroup B Let B = a, d ad = da. Then B is a special subgroup of A with Γ B depicted below. a d The generators of H (A) and H (B) are listed below. Dimension H (A) Gens H (B) Gens 3 abd 2 ab, ad, bd, bc ad 1 a, b, c, d a, d Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

29 Example 1: A with special subgroup B Sample Result TC(A, B) = 5 Proof: TC(A, B) TC(A) = 5 Consider a b c d ker[h (A) H (B) H (B)]. a b c d = (a 1 1 ι (a)) (b 1 1 ι (b)) (c 1 1 ι (c)) (d 1 1 ι ( = (a 1 1 a)(b 1)(c 1)(d 1 1 d = bc ad 0 Thus, TC(A, B) > 4 = TC(A, B) = 5. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

30 Building a cmpa for RAAGs In the next several slides, we go over the background needed to construct an optimal cmpa on RAAGs. The path we will take is as follows: T n S G (S G, S H ) S 1 Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

31 Stage 1: S 1 Theorem (Farber, 2003) TC(S 1 ) = 2 Proof: Let g H 1 (S 1 ). Then g 0, so TC(S 1 ) > 1. So, we need only construct a cmpa of size 2 on S 1 S 1. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

32 A cmpa on S 1 S 1 E S = {(x, y) S 1 S 1 y x} s 1 (x, y) = the unique shortest path x y. E C = {(x, x) S 1 S 1 } s 2 (x, x) = the counterclockwise path x x. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

33 Stage 2: T n Theorem (Farber, 2003) TC(T n ) = n + 1 Proof: Let g 1,..., g n generate H 1 (T n ). Then, g 1 g n 0, so TC(T n ) > n. So, we need only construct a cmpa of size n + 1 on T n T n. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

34 A cmpa on T n T n Think of T n T n = {(x 1,..., x n, y 1,..., y n ) (S 1 ) 2n } as (T 2 ) n = {((x 1, y 1 ), (x 2, y 2 ),..., (x n, y n )) (S 1 S 1 ) n }. Using this, we can see that T n T n can be partitioned into sets written as products of E S and E C from our cmpa on S 1 S 1. Example: E S E S E C T 3 T 3. Let F i = E E, where the union is over sets where exactly i of the components are E C and the remainder are E S. Each F i is a topologically disjoint union of these types of sets. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

35 Sample Partition: T 4 T 4 F 0 = F 1 = F 2 = F 3 = F 4 = E S E S E S E S E S E S E S E C E S E S E C E S E S E C E S E S E C E S E S E S E S E S E C E C E S E C E S E C E C E S E S E C E S E C E C E S E C E S E C E S E C E C E S E S E S E C E C E C E C E S E C E C E C E C E S E C E C E C E C E S E C E C E C E C Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

36 A cmpa on T n T n We can define sections over each F i by component-wise moving points along the path prescribed by the cmpa on S 1 S 1. Since each F i is a topologically disjoint union, this section is continuous over each F i. Example: On E S E S E C, take the shortest path in the first and second components and the counterclockwise path in the third component. So, {F 0, F 1,..., F n } forms a partition of T n T n and a cmpa of size n + 1 on T n T n. Thus, TC(T n ) = n + 1. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

37 Stage 3: G a RAAG Theorem (Cohen-Pruidze, 2008) If G is a RAAG, then TC(G) = 1 + max{ V(K 1 ) V(K 2 ) } where the max runs over cliques K 1, K 2 Γ G. Proof Idea: Let V = V(K 1 ) V(K 2 ) for cliques K 1, K 2 Γ G which exhibits the maximum size. v V v 0, so TC(G) > V. For a cmpa, consider {F i (S G S G )} n i=0. This is a cover of S G S G as desired, but some of these sets are empty! It turns out that F i (S G S G ) = i > V. This yields a cmpa of size V + 1 on S G S G. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

38 Stage 4: G a RAAG with special subgroup H Theorem (S., 2018) If G is a RAAG with special subgroup H, then TC(G, H)= 1 + max{ V(K G ) V(K H ) } where the max runs over cliques K G Γ G and K H Γ H. Proof Idea: Let V =V(K G ) V(K H ) for cliques K G Γ G and K H Γ H which exhibits the maximum size. v V v 0, so TC(G, H) > V. For a cmpa, consider {F i (S G S H )} n i=0. This is a cover of S G S H as desired, but some of these sets are empty! It turns out that F i (S G S H ) = i > V. This yields a cmpa of size V + 1 on S G S H. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

39 Thank you for your attention! b a c b d b a c a c d d Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40

40 References R. Charney, An introduction to right-angled Artin groups, Geometriae Dedicata 125 (2007) D. Cohen and G. Pruidze, Motion planning in tori, Bull. London Math. Soc. 40 (2008), S. Eilenberg and T. Ganea, On the Lusternik Schnirelmann category of abstract groups, Ann. of Math. (2) 65 (1957), M. Farber Topological Complexity of Motion Planning. Discrete and Computational Geometry 29(2003), R. Ghrist, Configuration spaces and braid groups on graphs in robotics, Knots, braids, and mapping class groups papers dedicated to Joan S. Birman (New York, 1998), AMS/IP Stud. Adv. Math., 24, Amer. Math. Soc., Providence, RI ( 2001), R. Short. Relative topological complexity of a pair. Submitted to Topology and its Applications. Preprint available at arxiv: [math.at]. Robert Short (Lehigh University) Rel TC of RAAGs CUNY Grad Center February 28, / 40