Homological and homotopical filling functions
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1 Homological and homotopical filling functions Robert Young University of Toronto Feb. 2012
2 The isoperimetric problem Q: Given a curve of length l in the plane, what s the maximum area it can enclose?
3 The isoperimetric problem Q: Given a curve of length l in the plane, what s the maximum area it can enclose? If γ : S 1 R 2, let F (γ) be the area of γ. Then we want to calculate: i(n) = sup α:s 1 R 2 l(α) n f (α).
4 Generalizing filling area Q: Given a curve α : S 1 X, what s its filling area?
5 The geometric group theory perspective Suppose that X is a simply connected complex and G acts geometrically (cocompactly, properly discontinuously, and by isometries) on X. Then:
6 The geometric group theory perspective Suppose that X is a simply connected complex and G acts geometrically (cocompactly, properly discontinuously, and by isometries) on X. Then: Paths in X correspond to words in G
7 The geometric group theory perspective Suppose that X is a simply connected complex and G acts geometrically (cocompactly, properly discontinuously, and by isometries) on X. Then: Paths in X correspond to words in G Loops in X correspond to words in G that represent the identity
8 The geometric group theory perspective Suppose that X is a simply connected complex and G acts geometrically (cocompactly, properly discontinuously, and by isometries) on X. Then: Paths in X correspond to words in G Loops in X correspond to words in G that represent the identity Discs in X correspond to proofs that a word represents the identity
9 The geometric group theory perspective Suppose that X is a simply connected complex and G acts geometrically (cocompactly, properly discontinuously, and by isometries) on X. Then: Paths in X correspond to words in G Loops in X correspond to words in G that represent the identity Discs in X correspond to proofs that a word represents the identity So by studying discs in X, we can get invariants related to the combinatorial group theory of G!
10 The geometric measure theory perspective The Plateau problem: Is every curve in X filled by some minimal surface?
11 The geometric measure theory perspective The Plateau problem: Is every curve in X filled by some minimal surface? Federer and Fleming constructed spaces of currents: Every curve corresponds to a 1-dimensional current, every surface corresponds to a 2-dimensional current.
12 The geometric measure theory perspective The Plateau problem: Is every curve in X filled by some minimal surface? Federer and Fleming constructed spaces of currents: Every curve corresponds to a 1-dimensional current, every surface corresponds to a 2-dimensional current. The currents form a vector space, and the map sending a current to its boundary is linear.
13 The geometric measure theory perspective The Plateau problem: Is every curve in X filled by some minimal surface? Federer and Fleming constructed spaces of currents: Every curve corresponds to a 1-dimensional current, every surface corresponds to a 2-dimensional current. The currents form a vector space, and the map sending a current to its boundary is linear. Currents satisfy nice compactness properties.
14 The geometric measure theory perspective The Plateau problem: Is every curve in X filled by some minimal surface? Federer and Fleming constructed spaces of currents: Every curve corresponds to a 1-dimensional current, every surface corresponds to a 2-dimensional current. The currents form a vector space, and the map sending a current to its boundary is linear. Currents satisfy nice compactness properties. So we can find minimal currents filling a curve by taking limits of surfaces whose area approaches the infimum!
15 Two filling area functions Let X be a simply-connected simplicial complex or manifold.
16 Two filling area functions Let X be a simply-connected simplicial complex or manifold. Let α : S 1 X be a closed curve. δ(α) = inf area β. β:d 2 X β S 1 =α
17 Two filling area functions Let X be a simply-connected simplicial complex or manifold. Let α : S 1 X be a closed curve. δ(α) = inf area β. β:d 2 X β S 1 =α Let a be a 1-cycle. FA(a) = inf β a 2-chain β=a area β.
18 Two filling area functions Let X be a simply-connected simplicial complex or manifold. Let α : S 1 X be a closed curve. δ(α) = inf area β. β:d 2 X β S 1 =α Let a be a 1-cycle. FA(a) = inf β a 2-chain β=a area β. δ X (n) = sup δ(α). α:s 1 X l(α) n
19 Two filling area functions Let X be a simply-connected simplicial complex or manifold. Let α : S 1 X be a closed curve. δ(α) = δ X (n) = inf area β. β:d 2 X β S 1 =α sup δ(α). α:s 1 X l(α) n Let a be a 1-cycle. FA(a) = FA X (n) = inf β a 2-chain β=a sup a a 1-cycle l(a) n area β. FA(a).
20 Two filling area functions Let X be a simply-connected simplicial complex or manifold. Let α : S 1 X be a closed curve. δ(α) = inf area β. β:d 2 X β S 1 =α Let a be a 1-cycle. FA(a) = inf β a 2-chain β=a area β. δ X (n) = sup δ(α). α:s 1 X l(α) n FA X (n) = sup a a 1-cycle l(a) n We call δ X the homotopical Dehn function and FA X the homological Dehn function. FA(a).
21 Question: Can we find nice spaces (say, spaces with a geometric action by some G) where these are different?
22 Question: Can we find nice spaces (say, spaces with a geometric action by some G) where these are different? Theorem (Abrams, Brady, Dani, Guralnik, Lee, Y.) Yes.
23 Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties:
24 Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties: Any discrete group acts geometrically on some discrete metric space.
25 Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties: Any discrete group acts geometrically on some discrete metric space. G acts geometrically on a connected complex G is finitely generated.
26 Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties: Any discrete group acts geometrically on some discrete metric space. G acts geometrically on a connected complex G is finitely generated. G acts geometrically on a simply-connected complex G is finitely presented.
27 Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties: Any discrete group acts geometrically on some discrete metric space. G acts geometrically on a connected complex G is finitely generated. G acts geometrically on a simply-connected complex G is finitely presented. G acts geometrically on a n 1-connected complex G is F n.
28 Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties: Any discrete group acts geometrically on some discrete metric space. G acts geometrically on a connected complex G is finitely generated. G acts geometrically on a simply-connected complex G is finitely presented. G acts geometrically on a n 1-connected complex G is F n. (Equivalently, G has a K(G, 1) with finite n-skeleton.)
29 Homological finiteness properties G is FP n if Z admits a projective resolution as a ZG-module which is finitely generated in dimensions n.
30 Homological finiteness properties G is FP n if Z admits a projective resolution as a ZG-module which is finitely generated in dimensions n. In particular, if G is F n, we can take a K(G, 1) with finite n-skeleton and consider its simplicial chain complex.
31 Homological finiteness properties G is FP n if Z admits a projective resolution as a ZG-module which is finitely generated in dimensions n. In particular, if G is F n, we can take a K(G, 1) with finite n-skeleton and consider its simplicial chain complex. Or if G acts geometrically on some homologically n-connected space (i.e., with trivial H i (X ; Z) for i n).
32 Homological finiteness properties G is FP n if Z admits a projective resolution as a ZG-module which is finitely generated in dimensions n. In particular, if G is F n, we can take a K(G, 1) with finite n-skeleton and consider its simplicial chain complex. Or if G acts geometrically on some homologically n-connected space (i.e., with trivial H i (X ; Z) for i n). So δ and FA are quantitative versions of F 2 and FP 2.
33 Theorem (Bestvina-Brady) Given a flag complex Y, there is a group K Y such that K Y acts geometrically on a space consisting of infinitely many scaled copies of Y. Indeed, this space is homotopy equivalent to an infinite wedge product of Y s.
34 Theorem (Bestvina-Brady) Given a flag complex Y, there is a group K Y such that K Y acts geometrically on a space consisting of infinitely many scaled copies of Y. Indeed, this space is homotopy equivalent to an infinite wedge product of Y s. K Y is finitely generated if and only if Y is connected
35 Theorem (Bestvina-Brady) Given a flag complex Y, there is a group K Y such that K Y acts geometrically on a space consisting of infinitely many scaled copies of Y. Indeed, this space is homotopy equivalent to an infinite wedge product of Y s. K Y is finitely generated if and only if Y is connected K Y is finitely presented if and only if Y is simply connected
36 Theorem (Bestvina-Brady) Given a flag complex Y, there is a group K Y such that K Y acts geometrically on a space consisting of infinitely many scaled copies of Y. Indeed, this space is homotopy equivalent to an infinite wedge product of Y s. K Y is finitely generated if and only if Y is connected K Y is finitely presented if and only if Y is simply connected K Y is F n if and only if Y is n 1-connected K Y is FP n if and only if Y is homologically n 1-connected
37 First version of construction Suppose X is a space (not necessarily with a group action) with large Dehn function,
38 First version of construction Suppose X is a space (not necessarily with a group action) with large Dehn function, and suppose Y is a complex such that H 1 (Y ) is trivial, π 1 (Y ) is nontrivial, π 1 (Y ) is generated by conjugates of γ.
39 First version of construction Suppose X is a space (not necessarily with a group action) with large Dehn function, and suppose Y is a complex such that H 1 (Y ) is trivial, π 1 (Y ) is nontrivial, π 1 (Y ) is generated by conjugates of γ. Then if we glue infinitely many scaled copies of Y to X, the result should have small homological Dehn function!
40 Modifying the construction for groups Let Y be a flag complex with trivial H 1 and nontrivial π 1, normally generated by a single element γ. Say γ is a path of length 4 in Y.
41 Modifying the construction for groups Let Y be a flag complex with trivial H 1 and nontrivial π 1, normally generated by a single element γ. Say γ is a path of length 4 in Y. Then, by Bestvina-Brady, the level set L Y is acted on by a subgroup K Y, and L Y is made up of copies of Y.
42 Modifying the construction for groups Let Y be a flag complex with trivial H 1 and nontrivial π 1, normally generated by a single element γ. Say γ is a path of length 4 in Y. Then, by Bestvina-Brady, the level set L Y is acted on by a subgroup K Y, and L Y is made up of copies of Y. Furthermore, there is a copy of F 2 F 2 in A Y corresponding to that square. Let E = K Y F 2 F 2.
43 Modifying the construction for groups Let Y be a flag complex with trivial H 1 and nontrivial π 1, normally generated by a single element γ. Say γ is a path of length 4 in Y. Then, by Bestvina-Brady, the level set L Y is acted on by a subgroup K Y, and L Y is made up of copies of Y. Furthermore, there is a copy of F 2 F 2 in A Y corresponding to that square. Let E = K Y F 2 F 2. Then E acts on a subset L E L Y made up of copies of γ.
44 Modifying the construction for groups, part 2 So if we can find a copy of E in some other group D, we can amalgamate D and K Y together along E.
45 Modifying the construction for groups, part 2 So if we can find a copy of E in some other group D, we can amalgamate D and K Y together along E. There are semidirect products D = F n (φ,φ) F 2 which contain copies of E and have large Dehn functions.
46 Modifying the construction for groups, part 2 So if we can find a copy of E in some other group D, we can amalgamate D and K Y together along E. There are semidirect products D = F n (φ,φ) F 2 which contain copies of E and have large Dehn functions. So we can construct an amalgam of D with several copies of K Y, glued along E. This is a group with large homotopical Dehn function, but small homological Dehn function.
47 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n.
48 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n. In fact, if δ k (n) is the k-th order homotopical Dehn function and FA k is the corresponding homological Dehn function, then:
49 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n. In fact, if δ k (n) is the k-th order homotopical Dehn function and FA k is the corresponding homological Dehn function, then: δ FA δ FA Yes!(ABDGLY) δ 2 FA 2 δ 2 FA 2 δ 3+ FA 3+ δ 3+ FA 3+
50 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n. In fact, if δ k (n) is the k-th order homotopical Dehn function and FA k is the corresponding homological Dehn function, then: δ FA δ FA Yes!(ABDGLY) δ 2 FA 2 δ 2 FA 2 Yes(Y) δ 3+ FA 3+ δ 3+ FA 3+
51 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n. In fact, if δ k (n) is the k-th order homotopical Dehn function and FA k is the corresponding homological Dehn function, then: δ FA δ FA Yes!(ABDGLY) δ 2 FA 2 δ 2 FA 2 Yes(Y) No(Gromov, White) δ 3+ FA 3+ δ 3+ FA 3+ No(Gromov, White)
52 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n. In fact, if δ k (n) is the k-th order homotopical Dehn function and FA k is the corresponding homological Dehn function, then: δ FA δ FA Yes!(ABDGLY) δ 2 FA 2 δ 2 FA 2 Yes(Y) No(Gromov, White) δ 3+ FA 3+ δ 3+ FA 3+ No(Brady-Bridson-Forester-Shankar) No(Gromov, White)
53 Theorem (ABDGLY) There is a subgroup of a CAT(0) group which has FA(n) n 5 but δ(n) n d for any d, or even δ(n) e n. In fact, if δ k (n) is the k-th order homotopical Dehn function and FA k is the corresponding homological Dehn function, then: δ FA δ FA??? Yes!(ABDGLY) δ 2 FA 2 δ 2 FA 2 Yes(Y) No(Gromov, White) δ 3+ FA 3+ δ 3+ FA 3+ No(Brady-Bridson-Forester-Shankar) No(Gromov, White)
54 Open question: Is there a finitely presented group with δ FA?
55 Open question: Is there a finitely presented group with δ FA? This would have to be a group where it s harder to fill two curves of length n/2 than to fill any curve of length n.
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