Today, we show that isoperimetric inequalities pass to asymptotic cones (this requires the theory of currents in metric spaces). I intend to prove the proposition about Lipschitz disks with positive area, and use it to prove non exactly polynomial Dehn function for certain nilpotent groups.

**1. Currents in metric spaces **

Started by Ambrosio and Kirchheim in 2000. Urs Lang, and indepedently Thierry de Pauw and Bob Hardt recently gave alternate versions.

The idea, going back to De Giorgi, is to replace differential forms with tuples of Lipschitz functions,

** 1.1. Definition **

Definition 1Let be a complete metric space. LetA function is a

metric -currentif

Multilinearity: is multilinear.Continuity: If converge pointwise to with uniformly bounded Lipschitz constants, then converges to .Locality: If some is constant on the support of , then .Finite mass: There exists a Borel measure , concentrated on a -compact set, such that, on ,

We denote by the space of metric -currents.

Remark 1

- Locality forces to depend on rather than on itself.
- Continuity axiom is stronger than in Euclidean geometric measure theory. For instance, the evaluation of a differential form at a -covector at some point is a current in the sense of Federer and Fleming, but it does not satisfy our continuity axiom (test on
- Finite mass implies that continuously extends to bounded functions .

Proposition 2Let . Denote by the collection of measures as in axiom (4). Define, for a Borel set ,This is a finite Borel measure, and

Moreover

Remark 2In Federer and Fleming’s theory,

Definition 3The measure , as well as its total mass , are called themassof current .

Corollary 4Mass is lower semi-continuous with respect to weak convergence, i.e. iffor all , then

** 1.2. Constructions **

\subsubsection{Boundary operator}

For , m, define

Then satisfies axioms (1) to (3). Say that is a *normal current* if has finite mass, i.e. is again a current. Notation

Remark 3This fits with the classical definition. Furthermore, by locality, .

\subsubsection{Push forward}

If is Lipschitz, , define

Then and, as positive measures,

\subsubsection{Multiplication with Borel functions}

Let be a bounded Borel function on . Define

Then . In particular, if is a Borel set, the restriction of to is .

\subsubsection{Standard example}

Let . Define

Then .

Remark 4

- Only continuity is non trivial.
- For and smooth, .
- If has bounded variation, then is a normal current, and in the notation of theory.

** 1.3. Compactness theorem **

Theorem 5Let be a compact metric space, a sequence of normal currents with

Then there is a subsequence which weakly converges to a normal current.

** 1.4. Integral currents **

Definition 6A -dimensional current is calledintegral rectifiableif there exist points and non zero integers such that on Lipschitz functions ,If , a current is called

integral rectifiableif

- is concentrated on a
countably rectifiable set, i.e. a countable disjoint union of biLipschitz images of subsets in , and vanishes on sets of vanishing -dimensional Hausdorff measure.- For every open and Lipschitz , there exists an integer valued function such that

We denote by the space of integral rectifiable -currents.

Theorem 7(Representation theorem).Let be an integral rectifiable -current. Then there exist subsets , biLipschitz maps and integer valued functions on such thatand

Definition 8Anintegral currentis a normal current which is integral rectifiable. The space of integral -currents is denoted by .

Note that . A *-cycle* is an integral -current with .

** 1.5. Slicing **

Let be a normal current and a Lipschitz function. The *slices* of with respect to are the currents

In some sense, this is the restriction of to the level set .

Theorem 9(Slicing Theorem). For almost every ,

- is a normal -current, with its mass concentrated on .
- .
- If is an integral current, so is almost every .

*Proof:* Think of coarea formula

** 1.6. Closure Theorem **

Theorem 10If is a weakly converging sequence of integral currents with uniformly bounded , then the limiting normal current is again an integral current.

This allows to solve Plateau’s problem in the class of integral currents.

**2. Isoperimetric inequalities and asymptotic cones **

Up to now, we insisted on filling cruves with disks. We need compactness, and for this we shall replace disks with currents. Since currents do not have prescribed topology, one may view the filling function for currents as a homological version of the disk filling function.

** 2.1. Homological filling function **

Definition 11For , defineA Lipschitz curve defines an integral current , again denote by , whose mass is the length of , and we define

A priori, , with for Riemannian manifolds, but only in general, since the mass of integral -currents is not exactly equal to area.

Theorem 12 (Wenger 2010)Let be a geodesic metric space, let be a geodesic thickening pf . Ifthen there exists such that for every asymptotic cone of , and for all ,

Remark 5Since every integral -current with is a countable sum of curves, quadratic filling for curves implies a quadrating filling inequality for all integral cycles.

Corollary 13Let be a finitely presented group with quadratic Dehn function. Then every asymptotic cone has quadratic filling for -cycles.

Panos Papasoglu shows that quadratic Dehn function implies that is simply connected. Does quadratic Dehn function imply that is quadratic ?

**3. Completion of proofs **

** 3.1. Proof of the proposition about cones of non hyperbolic spaces **

Proposition 14 (Wenger 2008)Let be a geodesic metric space, a geodesic thickening with quadratic filling. If is not hyperbolic, there exists an asymptotic cone of , a compact set and a Lipschitz map such that has positive -dimensional Hausdorff measure.

*Proof:* of Proposition from Theorem 12.

Let be an asymptotic cone which is not a tree. Then there is a closed Lipschitz curve in such that the corresponding current is not identically zero. Take for a constant speed parametrization of a geodesic triangle such that is not included in . Let denote distance to and the maximum of and . For small enough,

Theorem 12 gives an integral current filling . Then . According to the Representation Theorem 7, , , so there is such that .

** 3.2. Preparation for the proof of Theorem \ref **

}

In order to apply the Compactness Theorem 5, we must arrange a sequence of fillings in to sit in a fixed compact metric space. For this, thanks to Gromov’s compactness criterion for metric spaces (in the Gromov-Hausdorff distance), it suffices to control the size of nets on fillings. For this, one constructs fillings which leave a definite fraction of their mass in balls.

Proposition 15 (Ambrosio, Kirchheim)Let be a metric space. Assume that has a quadratic filling inequality for . Let be a cycle and . There is with such that

- .
- For each and all ,

*Proof:* Assume first there exists a mass minimizing with . Let and , the distance function to . For almost every ,

By minimality,

Integrate this differential inequality to get .

In general, consider the set of integral -currents filling with mass . For every , there exists in an such that for all ,

This is a very general fact which follows from completeness (and not compactness) of (known as Bishop-Phelps, or Ekeland variational principle).

For every competitor ,

Thus, using quadratic filling inequality,

and the proof ends as before.

** 3.3. Sketch of the proof of Theorem \ref **

}

We assume that has quadratic filling, and show that asymptotic cones do as well. **Step 1**. Show that there exists a geodesic thickening of which has for all . For this, cover with balls which sufficiently overlap, then replace them by their injective hulls.

**Step 2**. Let be a Lipschitz loop in an asymptotic cone . Pick a partition of . Show that there exists a Lipschitz loop , with , with shorter lengths between successive ‘s, and an integral -current filling , with . This suffices, since holes between and can be inductively filled, achieving a convergent series of currents whose sum fills .

To construct , let , view as a sequence . Complete into a geodesic polygon . Use Proposition 15. Since are uniformly Lipschitz, they can be filled with integral -currents in such a way that

- .
- .
- for all , .

The sequence of metric spaces is uniformly compact, since we have upper bounds for the size of -nets on it for all . Gromov’s compactness theorem implies that all these spaces simultaneously embed isometrically into a fixed compact metric space. Apply the compactness theorem 5 in that space to extract a subsequence such that

- converge in Gromov-Hausdorff sense to a compact metric space .
- The curves converge to a curve in .
- The currents converge to an integral -current in .
- The (inverse) isometric embeddings converge to an isometric embedding .

Then in , in and

**4. Carnot groups with non exactly polynomial Dehn function **

** 4.1. Superquadratic Dehn function **

Theorem 16 (Wenger 2010)Let be a -step nilpotent Lie group with Lie algebra graded as . Assume that there exists which is not a bracket for any vectors , . Consider the Lie group with Lie algebra . Endow with a left-invariant Riemannian metric. Then

Remark 6If , then there exists satisfying the assumption in Theorem 16.

** 4.2. Application to central products **

Let be a -step nilpotent Lie group with Lie algebra graded as . Given , let denote the Carnot group with Lie algebra

with Lie bracket . This will be called the *-fold central product* of .

Example 1If is the first Heisenberg group, is the -th Heisenberg group.

Corollary 17Let be a -step Carnot group with Lie algebra . Suppose and there exists which cannot be writtenLet be the Lie group with Lie algebra and the -fold central product of . Then

*Proof:* The central product can be viewed as where . So Theorem 16 applies provided is not a bracket in , i.e. is not a sum of brackets in .

Example 2Start with the free -step nilpotent Lie algebra with . Set . Then is not a sum of bracket of two vectors if .

Indeed, as an alternating bilinear form, its rank is , whereas a sum of brackets has rank at most .

Theorem 18 (Olshanskii, Sapir, Young)Let be a -step Carnot group containing a lattice. Then its -fold central product, , hasIf furthermore is free -step nilpotent, then

Corollary 19There exists a -step Carnot group such that

but is not .

** 4.3. Preparation for the proof of Theorem \ref **

}

A *Carnot group* has Lie algebra graded as . The map defined by

is an automorphism. It integrates into a group automorphism.

The Carnot-Carathéodory metric is the left-invariant geodesic metric obtained by minimizing length of curves tangent to the distribution of left-translates of . It is homogeneous of degree under .

It follows that every asymptotic cone of is biLipschitz to . In fact (Pansu 1983), the asymptotic cone is unique.

** 4.4. Proof of the lower bound on Dehn function **

Let be a -step Carnot group, let be the group with Lie algebra . We show that the second homology of the asymptotic cone does not vanish.

**Claim**: There are Lipschitz loops which do not bound any integral current in .

To construct , merely pick a Lipschitz horizontal curve in joining the origin to , and project it to . Assume that there exists an integral -current in such that . By the representation theorem 7,

where is Lipschitz, , .

Note that projection along defines a Lipschitz map . is an integral current in Euclidean filling . Let be a linear functional on which does not vanish on . Then defines a -form on as follows:

Then

where is the canonical basis of .

Let us show that for all and almost everywhere on . At almost every point of , all Lipschitz maps are differentiable. The differential is a group homomorphism . The image of the corresponding Lie algebra homomorphism is an abelian subspace in . Its image by is an subspace of on which the -bracket vanishes, i.e. the -bracket is colinear to . Since we assumed that is not a bracket of vectors from , the -bracket vanishes on the image of . So .

The contradiction proves the claim.