Notes of Stefan Wenger’s lecture nr 3

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,

$\displaystyle \begin{array}{rcl} f\,d\pi_1 \wedge \cdots \wedge d\pi_m \leftrightarrow (f,\pi_1 ,\ldots,\pi_m). \end{array}$

1.1. Definition

Definition 1 Let ${X}$ be a complete metric space. Let
$\displaystyle \begin{array}{rcl} \mathcal{D}^m (X):=Lip_b (X)\times Lip(X)^m . \end{array}$

A function ${T:\mathcal{D}^m (X)\rightarrow{\mathbb R}}$ is a metric ${m}$ -current if

Multilinearity: ${T}$ is multilinear.
Continuity: If ${\pi_i^n}$ converge pointwise to ${\pi_i}$ with uniformly bounded Lipschitz constants, then ${T(f,\pi_1^n ,\ldots,\pi_m^n)}$ converges to ${T(f,\pi_1 ,\ldots,\pi_m)}$ .
Locality: If some ${\pi}$ is constant on the support of ${f}$ , then ${T(f,\pi_1 ,\ldots,\pi_m)=0}$ .
Finite mass: There exists a Borel measure ${\mu}$ , concentrated on a ${\sigma}$ -compact set, such that, on ${\mathcal{D}^m}$ ,
$\displaystyle \begin{array}{rcl} |(T(f,\pi_1 ,\ldots,\pi_m)|\leq\prod_{i}Lip(\pi_i)\int_{X}|f|\,d\mu. \end{array}$

We denote by ${M_m (X)}$ the space of metric ${m}$ -currents.

Remark 1

Locality forces ${T(f,\pi_1 ,\ldots,\pi_m)}$ to depend on ${d\pi_i}$ rather than on ${\pi_i}$ itself.
Continuity axiom is stronger than in Euclidean geometric measure theory. For instance, the evaluation of a differential form at a ${m}$ -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 ${T}$ continuously extends to bounded functions ${\times Lip(X)^m}$ .

Proposition 2 Let ${T\in M_m (X)}$ . Denote by ${\Lambda}$ the collection of measures ${\mu}$ as in axiom (4). Define, for a Borel set ${B\subset X}$ ,
$\displaystyle \begin{array}{rcl} \|T\|(B)=\inf\{\sum_{i}\mu_i (B_i)\,;\, (\mu_i) \textrm{sequence in }\Lambda,\,B=\coprod_i B_i\}. \end{array}$

This is a finite Borel measure, and
$\displaystyle \begin{array}{rcl} |T(f,\pi_1 ,\ldots,\pi_m)|\leq \prod_{i=1}^m Lip(\pi_i)\,\int_{X}f\,d\mu. \end{array}$

Moreover
$\displaystyle \begin{array}{rcl} \|T\|(B)=\sup\{\sum_{n}|T(\chi_n ,\pi_1^n ,\ldots,\pi_m^n)|\;\,B=\coprod_i B_i,\,Lip(\pi_i^n)\leq 1\}. \end{array}$

Remark 2 In Federer and Fleming’s theory,
$\displaystyle \begin{array}{rcl} \|T\|({\mathbb R}^n)=\sup\{|T(\omega)|\,;\,\|\omega\|_{\infty}\leq 1\}. \end{array}$

Definition 3 The measure ${\|T\|}$ , as well as its total mass ${M(T)=\|T\|(X)}$ , are called the mass of current ${T}$ .

Corollary 4 Mass is lower semi-continuous with respect to weak convergence, i.e. if
$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}T_n (f,\pi_1 ,\ldots,\pi_m)=T(f,\pi_1 ,\ldots,\pi_m) \end{array}$

for all ${(f,\pi_1 ,\ldots,\pi_m)\in\mathcal{D}^m (X)}$ , then
$\displaystyle \begin{array}{rcl} M(T)\leq \liminf M(T_n). \end{array}$

1.2. Constructions

\subsubsection{Boundary operator}

For ${T\in M_m (X)}$ , m ${\geq 1}$ , define

$\displaystyle \begin{array}{rcl} (\partial T)(f,\pi_1 ,\ldots,\pi_{m-1})=T(1,f,\pi_1 ,\ldots,\pi_{m-1}). \end{array}$

Then ${T}$ satisfies axioms (1) to (3). Say that ${T}$ is a normal current if ${\partial T}$ has finite mass, i.e. is again a current. Notation

$\displaystyle \begin{array}{rcl} N_m (X)=\{T\in M_m (X)\,;\,\partial T\in M_{m-1}(X)\}. \end{array}$

Remark 3 This fits with the classical definition. Furthermore, by locality, ${\partial\partial=0}$ .

\subsubsection{Push forward}

If ${\phi:X\rightarrow Y}$ is Lipschitz, ${T\in M_m (X)}$ , define

$\displaystyle \begin{array}{rcl} (\phi_{\#}T)(f,\pi_1 ,\ldots,\pi_{m})=T(f\circ\phi,\pi_1 \circ\phi,\ldots,\pi_{m}\circ\phi). \end{array}$

Then ${\phi_{\#}T\in M_m (Y)}$ and, as positive measures,

$\displaystyle \begin{array}{rcl} \|\phi_{\#}T\|\leq Lip(\phi)^m \phi_{\#}\|T\|. \end{array}$

\subsubsection{Multiplication with Borel functions}

Let ${g}$ be a bounded Borel function on ${X}$ . Define

$\displaystyle \begin{array}{rcl} (T \llcorner g)(f,\pi_1 ,\ldots,\pi_{m})=T(gf,\pi_1 ,\ldots,\pi_{m}). \end{array}$

Then ${T \llcorner g\in M_m (X)}$ . In particular, if ${U}$ is a Borel set, the restriction of ${T}$ to ${U}$ is ${T\llcorner 1_U}$ .

\subsubsection{Standard example}

Let ${\Theta\in L^1 ({\mathbb R}^n)}$ . Define

$\displaystyle \begin{array}{rcl} [\Theta](f,\pi_1 ,\ldots,\pi_{m})=\int_{{\mathbb R}^n}\Theta fdet(\nabla\pi_1 ,\ldots,\nabla\pi_{m}). \end{array}$

Then ${[\Theta]\in M_m ({\mathbb R}^n)}$ .

Remark 4

Only continuity is non trivial.
For ${f}$ and ${\pi}$ smooth, ${[\Theta](f,\pi_1 ,\ldots,\pi_{m})=\int_{{\mathbb R}^n}\Theta f\,d\pi_1 ,\ldots,d\pi_{m}}$ .
If ${\Theta}$ has bounded variation, then ${[\Theta]}$ is a normal current, and ${\|\partial[\Theta]\|=|D\Theta|}$ in the notation of ${BV}$ theory.

1.3. Compactness theorem

Theorem 5 Let ${X}$ be a compact metric space, ${T_n}$ a sequence of normal currents with
$\displaystyle \begin{array}{rcl} \sup_{n}M(T_n)+M(\partial T_n) <\infty. \end{array}$

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

1.4. Integral currents

Definition 6 A ${0}$ -dimensional current ${T\in M_0 (X)}$ is called integral rectifiable if there exist points ${x_i \in X}$ and non zero integers ${m_i}$ such that on Lipschitz functions ${f}$ ,
$\displaystyle \begin{array}{rcl} T(f)=\sum_{i}m_i f(x_i). \end{array}$

If ${m\geq 1}$ , a current ${T\in M_m (X)}$ is called integral rectifiable if

${\|T\|}$ is concentrated on a countably rectifiable set, i.e. a countable disjoint union of biLipschitz images of subsets in ${{\mathbb R}^n}$ , and ${\|T\|}$ vanishes on sets of vanishing ${m}$ -dimensional Hausdorff measure.
For every open ${U\subset X}$ and Lipschitz ${\phi:U\rightarrow{\mathbb R}^m}$ , there exists an integer valued function ${\Theta\in L^1 ({\mathbb R}^m)}$ such that
$\displaystyle \begin{array}{rcl} \phi_{\#}(T\llcorner 1_{U})=[\Theta]. \end{array}$

We denote by ${\mathcal{I}_m (X)}$ the space of integral rectifiable ${m}$ -currents.

Theorem 7 (Representation theorem). Let ${T}$ be an integral rectifiable ${m}$ -current. Then there exist subsets ${K_i \subset {\mathbb R}^m}$ , biLipschitz maps ${\psi_i : K_i \rightarrow X}$ and integer valued ${L^1}$ functions ${\Theta_i}$ on ${K_i}$ such that
$\displaystyle \begin{array}{rcl} T=\sum_{i}\psi_{\#}[\Theta_i], \end{array}$

and
$\displaystyle \begin{array}{rcl} M(T)=\sum_{i}M(\psi_{\#}[\Theta_i]). \end{array}$

Definition 8 An integral current is a normal current which is integral rectifiable. The space of integral ${m}$ -currents is denoted by ${\mathbb{I}_{m}(X)}$ .

Note that ${\partial\mathbb{I}_{m}(X)\subset \mathbb{I}_{m-1}(X)}$ . A ${m}$ -cycle is an integral ${m}$ -current ${T}$ with ${\partial T=0}$ .

1.5. Slicing

Let ${T}$ be a normal current and ${\phi:X\rightarrow{\mathbb R}}$ a Lipschitz function. The slices of ${T}$ with respect to ${u}$ are the currents

$\displaystyle \begin{array}{rcl} \langle T,u,r \rangle=\partial(T\llcorner1_{\{u\leq r\}})-(\partial T)\llcorner 1_{\{u\leq r\}}. \end{array}$

In some sense, this is the restriction of ${T}$ to the level set ${\{u=r\}}$ .

Theorem 9 (Slicing Theorem). For almost every ${r}$ ,

${\langle T,u,r \rangle}$ is a normal ${m-1}$ -current, with its mass concentrated on ${\mathrm{support}(\|T\|)\cap\{u=r\}}$ .
${M(\langle T,u,r \rangle)\leq Lip(u)\frac{d}{dr}\|T\|(\{u\leq r\})}$ .
If ${T}$ is an integral current, so is almost every ${\langle T,u,r \rangle}$ .

Proof: Think of coarea formula

$\displaystyle \begin{array}{rcl} \int_{{\mathbb R}}\mathcal{H}^{m-1}(\{u=r\})\,dr=\int_{{\mathbb R}^n}|\nabla u| . \end{array}$

$\Box$

1.6. Closure Theorem

Theorem 10 If ${T_m}$ is a weakly converging sequence of integral currents with uniformly bounded ${M(T_n)+M(\partial T_n)}$ , 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 11 For ${T\in\mathbb{I}_{1}(X)}$ , define
$\displaystyle \begin{array}{rcl} Fillarea^{Y}(T)=\inf\{M(S)\,;\,S\in\mathbb{I}_{2}(Y),\,\partial S=T\}. \end{array}$

A Lipschitz curve ${c:[0,1]\rightarrow X}$ defines an integral current ${c_{\#}1_{[0,1]}}$ , again denote by ${c}$ , whose mass is the length of ${c}$ , and we define
$\displaystyle \begin{array}{rcl} FA_{X,Y}=\sup\{Fillarea^{Y}(c)\,;\,c \textrm{ closed curve }, \mathrm{length}(c)\leq r\}. \end{array}$

A priori, ${FA^{X,Y}\leq \mathrm{const.}\,FA_0^{X,Y}}$ , with ${const.=1}$ for Riemannian manifolds, but only ${const.\leq\sqrt{2}}$ in general, since the mass of integral ${2}$ -currents is not exactly equal to area.

Theorem 12 (Wenger 2010) Let ${X}$ be a geodesic metric space, let ${Y}$ be a geodesic thickening pf ${X}$ . If
$\displaystyle \begin{array}{rcl} FA^{X,Y}(r)\preceq r^2 , \end{array}$

then there exists ${C}$ such that for every asymptotic cone ${X_{\omega}}$ of ${X}$ , and for all ${r\geq 0}$ ,
$\displaystyle \begin{array}{rcl} FA^{X_{\omega}}(r)\leq C\,r^2. \end{array}$

Remark 5 Since every integral ${1}$ -current ${T}$ with ${\partial T=0}$ is a countable sum of curves, quadratic filling for curves implies a quadrating filling inequality for all integral cycles.

Corollary 13 Let ${\Gamma}$ be a finitely presented group with quadratic Dehn function. Then every asymptotic cone ${\Gamma_{\omega}}$ has quadratic filling for ${1}$ -cycles.

Panos Papasoglu shows that quadratic Dehn function implies that ${\Gamma_{\omega}}$ is simply connected. Does quadratic Dehn function imply that ${FA_0^{\Gamma_{\omega}}}$ is quadratic ?

3. Completion of proofs

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

Proposition 14 (Wenger 2008) Let ${X}$ be a geodesic metric space, ${Y}$ a geodesic thickening with quadratic filling. If ${X}$ is not hyperbolic, there exists an asymptotic cone ${X_{\omega}}$ of ${X}$ , a compact set ${K\subset {\mathbb R}^2}$ and a Lipschitz map ${\psi:K \rightarrow X_{\omega}}$ such that ${\psi(K)}$ has positive ${2}$ -dimensional Hausdorff measure.

Proof: of Proposition from Theorem 12.

Let ${X_{\omega}}$ be an asymptotic cone which is not a tree. Then there is a closed Lipschitz curve ${c}$ in ${X_{\omega}}$ such that the corresponding current is not identically zero. Take for ${c}$ a constant speed parametrization of a geodesic triangle ${xyz}$ such that ${[x,y]}$ is not included in ${[y,z]\cup[z,x]}$ . Let ${\pi}$ denote distance to ${x}$ and ${f}$ the maximum of ${0}$ and ${1-\frac{1}{\epsilon}d(\cdot,[x,y])}$ . For ${\epsilon}$ small enough,

$\displaystyle \begin{array}{rcl} c(f,\pi)=\int_{0}^{1}f\circ c(t) (\pi\circ c)'(t)\,dt\not=0. \end{array}$

Theorem 12 gives an integral current ${S}$ filling ${c}$ . Then ${S\not=0}$ . According to the Representation Theorem 7, ${S=\sum_{i}\psi_{\#}[\Theta_i]}$ , ${M(T)=\sum_{i}M(\psi_{\#}[\Theta_i])>0}$ , so there is ${i}$ such that ${M(\psi_{\#}[\Theta_i])>0}$ . $\Box$

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 ${X}$ 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 ${Y}$ be a metric space. Assume that ${Y}$ has a quadratic filling inequality for ${\mathbb{I}_1 (Y)}$ . Let ${T\in \mathbb{I}_1 (Y)}$ be a cycle and ${\epsilon>0}$ . There is ${S\in\mathbb{I}_2 (Y)}$ with ${\partial S=T}$ such that

${M(S)\leq\min\{C\,M(T)^2 ,(1+\epsilon)Fillarea^{Y}(T)\}}$ .
For each ${y\in \mathrm{support}(\|S\|)}$ and all ${r\in [0,d(x,\mathrm{support}(\|T\|)}$ ,
$\displaystyle \begin{array}{rcl} \|S\|(B(x,r))\geq \frac{1}{4C}r^2 . \end{array}$

Proof: Assume first there exists a mass minimizing ${S}$ with ${\partial S=T}$ . Let ${g(r)=\|S\|(B(x,r))}$ and ${d_x}$ , the distance function to ${x}$ . For almost every ${r}$ ,

$\displaystyle \begin{array}{rcl} \langle S,d_x ,r \rangle=\partial(S\llcorner 1_{B(x,r)})\in \mathbb{I}_1 (Y). \end{array}$

By minimality,

$\displaystyle \begin{array}{rcl} g(r)&=&\|S\|(B(x,r))\|\\ &=&M(S\llcorner 1_{B(x,r)})\\ &\leq& C\,M(\langle S,d_x ,r \rangle)^2 \\ &\leq&C\,(\frac{d}{dr}\|S\|(B(x,r)))^2 \\ &=&C\,g'(r)^2 . \end{array}$

Integrate this differential inequality to get ${g(r)\geq\frac{1}{C}r^2}$ .

In general, consider the set ${Z}$ of integral ${2}$ -currents filling ${T}$ with mass ${\leq L}$ . For every ${\delta\in(0,1)}$ , there exists in ${Z}$ an ${S}$ such that for all ${S'\in Z}$ ,

$\displaystyle \begin{array}{rcl} M_{\delta}(S):=M(S)+\delta M(S'-S)\geq M(S). \end{array}$

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

For every competitor ${R}$ ,

$\displaystyle \begin{array}{rcl} M(S)=M_{\delta}(S)&\leq& M_{\delta}(S\llcorner B(x,r)^{c}+R)\\ &\leq&\|S\|(B(x,r)^c )+M(R)+\delta \|S\|(B(x,r))+\delta M(R). \end{array}$

Thus, using quadratic filling inequality,

$\displaystyle \begin{array}{rcl} \|S\|(B(x,r))&\leq&\frac{1+\delta}{1-\delta}M(R)\\ &\leq&\frac{1+\delta}{1-\delta}C\,M(\langle S,d_x ,r \rangle)^2 , \end{array}$

and the proof ends as before. $\Box$

3.3. Sketch of the proof of Theorem \ref

}

We assume that ${X}$ has quadratic filling, and show that asymptotic cones do as well. Step 1. Show that there exists a geodesic thickening ${Y}$ of ${X}$ which has ${Fillarea^{Y}(r)\leq C\, r^2}$ for all ${r}$ . For this, cover ${X}$ with balls which sufficiently overlap, then replace them by their injective hulls.

Step 2. Let ${c}$ be a Lipschitz loop in an asymptotic cone ${X_{\omega}}$ . Pick a partition ${(t_i)}$ of ${[0,1]}$ . Show that there exists a Lipschitz loop ${c'}$ , with ${c(t_i)=c'(t_i)}$ , with shorter lengths between successive ${t_i}$ ‘s, and an integral ${2}$ -current ${S}$ filling ${c'}$ , with ${M(S)\leq C\,\mathrm{length}(c)^2}$ . This suffices, since holes between ${c}$ and ${c'}$ can be inductively filled, achieving a convergent series of currents whose sum fills ${c}$ .

To construct ${S}$ , let ${x^i =c(t_i)}$ , view ${x^i}$ as a sequence ${(x_n^i)\in \hat{X}}$ . Complete ${x_n^1 ,\ldots,x_n^m}$ into a geodesic polygon ${c_n :[0,1]\rightarrow X_n =(X,\frac{1}{n}d)}$ . Use Proposition 15. Since ${c_n}$ are uniformly Lipschitz, they can be filled with integral ${2}$ -currents ${S_n \in \mathbb{I}_2 (Y,\frac{1}{n}d)}$ in such a way that

${\partial S_n =c_n}$ .
${M(S_n)\leq C\,\mathrm{length}(c_n)^2}$ .
${\|S_n\|(B(x,r))\geq \frac{1}{4C}r^2}$ for all ${x\in\mathrm{support}(S_n)}$ , ${r\leq d(x,c_n)}$ .

The sequence of metric spaces ${A_n =(\mathrm{support}(S_n),\frac{1}{n}d)}$ is uniformly compact, since we have upper bounds for the size of ${\epsilon}$ -nets on it for all ${\epsilon>0}$ . 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

${A_n}$ converge in Gromov-Hausdorff sense to a compact metric space ${A}$ .
The curves ${c_n}$ converge to a curve ${c''}$ in ${A}$ .
The currents ${S_n}$ converge to an integral ${2}$ -current ${S}$ in ${A}$ .
The (inverse) isometric embeddings ${\psi_n : A_n \rightarrow X_n}$ converge to an isometric embedding ${\psi:A'\rightarrow X_{\omega}}$ .

Then ${\partial S=c''}$ in ${A}$ , ${\partial\psi_{\#}(S)=\psi_{\#}(c'')=c'}$ in ${X_{\omega}}$ and

$\displaystyle \begin{array}{rcl} M(\psi_{\#}(S))&=&M(S)\\ &\leq& \liminf_{n\rightarrow\infty}M(S_n)\\ &\leq&C\,\liminf_{n\rightarrow\infty}\mathrm{length}(c_n)^2\\ &=&C\,\mathrm{length}(c')^2 . \end{array}$

4. Carnot groups with non exactly polynomial Dehn function

4.1. Superquadratic Dehn function

Theorem 16 (Wenger 2010) Let ${G}$ be a ${2}$ -step nilpotent Lie group with Lie algebra graded as ${\mathfrak{g}=V_1 \oplus V_2}$ . Assume that there exists ${u\in V_2}$ which is not a bracket ${u=[v,w]}$ for any vectors ${v}$ , ${w\in V_1}$ . Consider the Lie group ${H}$ with Lie algebra ${\mathfrak{h}=V_1 \oplus (V_2 /\langle u \rangle)}$ . Endow ${H}$ with a left-invariant Riemannian metric. Then
$\displaystyle \begin{array}{rcl} \lim_{r\rightarrow \infty}\frac{FA^{H}(r)}{r^2}=+\infty. \end{array}$

Remark 6 If ${\mathrm{dim}(V_2)\geq 2\,\mathrm{dim}(V_1)}$ , then there exists ${u}$ satisfying the assumption in Theorem 16.

4.2. Application to central products

Let ${G}$ be a ${2}$ -step nilpotent Lie group with Lie algebra graded as ${\mathfrak{g}=V_1 \oplus V_2}$ . Given ${m\geq 2}$ , let ${H=G\times_{Z}\cdots\times_{Z}G}$ denote the Carnot group with Lie algebra

$\displaystyle \begin{array}{rcl} \mathfrak{g}'=(V_1^1 \oplus \cdots \oplus V_1^m) \oplus V_2 \end{array}$

with Lie bracket ${[(v_1 ,\ldots, v_n),(w_1 ,\ldots, w_n)]=\sum_{i=1}^{m}[v_i ,w_i]}$ . This will be called the ${m}$ -fold central product of ${G}$ .

Example 1 If ${G=H^1}$ is the first Heisenberg group, ${G\times_{Z}\cdots\times_{Z}G =H^m}$ is the ${m}$ -th Heisenberg group.

Corollary 17 Let ${G'}$ be a ${2}$ -step Carnot group with Lie algebra ${\mathfrak{g}'=V_1 \oplus V_2}$ . Suppose ${m\geq 2}$ and there exists ${u'\in V_2}$ which cannot be written
$\displaystyle \begin{array}{rcl} u'=\sum_{i=1}^{m}[v_i ,w_i]. \end{array}$

Let ${H'}$ be the Lie group with Lie algebra ${\mathfrak{h}=V_1 \oplus (V_2 /\langle u' \rangle)}$ and ${H}$ the ${m}$ -fold central product of ${H}$ . Then
$\displaystyle \begin{array}{rcl} \lim_{r\rightarrow \infty}\frac{FA^{H}(r)}{r^2}=+\infty. \end{array}$

Proof: The central product ${H=H'\times_Z H'}$ can be viewed as ${G/\exp(u')}$ where ${G=G'\times_Z \cdots \times_Z G'}$ . So Theorem 16 applies provided ${u'}$ is not a bracket in ${\mathfrak{g}}$ , i.e. ${u'}$ is not a sum of ${m}$ brackets in ${\mathfrak{g}'}$ . $\Box$

Example 2 Start with the free ${2}$ -step nilpotent Lie algebra with ${\mathrm{dim}(V_1)=2k}$ . Set ${u=e_1 \wedge e_2 +\cdots+e_{2k-1}\wedge e_{2k}}$ . Then ${u}$ is not a sum of ${m}$ bracket of two vectors if ${m<k}$ .

Indeed, as an alternating bilinear form, its rank is ${2k}$ , whereas a sum of ${m}$ brackets has rank at most ${2m}$ .

Theorem 18 (Olshanskii, Sapir, Young) Let ${H}$ be a ${2}$ -step Carnot group containing a lattice. Then its ${m}$ -fold central product, ${m\geq 2}$ , has
$\displaystyle \begin{array}{rcl} FA_{0}^{H}(r)\preceq r^2 \log r . \end{array}$

If furthermore ${H}$ is free ${2}$ -step nilpotent, then
$\displaystyle \begin{array}{rcl} FA_{0}^{H}(r)\sim r^2 . \end{array}$

Corollary 19 There exists a ${2}$ -step Carnot group ${H}$ such that
$\displaystyle \begin{array}{rcl} FA_{0}^{H}(r)\preceq r^2 \log r . \end{array}$

but ${FA_{0}^{H}(r)}$ is not ${\leq O(r^2)}$ .

4.3. Preparation for the proof of Theorem \ref

}

A Carnot group has Lie algebra graded as ${\mathfrak{g}=V_1 \oplus \cdots \oplus V_k}$ . The map ${\delta_r :\mathfrak{g}\rightarrow\mathfrak{g}}$ defined by

$\displaystyle \begin{array}{rcl} \delta_r (v_1 +\cdots v_k)=\sum_{i=1}^{k}r^i v_i \end{array}$

is an automorphism. It integrates into a group automorphism.

The Carnot-Carathéodory metric ${d_c}$ is the left-invariant geodesic metric obtained by minimizing length of curves tangent to the distribution of left-translates of ${V_1}$ . It is homogeneous of degree ${1}$ under ${\delta_r}$ .

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

4.4. Proof of the lower bound on Dehn function

Let ${G}$ be a ${2}$ -step Carnot group, let be ${H}$ the group with Lie algebra ${\mathfrak{h}=V_1 \oplus (V_2 /\langle u \rangle)}$ . We show that the second homology of the asymptotic cone ${(H,d_c)}$ does not vanish.

Claim: There are Lipschitz loops ${c}$ which do not bound any integral current in ${H}$ .

To construct ${c}$ , merely pick a Lipschitz horizontal curve in ${G}$ joining the origin to ${\exp(u)}$ , and project it to ${H}$ . Assume that there exists an integral ${2}$ -current ${S}$ in ${H}$ such that ${\partial S=c}$ . By the representation theorem 7,

$\displaystyle \begin{array}{rcl} S=\sum_{i}(\psi_i)_{\#}[\Theta_i] \end{array}$

where ${\psi_i :K_i \rightarrow (H,d_c)}$ is Lipschitz, ${K_i \in{\mathbb R}^2}$ , ${\Theta_i \in L^1 (K_i)}$ .

Note that projection along ${V_2}$ defines a Lipschitz map ${\eta:H\rightarrow V_1}$ . ${\eta_{\#}S}$ is an integral current in Euclidean ${V_1}$ filling ${c_1 :=\eta(c)}$ . Let ${Q}$ be a linear functional on ${{\mathbb R}}$ which does not vanish on ${u}$ . Then ${Q}$ defines a ${1}$ -form on ${V_1}$ as follows:

$\displaystyle \begin{array}{rcl} \alpha(x)(v)=Q([x,v]). \end{array}$

Then

$\displaystyle \begin{array}{rcl} d\alpha(v,w)=Q([v,w]), \end{array}$

$\displaystyle \begin{array}{rcl} 0\not= Q(u)&=&c_1 (\alpha)\\ &=&(\partial\eta_{\#}S)(\alpha)\\ &=&(\eta_{\#}S)(d\alpha)\\ &=&\sum_{i}(\eta\circ\psi_i)_{\#}[\Theta_i](d\alpha)\\ &=&\sum_{i}\int_{K_i}[d(\eta\circ\psi_i)(e_1),d(\eta\circ\psi_i)(e_2)] \end{array}$

where ${(e_1 ,e_2)}$ is the canonical basis of ${{\mathbb R}^2}$ .

Let us show that ${[d(\eta\circ\psi_i)(e_1),d(\eta\circ\psi_i)(e_2)]=0}$ for all ${i}$ and almost everywhere on ${K_i}$ . At almost every point of ${K_i}$ , all Lipschitz maps ${\psi_i}$ are differentiable. The differential ${d^P \psi}$ is a group homomorphism ${{\mathbb R}^2 \rightarrow H}$ . The image of the corresponding Lie algebra homomorphism is an abelian subspace in ${\mathfrak{h}}$ . Its image by ${d^P \eta}$ is an subspace of ${V_1}$ on which the ${\mathfrak{h}}$ -bracket vanishes, i.e. the ${\mathfrak{g}}$ -bracket is colinear to ${u}$ . Since we assumed that ${u}$ is not a bracket of ${2}$ vectors from ${V_1}$ , the ${\mathfrak{g}}$ -bracket vanishes on the image of ${d(\eta\circ\psi_i)}$ . So ${[d(\eta\circ\psi_i)(e_1),d(\eta\circ\psi_i)(e_2)]=0}$ .