Homological and homotopical filling functions

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Homological and homotopical filling functions Robert Young University of Toronto Feb. 2012

The isoperimetric problem Q: Given a curve of length l in the plane, what s the maximum area it can enclose?

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).

Generalizing filling area Q: Given a curve α : S 1 X, what s its filling area?

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:

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

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

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

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!

The geometric measure theory perspective The Plateau problem: Is every curve in X filled by some minimal surface?

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 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.

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.

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!

Two filling area functions Let X be a simply-connected simplicial complex or manifold.

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

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.

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

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).

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).

Question: Can we find nice spaces (say, spaces with a geometric action by some G) where these are different?

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.

Homotopical finiteness properties Finitely generated and finitely presented are part of a spectrum of properties:

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.

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.

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.

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.

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.)

Homological finiteness properties G is FP n if Z admits a projective resolution as a ZG-module which is finitely generated in dimensions n.

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.

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).

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.

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.

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

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

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

First version of construction Suppose X is a space (not necessarily with a group action) with large Dehn function,

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.

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!

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.

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.

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.

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.

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.

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.

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.

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.

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:

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+

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+

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)

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)

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)

Open question: Is there a finitely presented group with δ FA?

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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