aa r X i v : . [ m a t h . G R ] A p r NILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHNFUNCTION
STEFAN WENGERA bstract . We prove super-quadratic lower bounds for the growth of the filling area func-tion of a certain class of Carnot groups. This class contains groups for which it is knownthat their Dehn function grows no faster than n log n . We therefore obtain the existence of(finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomialgrowth and we thus answer a well-known question about the possible growth rate of Dehnfunctions of nilpotent groups.
1. I ntroduction
Dehn functions have played an important role in geometric group theory. They measurethe complexity of the word problem in a given finitely presented group and provide a quasi-isometry invariant of the group. A class of groups for which the Dehn function has beenwell-studied is that of nilpotent groups. Many results on upper bounds for the growth ofthe Dehn function are known, while fewer techniques have been found for obtaining lowerbounds. Here are some known results: if Γ is a finitely generated nilpotent group of step c then its Dehn function δ Γ ( n ) grows no faster than n c + , in short δ Γ ( n ) (cid:22) n c + , [12, 20, 9]. If Γ is a free nilpotent group of step c then δ Γ ( n ) ∼ n c + , [5, 21]. This includes the particularcase of the first Heisenberg group, which was known before, see [8]. For the definitionof δ Γ ( n ) and the meaning of (cid:22) and ∼ see Section 2. The higher Heisenberg groups havequadratic Dehn functions, [12, 1, 16]. Ol’shanskii-Sapir used the arguments in their paper[16] to prove that the Dehn function of the central product of m ≥ n log n . This has also recently been proved in [27]with di ff erent methods.The following question about the possible growth rate of the Dehn function of nilpotentgroups has been raised by many authors, see e.g. [21, 4, 6], but has remained open so far. Question 1.1.
Does the Dehn function of every finitely generated nilpotent group Γ growexactly polynomially, that is, δ Γ ( n ) ∼ n α for some (integer) α ? The primary purpose of this article is to prove super-quadratic lower bounds for the growthof the Dehn function of certain classes of nilpotent groups and to combine these with theresults from [16, 27] to give a negative answer to the above question. We establish:
Theorem 1.2.
There exist finitely generated nilpotent groups Γ of step whose Dehn func-tion δ Γ ( n ) satisfies (1) n ̺ ( n ) (cid:22) δ Γ ( n ) (cid:22) n log nfor some function ̺ with ̺ ( n ) → ∞ as n → ∞ . Date : April 16, 2010.Partially supported by NSF grant DMS 0956374.
We can in fact produce a whole family of groups satisfying (1). They all arise as lattices of(central powers) of Carnot groups of step 2, and the upper bound comes as a consequenceof [16, 27]. Recall that a simply connected nilpotent Lie group is called Carnot group if itsLie algebra admits a grading, the first layer of which generates the whole Lie algebra, seeSection 2.3 for definitions. In Theorem 4.3 we will show that under suitable conditions onthe Lie algebra of a Carnot group G of step 2 the m -th central power of G has filling areafunction which grows strictly faster than quadratically. Recall that the filling area functionFA( r ) of a simply connected Riemannian manifold X is the smallest function such thatevery closed curve in X of length at most r bounds a singular Lipschitz chain of area atmost FA( r ). As is well-known, FA( r ) bounds from below the Dehn function of any finitelygenerated group acting properly discontinuously and cocompactly by isometries on X .In preparation of the proof of the super-quadratic lower bounds and of Theorem 1.2, wewill prove, in Section 3, the following theorem which should be of independent interest. Theorem 1.3.
Let X be a simply connected Riemannian manifold with quadratic fillingarea function,
FA( r ) (cid:22) r . Then every asymptotic cone X ω of X admits a quadratic isoperi-metric inequality for integral -currents in X ω . In particular, there exists C > such thatevery closed Lipschitz curve c in X ω bounds an integral -current in X ω of mass (“area”)at most C length( c ) . The above theorem holds more generally for complete metric length spaces X , see The-orem 3.1. The theory of integral currents in metric spaces was developed by Ambrosio-Kirchheim in [3]. We refer to Section 2 for definitions. Integral 2-currents should bethought of as generalized singular Lipschitz chains and the mass is proportional to theHausdor ff area. A consequence of Theorem 1.3 is the following: Corollary 1.4.
Let Γ be a finitely presented group with quadratic Dehn function. Thenevery asymptotic cone of Γ admits a quadratic isoperimetric inequality for integral -currents. It is known that finitely presented groups with quadratic Dehn function have simply con-nected asymptotic cones, [19]. Many groups, however, have simply connected asymptoticcones but do not have quadratic Dehn function. Corollary 1.4 does not yield simple con-nectedness but instead provides strong metric information. This can be used to prove thatcertain groups cannot admit quadratic Dehn function, see below.We now briefly describe how Theorem 1.3 will be used to obtain super-quadratic growthestimates for filling area functions. Let H be a Carnot group and let d be the distance in-duced by a left-invariant Riemannian metric on H . Denote by d c the Carnot-Carath´eodorydistance on H associated with d , see Section 2.3. By [17], the metric space ( H , d c ) is theunique asymptotic cone of ( H , d ). In order to prove that the filling area function of ( H , d )grows strictly faster than quadratically for a given ( H , d ) it thus su ffi ces, by Theorem 1.3,to prove that the metric space ( H , d c ) cannot admit a quadratic isoperimetric inequality forintegral 1-currents. In Sections 4 and 5 we will show that under suitable conditions on theLie algebra of H , the metric space ( H , d c ) has non-trivial first homology group for inte-gral currents, and in particular ( H , d c ) does not admit a quadratic isoperimetric inequalityfor integral 1-currents. The conditions exhibited in Section 4 will be used to establish thelower bounds needed to prove Theorem 1.2; they can roughly be described as follows. Let G be a Carnot group of step 2 with grading g = V ⊕ V of its Lie algebra. Given a propersubspace U ⊂ V , one obtains a new Lie algebra g U = V ⊕ ( V / U ) and hence a Carnotgroup G U with Lie algebra g U . Note that every simply connected nilpotent Lie group ofstep 2 arises this way with g = V ⊕ V a free nilpotent Lie algebra of step 2 and U ⊂ V a ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 3 suitable subspace. In the proof of Theorem 4.3 we will show that under a suitable condi-tion on U , which we term m -inaccessibility, the m -th central power H : = G U × Z · · · × Z G U ,endowed with a Carnot-Carath´eodory distance, has non-trivial first Lipschitz homology.In view of the above, this is enough to conclude that the filling area function of H growssuper-quadratically when H is endowed with a left-invariant Riemannian metric. The m -inaccessibility condition is given in Definition 4.1; it is not di ffi cult to provide examples of m -inaccessible subspaces U , see Example 4.2 and the remark following it. For the proofsof Theorems 4.3 and 1.2, we only need the special case of Theorem 1.3 when X is a Carnotgroup endowed with a left-invariant Riemannian metric. In this case, the proof can besimplified and we can in fact prove an analog for cycles of any dimension, see Proposi-tion 3.6. We believe that, nevertheless, the generality of Theorem 1.3 should be of interest,especially since not much is known about groups with quadratic Dehn function.The paper is structured as follows: Section 2 contains definitions of and facts about Dehnfunctions, filling area functions, asymptotic cones, and Carnot groups. We will also recallall definitions from Ambrosio-Kirchheim’s theory of integral currens [3] which we willneed later. In Section 3 we will use the theory of currents to prove Theorem 1.3 andthe more general Theorem 3.1. In Section 4, Theorem 3.1 will be used to prove the super-quadratic lower bounds in Theorem 4.3 and to establish Theorem 1.2. Finally, in Section 5,we use Theorem 3.1 to exhibit super-quadratic lower bounds for another class of Carnotgroups, not necessarily of step 2, see Theorem 5.1. The results obtained in Section 5 arenot needed for the proof of Theorem 1.2. Acknowledgment:
I am indebted to Robert Young for inspiring discussions and for bring-ing Question 1.1 to my attention. Parts of the research underlying this paper was carriedout during a research visit to the Universit´e Catholique de Louvain, Belgium. I wish tothank the Mathematics Department and Thierry de Pauw for the hospitality.2. P reliminaries
The purpose of this section is to collect some definitions and facts which will be used inthe sequel.2.1.
Dehn and filling area functions.
Let
Γ = < S | R > be a finitely presented group.The Dehn function δ Γ ( n ) of Γ with respect to the given presentation is defined by δ Γ ( n ) : = max w = Γ , | w |≤ n min k : w = k Y i = g i r ± i g − i , r i ∈ R , g i word in S . Here, w is a word in the alphabet S and | w | denotes the word length of w . The equality w = Q ki = g i r ± i g − i is in the free group generated by S . For functions f , g : N → N or f , g : [0 , ∞ ] → (0 , ∞ ) one writes f (cid:22) g if there exists C such that f ( r ) ≤ Cg ( Cr + C ) + Cr + C for all r >
0. One furthermore writes f ∼ g if f (cid:22) g and g (cid:22) f . If Γ acts properly discon-tinuously and cocompactly by isometries on a simply connected Riemannian manifold X then δ Γ ( n ) ∼ FA ( n ) , where the Dehn function FA ( r ) on X is defined byFA ( r ) : = sup { Fillarea ( c ) : c closed Lipschitz curve in X of length( c ) ≤ r } , STEFAN WENGER and Fillarea ( c ) of c : S → X is given byFillarea ( c ) : = inf n Area( ϕ ) : ϕ : D → X Lipschitz, ϕ | S = c o . Here, Area( ϕ ) is the integral over the disc D of the jacobian of ϕ . The filling area functionFA( r ) of X is defined byFA( r ) : = sup { Fillarea( c ) : c closed Lipschitz curve in X of length( c ) ≤ r } , whereFillarea( c ) : = inf (cid:8) Area( z ) : z is a singular Lipschitz 2-chain in X with boundary c (cid:9) . Recall that a singular Lipschitz 2-chain is a formal finite sum z = P m i ϕ i with m i ∈ Z and ϕ i : ∆ → X Lipschitz and its area isArea( z ) : = X | m i | Area( ϕ i ) , see [11]. It is clear that FA( r ) ≤ FA ( r )for all r ≥ Ultralimits and asymptotic cones of metric spaces.
Recall that a non-principal ul-trafilter on N is a finitely additive probability measure ω on N (together with the σ -algebraof all subsets) such that ω takes values in { , } only and ω ( A ) = A ⊂ N isfinite. The existence of non-principal ultrafilters on N follows from Zorn’s lemma. It is notdi ffi cult to prove the following fact. If ( Y , τ ) is a compact topological Hausdor ff space thenfor every sequence ( y n ) n ∈ N ⊂ Y there exists a unique point y ∈ Y such that ω ( { n ∈ N : y n ∈ U } ) = U ∈ τ containing y . We will denote this point by lim ω y n .Let now ( X n , d n , p n ) be pointed metric spaces, n ≥
1, and fix a non-principal ultrafilter ω on N . A sequence ( x n ), where x n ∈ X n for each n , will be called bounded ifsup n d n ( p n , x n ) < ∞ . Define an equivalence relation on bounded sequences by( x n ) ∼ ( x ′ n ) if and only if lim ω d n ( x n , x ′ n ) = . Let X ω be the set of equivalence classes of bounded sequences, d ω the metric on X ω givenby d ω ([( x n )] , [( x ′ n )]) : = lim ω d n ( x n , x ′ n ) , and p ω : = [( p n )]. Then the pointed metric space ( X ω , d ω , p ω ) is called the ultralimit of thesequence ( X n , d n , p n ) with respect to ω . If ( X , d ) is a metric space, ( p n ) ⊂ X a sequence ofbasepoints, and r n > r n → ∞ , then the ultralimit of the sequence ( X , r − n d , p n )with respect to ω is called the asymptotic cone of ( X , ( r n ) , ( p n )) with respect to ω .It is not di ffi cult to show that ultralimits are always complete. Furthermore, if every ( X n , d n )is a length space, then the ultralimits are all length spaces as well. ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 5
Carnot groups and their asymptotic cones.
Recall that a connected and simplyconnected nilpotent Lie group G of step k is called Carnot group if its Lie algebra g admitsa grading g = V ⊕ · · · ⊕ V k such that [ V , V i ] = V i + for all i = , . . . , k − V , V k ] =
0, where [ V , V i ] is thesmallest subspace spanned by the elements [ v , v ′ ] with v ∈ V and v ′ ∈ V i . In other words,a Carnot group is a homogeneous nilpotent Lie group such that the first layer V of its Liealgebra generates the entire Lie algebra. Clearly, every connected and simply connectednilpotent Lie group of step 2 is a Carnot group. Let now G be a Carnot group of step k andlet g = V ⊕ · · · ⊕ V k be a grading of its Lie algebra g . Note that the exponential map exp : g → G is a di ff eo-morphism. G comes with a family of dilation homomorphisms δ r : G → G , r ≥
0, which,on the level of Lie algebras, take the form δ r ( v ) = k X i = r i v i for v = v + · · · + v k with v i ∈ V i .Let g be a left-invariant Riemannian metric on G and denote by d the distance on G coming from g . A new distance, called Carnot-Carath´eodory distance, can be associatedwith g as follows. Define the horizontal subbundle T H of TG by left-translating V . Acurve c : [0 , → G , absolutely continuous (ac for short) with respect to d , is calledhorizontal if ˙ c ( t ) ∈ T c ( t ) H for almost every t ∈ [0 , d c on G associated with g is then defined by d c ( x , y ) = inf { length g ( c ) : c horizontal ac curve joining x to y } , where length g ( c ) denotes the length of c with respect to g . It can be shown that d c definesa metric, ie. that d c ( x , y ) is always finite. Important properties of the Carnot-Carath´eodorydistance are that it is left invariant and 1-homogeneous with respect to the dilations, i.e., d c ( δ r ( x ) , δ r ( y )) = r d c ( x , y ) for all x , y ∈ G and all r ≥
0. We also obviously have therelationship d ≤ d c , and it is well-known that d and d c are not bi-Lipschitz equivalent unless G is Euclidean.Note however, that the topologies induced by d and d c are the same.The following theorem, which is a special case of a more general result due to Pansu [17],gives a link between left-invariant Riemannian metrics and Carnot-Carath´eodory metricson Carnot groups. Theorem 2.1.
Let G be a Carnot group and let d be the distance associated with aleft-invariant Riemannian metric g on G. Then the pointed spaces ( G , r d , e ) convergein the pointed Gromov-Hausdor ff sense to ( G , d c , e ) as r → ∞ , where d c is the Carnot-Carath´eodory distance on G associated with g . Here, e denotes the identity element ofG. In particular, it follows that ( G , d ) has a unique asymptotic cone, which moreover is iso-metric to ( G , d c ). This will be used in the proofs of Theorem 4.3 and Theorem 5.1. STEFAN WENGER
Integral currents in metric spaces.
The theory of integral currents in metric spaceswas developed by Ambrosio and Kirchheim in [3] and provides a suitable notion of sur-faces and area / volume in the setting of metric spaces. In the following we recall the defi-nitions that are needed for our purposes.Let ( Y , d ) be a complete metric space and m ≥ D m ( Y ) be the set of ( m + f , π , . . . , π m ) of Lipschitz functions on Y with f bounded. The Lipschitz constant of aLipschitz function f on Y will be denoted by Lip( f ). Definition 2.2.
An m-dimensional metric current T on Y is a multi-linear functional on D m ( Y ) satisfying the following properties: (i) If π ji converges point-wise to π i as j → ∞ and if sup i , j Lip( π ji ) < ∞ thenT ( f , π j , . . . , π jm ) −→ T ( f , π , . . . , π m ) . (ii) If { y ∈ Y : f ( y ) , } is contained in the union S mi = B i of Borel sets B i and if π i isconstant on B i then T ( f , π , . . . , π m ) = . (iii) There exists a finite Borel measure µ on Y such that (2) | T ( f , π , . . . , π m ) | ≤ m Y i = Lip( π i ) Z Y | f | d µ for all ( f , π , . . . , π m ) ∈ D m ( Y ) . The space of m -dimensional metric currents on Y is denoted by M m ( Y ) and the minimalBorel measure µ satisfying (2) is called mass of T and written as k T k . We also call massof T the number k T k ( Y ) which we denote by M ( T ). The support of T is, by definition, theclosed set spt T of points y ∈ Y such that k T k ( B ( y , r )) > r > θ ∈ L ( K , R ) with K ⊂ R m Borel measurable induces an element of M m ( R m )by [ θ ℄ ( f , π , . . . , π m ) : = Z K θ f det ∂π i ∂ x j ! d L m for all ( f , π , . . . , π m ) ∈ D m ( R m ).The restriction of T ∈ M m ( Y ) to a Borel set A ⊂ Y is given by( T A )( f , π , . . . , π m ) : = T ( f χ A , π , . . . , π m ) . This expression is well-defined since T can be extended to a functional on tuples for whichthe first argument lies in L ∞ ( Y , k T k ).If m ≥ T ∈ M m ( Y ) then the boundary of T is the functional ∂ T ( f , π , . . . , π m − ) : = T (1 , f , π , . . . , π m − ) . It is clear that ∂ T satisfies conditions (i) and (ii) in the above definition. If ∂ T also satisfies(iii) then T is called a normal current. By convention, elements of M ( Y ) are also callednormal currents.The push-forward of T ∈ M m ( Y ) under a Lipschitz map ϕ from Y to another completemetric space Z is given by ϕ T ( g , τ , . . . , τ m ) : = T ( g ◦ ϕ, τ ◦ ϕ, . . . , τ m ◦ ϕ )for ( g , τ , . . . , τ m ) ∈ D m ( Z ). This defines a m -dimensional current on Z . It follows directlyfrom the definitions that ∂ ( ϕ T ) = ϕ ( ∂ T ). ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 7
We will mainly be concerned with integral currents. Let H m denote Hausdor ff m -measureon Y and recall that an H m -measurable set A ⊂ Y is said to be countably H m -rectifiable ifthere exist countably many Lipschitz maps ϕ i : B i −→ Y from subsets B i ⊂ R m such that H m (cid:16) A \ [ ϕ i ( B i ) (cid:17) = . An element T ∈ M ( Y ) is called integer rectifiable if there exist finitely many points y , . . . , y n ∈ Y and θ , . . . , θ n ∈ Z \{ } such that T ( f ) = n X i = θ i f ( y i )for all bounded Lipschitz functions f . A current T ∈ M m ( Y ) with m ≥ k T k is concentrated on a countably H m -rectifiable set and vanishes on H m -negli-gible Borel sets.(ii) For any Lipschitz map ϕ : Y → R m and any open set U ⊂ Y there exists θ ∈ L ( R m , Z ) such that ϕ ( T U ) = [ θ ℄ .Integer rectifiable normal currents are called integral currents. The corresponding spaceis denoted by I m ( Y ). If A ⊂ R m is a Borel set of finite measure and finite perimeter then [ χ A ℄ ∈ I m ( R m ). Here, χ A denotes the characteristic function. If T ∈ I m ( Y ) and if ϕ : Y → Z is a Lipschitz map into another complete metric space then ϕ T ∈ I m ( Z ).We close this section with a few remarks. A Lipschitz curve c : [ a , b ] → Y gives riseto the element c [ χ [ a , b ] ℄ ∈ I ( Y ). If ϕ is one-to-one then M ( c [ χ [ a , b ] ℄ ) = length( c ). Itwas shown in [25, Lemma 2.3] that 1-dimensional integral currents are essentially inducedby (countably many) Lipschitz curves. A Lipschitz map ϕ : D → Y gives rise to the2-dimensional integral current S : = ϕ [ χ D ℄ . If ϕ is one-to-one then1 c Area( ϕ ) ≤ M ( S ) ≤ c Area( ϕ )for some universal c ; if Y is a Riemannian manifold then M ( S ) = Area( ϕ ). A singularLipschitz chain c = P m i ϕ i gives rise to the integral current P m i ϕ i [ χ ∆ ℄ . An element T ∈ I ( Y ) can be thought of as a generalized singular Lipschitz chain whose boundaryconsists of a union of closed Lipschitz curves of finite total length.3. Q uadratic isoperimetric inequalities and asymptotic cones A complete metric space X will be said to admit a quadratic isoperimetric inequality for I ( X ) if there exists C > T ∈ I ( X ) with ∂ T = S ∈ I ( X )with M ( S ) ≤ C M ( T ) . The number C will be called isoperimetric constant. The main result of this section is thefollowing theorem which will be needed in the proof of Theorem 1.2 and which should beof independent interest. Theorem 3.1.
Let X be a complete metric length space. If X admits a quadratic isoperi-metric inequality for I ( X ) with isoperimetric constant C then every asymptotic cone X ω ofX admits a quadratic isoperimetric inequality for I ( X ω ) with isoperimetric constant C. For the proof of Theorem 1.2 we will actually only need Theorem 3.1 for X a Carnot groupendowed with a left-invariant Riemannian metric. In this case, the proof can be simplified, STEFAN WENGER see Proposition 3.6. We believe that Theorem 3.1 should be of independent interest. As aconsequence of Theorem 3.1 we obtain the following result.
Corollary 3.2.
Let X be a metric length space which admits a coarse homological isoperi-metric inequality for curves. Then every asymptotic cone X ω of X admits a quadraticisoperimetric inequality for I ( X ω ) . The definition of coarse homological quadratic isoperimetric inequality for curves is givenin [25]. It is a homological analog of the notion introduced in [7, III.H.2].
Proof of Corollary 3.2.
By Proposition 3.2 and Lemma 2.3 of [25], X has a thickening X δ which is a complete metric length space admitting a quadratic isoperimetric inequality for I ( X δ ). By definition, a thickening of X is a metric space which contains X isometricallyand which is at finite Hausdor ff distance from X . Now, since asymptotic cones of X δ and X are the same, Theorem 3.1 shows that every asymptotic cone of X has a quadraticisoperimetric inequality for integral 1-currents. (cid:3) A special case of the above corollary is the following result.
Corollary 3.3.
Let Γ be a finitely presented group with quadratic Dehn function. Thenevery asymptotic cone of Γ admits a quadratic isoperimetric inequality for integral -currents. It is known that every asymptotic cone of a finitely presented group with quadratic Dehnfunction is simply connected, [19]. However, many groups have simply connected asymp-totic cones even though their Dehn function is not quadratic; for example, the asymptoticcone of any Carnot group is simply connected. Theorem 3.1 does not yield simple con-nectedness but instead gives strong metric information about the asymptotic cones.We turn to the proof of Theorem 3.1 for which we need two simple lemmas.
Lemma 3.4.
Let Y be a complete metric length space and D > and suppose Y satisfiesthe following condition: for every closed Lipschitz curve c : [0 , → Y and every finitepartition = t < t < · · · < t m = of [0 , there exist a Lipschitz curve c ′ : [0 , → Yand S ∈ I ( Y ) satisfying c ′ ( t i ) = c ( t i ) and length (cid:0) c ′ | [ t i , t i + ] (cid:1) ≤ length (cid:0) c | [ t i , t i + ] (cid:1) for all i, as well as ∂ S = c ′ [ χ [0 , ℄ and M ( S ) ≤ D length( c ) . Then Y admits a quadratic isoperimetric inequality for I ( Y ) with isoperimetric constant D.Proof.
Let c : [0 , → Y be a closed Lipschitz curve and set T : = c [ χ [0 , ℄ . We will find,for each n ≥
0, closed Lipschitz curves c n , i : [0 , → Y , i = , . . . , n , and S n ∈ I ( Y ) suchthat length( c n , i ) ≤ n length( c ) and M ( S n ) ≤ n − D length( c ) , as well as T = ∂ ( S + · · · + S n ) + n X i = c n , i [ χ [0 , ℄ . ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 9
For n = c , : = c and S : =
0. Suppose we have found S , . . . , S n − and c n − , i for some n ≥ c n , j and S n , weproceed as follows. First, fix i : = = t < t < · · · < t = c n − , i | [ t j , t j + ] ) =
18 length( c n − , i )for j = , . . . ,
8. By assumption, there exist a closed Lipschitz curve c ′ : [0 , → Y and S ∈ I ( Y ) such that c ′ ( t j ) = c n − , i ( t j ) andlength (cid:16) c ′ | [ t j , t j + ] (cid:17) ≤ length (cid:16) c n − , i | [ t j , t j + ] (cid:17) for all j , as well as ∂ S = c ′ [ χ [0 , ℄ and M ( S ) ≤ D length( c n − , i ) . Set S n , : = S and, for j = , . . . ,
8, let c n , j be the concatenation of c n − , i | [ t j , t j + ] with( c ′ | [ t j , t j + ] ) − . It is clear that(3) c n − , [ χ [0 , ℄ = X j = c n , j [ χ [0 , ℄ + ∂ S n , . Now, do the same for i = , . . . , n − in order to obtain, after relabeling indices, curves c n , j , j = , . . . , n , and currents S n , j , j = , . . . , n − . Clearly, we have length( c n , j ) ≤ − n length( c ) and, for S n : = S n , + · · · + S n , n − , we have M ( S n ) ≤ D n − X i = length( c n − , i ) ≤ n − D · n − length( c ) ! = n − D length( c ) . Finally, the fact that T = ∂ ( S + · · · + S n ) + n X i = c n , i [ χ [0 , ℄ follows from (3). This proves the existence of c n , i and S n with the properties stated above.Now, set S : = P ∞ n = S n and note that S is an integer rectifiable 2-current satisfying(4) M ( S ) ≤ ∞ X n = M ( S n ) ≤ D length( c ) . It is not di ffi cult to show that ∂ S = T . Indeed, view Y as a subset of Y ′ : = l ∞ ( Y ). It iswell-known, see [23], that Y ′ has a quadratic isoperimetric inequality for I ( Y ′ ). Let D ′ bethe isoperimetric constant. For each n and i = , . . . , n , there thus exists Q n , i ∈ I ( Y ′ ) with ∂ Q n , i = c n , i [ χ [0 , ℄ and such that M ( Q n , i ) ≤ D ′ length( c n , i ) . Set Q n : = Q n , + · · · + Q n , n ,note that Q n ∈ I ( Y ′ ), M ( Q n ) ≤ D ′ n X i = length( c n , i ) ≤ n D ′ length( c ) , and T = ∂ ( S + · · · + S n ) + ∂ Q n . Since Q n converges to 0 in mass and S + · · · + S n converges in mass to S , their boundaries converge weakly to 0 and ∂ S , respectively. Thisshows that indeed ∂ S = T . Since S satisfies (4) and since c was arbitrary, Lemma 2.3 of[25] shows that Y admits a quadratic isoperimetric inequality for I ( Y ) with isoperimetricconstant 2 D . (cid:3) Lemma 3.5.
Let Z be a compact metric space, m ∈ N , and ω a non-principal ultrafilter on N . Suppose A n ⊂ Z are closed subsets and a n , . . . , a mn ∈ A n for n ≥ . Then there exists asubsequence ( A n j ) such that A n j converges in the Hausdor ff distance to a closed subset ofA : = { lim ω a n : a n ∈ A n for every n } and a in j → lim ω a in as j → ∞ , for all i = , . . . , m.Proof. Since Z is compact, there exist integers m ≤ m < m < . . . and, for each n ∈ N ,a sequence ( a jn ) j ≥ m + ⊂ A n of points such that { a jn : j = , . . . , m i } is 2 − i -dense in A n . Foreach i set a i : = lim ω a in , and denote by C the closure of { a i : i ∈ N } ; clearly C ⊂ A . Definefor each j ∈ N Ω j : = { n : d ( a in , a i ) ≤ − j for i = , . . . , m j } and note that Ω ⊃ Ω ⊃ . . . and that ω ( Ω j ) = j ; in particular, Ω j is not finite.Choose n < n < . . . with n j ∈ Ω j for all j . It follows that A n j converges in the Hausdor ff sense to C . Furthermore, we have that a in j converges to a i as j → ∞ for each i and, inparticular, for i = , . . . , m . This concludes the proof. (cid:3) We now use the lemmas above to prove the main theorem of this section.
Proof of Theorem 3.1.
Let ω be a non-principal ultrafilter on N , let p n ∈ X , and r n > r n → ∞ . Let X ω be the asymptotic cone associated with the pointed sequence ( X , r − n d , p n )and ω . Denote by X n the space X endowed with the metric d n : = r − n d . Let c : [0 , → X ω be a closed Lipschitz curve and 0 = t < t < · · · < t m = , x i : = c ( t i ). Then x i = [( x in )] for some x in ∈ X withsup n d n ( x in , p n ) < ∞ , and we may assume that x mn = x n for all n . Let c n : [0 , → X n be a Lipschitz curve suchthat for all i we have c n ( t i ) = x in and c n | [ t i , t i + ] is parametrized proportional to arc-lengthwith length (cid:0) c n | [ t i , t i + ] (cid:1) ≤ d n ( x in , x i + n ) + mn . Note that sup n Lip( c n ) < ∞ and that there exists R > c n has image in the ball B X n ( p n , R ) for all n . It followsin particular that sup n length( c n ) < ∞ . Set T n : = c n [ χ [0 , ℄ and note that T n ∈ I ( X n )satisfies ∂ T n = M ( T n ) ≤ length( c n ) + / n . Set D : = C where C is the isoperimetricconstant for X . By Lemma 3.4 in [23], see also Theorem 10.6 in [3], there exists S n ∈ I ( X n ) such that ∂ S n = T n and M ( S n ) ≤ D M ( T n ) and such that the sequence ( Y n ) of metric spaces Y n given by Y n : = (spt S n ∪ c n ([0 , ∪ { p n } , d n )is uniformly compact in the sense of Gromov. It thus follows from Gromov’s compactnesstheorem [10] that there exist a compact metric space ( Z , d Z ) and isometric embeddings ϕ n : Y n → Z for all n . Set A n : = ϕ n ( Y n ) and a in : = ϕ n ( c n ( t i )) for i = , . . . , m . ByLemma 3.5 there exists a subsequence ( A n j ) such that A n j converges in the Hausdor ff senseto a closed subset of A : = { lim ω a n : a n ∈ A n for all n } and a in j converges to a i : = lim ω a in forevery i . After possibly passing to a further subsequence, we may assume that ψ j : = ϕ n j ◦ c n j converges uniformly to a Lipschitz curve ψ : [0 , → Z and that ϕ n j S n j converges weakly ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 11 to some S ′ ∈ I ( Z ). The second assertion is a consequence of the closure and compactnesstheorems for integral currents in compact metric spaces, see Theorems 5.2 and 8.5 in [3].Note also that ∂ S ′ = ψ [ χ [0 , ℄ ; furthermore ψ ( t i ) = a i andlength( ψ j ) ≤ n j + m − X i = d Z ( a in j , a i + n j ) . Since a in j → a i and d Z ( a i , a i + ) = d ω ( x i , x i + ), we concludelim sup j →∞ length( ψ j ) ≤ m − X i = d ω ( x i , x i + ) ≤ length( c ) , hence also length( ψ ) ≤ length( c ) and M ( S ′ ) ≤ D lim inf n →∞ M ( ϕ n j T n ) ≤ lim inf n →∞ length( ψ j ) ≤ D length( c ) . Since A n j converges in the Hausdor ff sense to a closed subset of A we furthermore have ψ ([0 , ∪ spt S ′ ⊂ A . Let now Λ : A → X ω be the isometric embedding defined by Λ ( a ) : = [( y n )], where( y n ) ⊂ X is a sequence with y n ∈ Y n and a = lim ω ϕ n ( y n ). It is not di ffi cult to show that Λ iswell-defined and is an isometric embedding. Furthermore, the Lipschitz curve c ′ : = Λ ◦ ψ satisfies c ′ ( t i ) = x i = c ( t i ) and length( c ′ ) ≤ length( c ) . We set S : = Λ S ′ and note that ∂ S = c ′ [ χ [0 , ℄ and M ( S ) ≤ D length( c ) . In view ofLemma 3.4 the proof is complete. (cid:3) As mentioned above, in the special case when X is a Carnot group endowed with a left-invariant Riemannian metric, the proof of Theorem 3.1 can be simplified. In fact, we canprove an analog for higher-dimensional cycles. Let m ≥
1. A complete metric space X issaid to admit an isoperimetric inequality of Euclidean type for I m ( X ) if there exists C > T ∈ I m ( X ) with ∂ T = S ∈ I m + ( X ) with M ( S ) ≤ C M ( T ) m + m . If the above holds only for T with spt T compact, then X will be said to admit an isoperi-metric inequality of Euclidean type for compactly supported integral m -currents.Let G be a Carnot group and d the metric on G coming from a left-invariant Riemannianmetric, and let d c be the associated Carnot-Carath´eodory metric. Proposition 3.6.
Let m ≥ . If X : = ( G , d ) admits an isoperimetric inequality of Eu-clidean type for I m ( X ) then Y : = ( G , d c ) admits an isoperimetric inequality of Euclideantype for compactly supported integral m-currents. Note that in the case m =
1, Proposition 3.6 together with [25, Lemma 2.3] yield a qua-dratic isoperimetric inequality for I ( Y ), that is, also for integral 1-currents whose supportsare not compact. Proof.
Let T ∈ I m ( Y ) with spt T compact and ∂ T =
0. Denote by ϕ : Y → X the identitymap and note that ϕ is 1-Lipschitz. For each n ≥ T n : = ( ϕ ◦ δ n ) T ∈ I m ( X ). Clearly,we have ∂ T n = M ( T n ) ≤ n m M ( T ) . By Lemma 3.4 in [23] (see also Theorem 10.6 in[3]), there exists S n ∈ I m + ( X ) such that ∂ S n = T n , M ( S n ) ≤ C M ( T n ) m + m ≤ Cn m + M ( T ) m + m , and k S n k ( B ( x , r )) ≥ C ′ r m + for all x ∈ spt S n and 0 ≤ r ≤ d ( x , spt T n ). Here, C and C ′ are constants only dependingon the isoperimetric constant for I m ( X ). It follows that there exists L such that spt S n ⊂ B ( e , nL ) for every n , where e is the identity in G . Define metric spaces by Y n : = (spt S n , n d ) . It follows that ( Y n ) is uniformly compact and thus, by Gromov’s compactness theorem,there exists a compact metric space Z and isometric embeddings ψ n : Y n → Z for every n . Define subsets A n : = ψ n (spt S n ) of Z and maps ̺ n : = ψ n ◦ ϕ ◦ δ n : (spt T , d c ) → Z .After possibly passing to a subsequence, we may assume that ( A n ) converges to a closedsubset A ⊂ Z in the Hausdor ff sense and ̺ n converges uniformly to a 1-Lipschitz map ̺ : (spt T , d c ) → Z . After possibly passing to a further subsequence, we may assumeby the compactness and closure theorems (see Theorems 5.2 and 8.5 in [3]) that ψ n S n converges weakly to some ˆ S ∈ I m + ( Z ). Let ω be a non-principal ultrafilter on N anddefine a map η : A → Y as follows. Given a ∈ A , let x n ∈ spt S n such that ψ n ( x n ) → a , anddefine η ( a ) : = lim ω δ n ( x n ) . It is not di ffi cult to show that η is well-defined, an isometric embedding, and satisfies η ◦ ̺ = id spt T . Set S : = η ˆ S and note that ∂ S = T as well as M ( S ) ≤ lim inf n →∞ n m + M ( S n ) ≤ C M ( T ) m + m . This concludes the proof. (cid:3)
4. A lower bound for the filling area function of C arnot groups of step G be a Carnot group of step 2 with grading g = V ⊕ V of its Lie algebra g and Liebracket [ · , · ]. If U is a subspace of V then [ · , · ] naturally induces a Lie bracket [ · , · ] U on g U = V ⊕ V ′ , where V ′ : = V / U is the quotient space. Let G U be the Carnot group whoseLie algebra is g U . Note that every connected and simply connected nilpotent Lie group ofstep 2 is of the form G U with g = V ⊕ V a free nilpotent Lie algebra of step 2 and U ⊂ V a suitable subspace. We will use the following terminology. Definition 4.1.
Let m ≥ . A non-trivial subspace U of V is called m-inaccessible if thereexists a proper subspace U ′ of U, possibly U ′ = { } , such that if v , . . . , v m , w , . . . , w m ∈ V then the vector [ v , w ] + · · · + [ v m , w m ] is contained in U if and only it is contained in U ′ . It is not di ffi cult to give examples of graded nilpotent Lie algebras g = V ⊕ V such that V has a non-trivial m -inaccessible subspace. Example 4.2.
Let k , m satisfy k ≥ m +
1) and let g = V ⊕ V be the free nilpotent Liealgebra of step 2 with dim V = k . Then there exist a basis { e , . . . , e k } of V and a basis { e i , j : 1 ≤ i < j ≤ k } of V such that the Lie bracket on g satisfies [ e i , e j ] = e i , j whenever i < j . It is straightforward to check that the one-dimensional subspace U : = span { e , + e , + · · · + e m + , m + } ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 13 of V is m -inaccessible. One may in fact take U ′ = { } . Note that here g U has a basis withrational structure constants.More generally, if g = V ⊕ V is a stratified nilpotent Lie algebra of step 2 with(5) dim V > (2 dim V − m then V possesses an m -inaccessible subspace U ; furthermore, U may be chosen such that g U has a basis with rational structure constants. Indeed, the subset C m ⊂ V given by C m : = { [ v , w ] + · · · + [ v m , w m ] : v i , w i ∈ V } is a cone containing 0 and is the image of a smooth map ψ : ( R × S k × S k ) m → V , where k = dim V − C m has Hausdor ff dimension at most (2 dim V − m andhence C m , V as soon as (5) holds. Clearly, for v ∈ V \ C m , the subspace U : = span { v } is m -inaccessible.We turn to the main result of this section. Theorem 4.3.
Let G be a Carnot group of step with grading g = V ⊕ V of its Liealgebra. Suppose U ⊂ V is an m-inaccessible subspace where m ≥ . Let G U be theCarnot group of step whose Lie algebra is g U = V ⊕ V ′ , where V ′ : = V / U, and letH : = G U × Z · · · × Z G U be the central product of m copies of G U . Then, endowed with a left-invariant Riemannianmetric, H has filling area function which grows strictly faster than quadratically: FA( r ) r → ∞ as r → ∞ . For the definition of FA( r ) see Section 2.1. Recall that the central product of m copies ofa group Γ is the quotient of the m -fold direct product of Γ by the normal subgroup N oftuples ( g , . . . , g m ) with g i ∈ [ Γ , Γ ] and g · · · g m = e , where e is the identity element of Γ .In order to prove Theorem 4.3 we will actually show that there exists a closed Lipschitzcurve in ( H , d c ) which does not bound an integral 2-current in ( H , d c ), where d c is theCarnot-Carath´eodory distance associated with the left-invariant Riemannian metric. Thesuper-quadratic growth of FA( r ) then follows from Theorem 3.1 or Proposition 3.6.From Theorem 4.3 and the fact that central products of finitely generated nilpotent groupsof step 2 have Dehn function bounded above by n log n , see [16, 27], we obtain: Corollary 4.4.
Let G U and H be as in Theorem 4.3, for some m ≥ , and suppose the Liealgebra of G U has a basis with rational structure constants. Then the filling area and Dehnfunctions of H satisfy (6) r ̺ ( r ) ≤ FA( r ) ≤ FA ( r ) ≤ Cr log rfor all r ≥ , where ̺ is a function satisfying ̺ ( r ) → ∞ as r → ∞ . Note that if g and U are as in Example 4.2 for some m ≥ G U andits m -th central power H = G U × Z · · · × Z G U satisfy the hypotheses of Corollary 4.4.Consequently, the filling area and Dehn functions of H satisfy (6). This answers in thenegative the question raised in [27] whether the Carnot group Γ = G U × Z G U , where g and U are as in Example 4.2 with k =
10 and m =
2, has quadratic Dehn function.
Proof of Corollary 4.4.
The lower bound for FA( r ) in (6) comes from Theorem 4.3. Inorder to prove the upper bound for FA ( r ) let g U denote the Lie algebra of G U with grading g U = V ⊕ V ′ . Since g U has a basis with rational structure constants, there exists a basis of vectors in V which generate a lattice Γ in G U . Now, there exists an injective homomor-phism from the central product Γ ′ : = Γ × Z · · · × Z Γ of m copies of Γ to H whose image is alattice in H . Therefore, the Dehn function δ Γ ′ ( n ) of Γ ′ satisfies δ Γ ′ ( n ) ∼ FA ( n ) . Since Γ ′ is a central product of m ≥ δ Γ ′ ( n ) (cid:22) n log n . This was first proved by Ol’shanskii-Sapir using the techniques of [16],see the remark on page 927 in [16]. More recently, Young gave a proof of this in [27]using di ff erent methods. It follows that there exists a suitable constant C such that FA ( r )satisfies FA ( r ) ≤ Cr log r for all r ≥
2. This concludes the proof. (cid:3)
Theorem 1.2 is a direct consequence of (the proof) of Corollary 4.4. Indeed, for the finitelygenerated Γ ′ : = Γ × Z · · · × Z Γ appearing in the above proof, we have n ̺ ( n ) (cid:22) δ Γ ( n ) (cid:22) n log n for some function ̺ with ̺ ( n ) → ∞ as n → ∞ .4.1. The proof of Theorem 4.3.
In the proof of the theorem it will often be useful toidentify a given Carnot group G via the exponential map with ( g , ∗ ), where g is the Liealgebra of G (viewed as a vector space) and ∗ is the multiplication on g given by the Baker-Campbell-Hausdor ff formula. The Lie algebra of ( g , ∗ ) is g and the exponential map issimply the identity map on g . If G is of step 2 then ∗ is given by v ∗ v ′ = v + v ′ +
12 [ v , v ′ ] . Proof of Theorem 4.3.
Let G , g = V ⊕ V , U , g U , G U , and H satisfy the hypotheses ofthe theorem. Suppose U ′ ⊂ U is as in Definition 4.1 and let U ′′ ⊂ U be a subspacecomplementary to U ′ , thus U = U ′ + U ′′ and U ′ ∩ U ′′ = { } . Let u ∈ U ′′ with u , v , . . . , v k ∈ V with u = [ v , v ] + [ v , v ] + · · · + [ v k − , v k ] , where [ · , · ] is the Lie bracket on g . Define, for every j = , . . . , k , a piecewise a ffi ne curve c j : [0 , → V by c j ( t ) = tv j − ≤ t < v j − + ( t − v j ≤ t < − t ) v j − + v j ≤ t < − t ) v j ≤ t ≤ , and let c : [0 , → V be the concatenation of the curves c j , j = , . . . , k , parametrizedon [0 , c is closed and piecewise a ffi ne. Define ˆ c : [0 , → V byˆ c ( t ) : = Z t [ c ( s ) , ˙ c ( s )] ds and observe that ˆ c (1) = u . Let now W : = V ⊕ · · · ⊕ V be the direct sum of m copies of V . Define quotient spaces W : = V / U and W ′ : = V / U ′ and denote by P : V → W and P ′ : V → W ′ the natural projections. The Lie bracket ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 15 [ · , · ] on g gives rise to the bilinear map W × W → V , denoted by the same symbol [ · , · ],defined by(7) [ w , w ′ ] = [ v , v ′ ] + · · · + [ v m , v ′ m ] , where w , w ′ ∈ W are of the form w = v + · · · + v m and w ′ = v ′ + · · · + v ′ m with v i and v ′ i inthe i -th copy of V . Clearly, H has Lie algebra h = W ⊕ W with Lie bracket[ w + w , w ′ + w ′ ] h : = P ([ w , w ′ ]) , where w i , w ′ i ∈ W i for i = ,
2. It follows from the m -inaccessibility of U that for any w , w ′ ∈ W we have(8) [ w , w ′ ] h = P ′ ([ w , w ′ ]) = . Identifying V with the first of the m copies of V in W , we may view c as a curve in W .Define a curve c : = [0 , → H by c : = c + P ◦ ˆ c and note that c (1) = c (1) + P ( u ) = k · k be a Euclidean norm on W and let d be the distance coming from a left-invariantRiemannian metric on H which, restricted to W ⊂ T H , induces k · k . Let furthermore d c be the Carnot-Carath´eodory metric on H associated with k · k . Set Y : = ( H , d c ). Clearly, c is a closed Lipschitz curve in Y . Set T : = c [ χ [0 , ℄ . We claim that there does not exist S ∈ I ( Y ) with ∂ S = T . We argue by contradiction and assume such an S ∈ I ( Y ) with ∂ S = T exists. By [3, Theorem 4.5], S is of the form S = ∞ X i = ϕ i [ θ i ℄ for some Lipschitz maps ϕ i : K i → Y with K i ⊂ R compact and θ i ∈ L ( K i , Z ). Denote by π : Y → ( W , k·k ) the projection given by π ( w ) : = w for w = w + w with w i ∈ W i , i = , H is identified with W ⊕ W . It is straightforward to check that π is 1-Lipschitzand Pansu-di ff erentiable at every point y ∈ Y with Pansu-di ff erential d P π ( y ) : H → W given by(9) d P π ( x )( w ) = w for w = w + w with w i ∈ W i , i = ,
2. Set ¯ S : = π S and note that ¯ S ∈ I (( W , k · k )) with ∂ ¯ S = π T = c [ χ [0 , ℄ ;furthermore ¯ S = ∞ X i = ψ i [ θ i ℄ where ψ i : = π ◦ ϕ i . Fix i . We claim that at almost every x ∈ K i (10) [ d x ψ i ( u ) , d x ψ i ( v )] h = u , v ∈ R , where d x ψ i is the classical derivative (which exists almost everywhereby Rademacher’s theorem). Indeed, by Pansu’s Rademacher-type theorem [18] and itsgeneralization to Lipschitz maps defined only on measurable sets, see [22] and [14], thePansu-di ff erential d P ϕ i ( x ) : R → H of ϕ i at x exists for almost every x ∈ K i and is a Liegroup homomorphism, equivariant with respect to the dilations. Viewed as a map betweenLie algebras, d P ϕ i ( x ) : R → h is a Lie algebra homomorphism and therefore(11) h d P ϕ i ( x )( u ) , d P ϕ i ( x )( v ) i h = d P ϕ i ( x )([ u , v ] R ) = u , v ∈ R . Since π is Pansu-di ff erentiable at ϕ i ( x ), the chain rule yields d x ψ i = d P π ( ϕ i ( x )) ◦ d P ϕ i ( x ) . This together with (9) and (11) yields (10), as claimed. It now follows from (10) and (8)that for almost every x ∈ K i and all u , v ∈ R (12) P ′ ([ d x ψ i ( u ) , d x ψ i ( v )]) = . Finally, let { ξ , . . . , ξ n } be a basis for W and let π j : W → R be the correspondingcoordinate functions, that is, π j ( r ξ + · · · + r n ξ n ) = r j , for j = , . . . , n . Let furthermore Q : W ′ → R be a linear functional and set Q ′ : = Q ◦ P ′ . Define functions f j : W → R by f j ( x ) : = Q ′ ([ x , ξ j ]) . Clearly, the functions f j and π j are Lipschitz when W is equipped with the norm k · k .Furthermore, f j is bounded on spt ∂ ¯ S . We calculate Z Q ′ ([ c ( t ) , ˙ c ( t )]) dt = n X j = ∂ ¯ S ( f j , π j ) = n X j = ¯ S (1 , f j , π j ) = ∞ X i = n X j = Z K i θ i ( x ) det (cid:16) ∇ (( f j , π j ) ◦ ψ i )( x ) (cid:17) dx . Since, by an easy computation and (12), n X j = det (cid:16) ∇ (( f j , π j ) ◦ ψ i )( x ) (cid:17) = Q ′ ([ d x ψ i ( e ) , d x ψ i ( e )]) = x ∈ K i , where e and e are the standard basis vectors of R , we obtain Z Q ′ ([ c ( t ) , ˙ c ( t )]) dt = . Since Q was arbitrary this shows that P ′ (ˆ c (1)) = , a contradiction since P ′ (ˆ c (1)) = P ′ ( u ) ,
0. This shows that there does not exist S ∈ I ( Y ) with ∂ S = T . In particular, Y does not admit a quadratic isoperimetric inequality for I ( Y ). Since, by Pansu’s result The-orem 2.1, the unique asymptotic cone of X : = ( H , d ) is Y it follows from Theorem 3.1 that X does not admit a quadratic isoperimetric inequality for I ( X ). This follows alternativelyfrom Proposition 3.6. Consequently, by Lemma 2.3 in [25], the filling area function FA( r )of X cannot be bounded by Cr for any C . This completes the proof. (cid:3)
5. A nother lower bound
In this section we use similar arguments as above to prove super-quadratic lower boundsfor the growth of the filling area functions for another class of Carnot groups. We prove:
Theorem 5.1.
Let G be a Carnot group of step k, endowed with a left-invariant Riemann-ian metric. Let g = V ⊕ · · · ⊕ V k be a grading of the Lie algebra g of G. If V does not contain a -dimensional subalgebrathen the filling area function FA( r ) of G grows strictly faster than quadratically: FA( r ) r → ∞ as r → ∞ . ILPOTENT GROUPS WITHOUT EXACTLY POLYNOMIAL DEHN FUNCTION 17
Proof.
By Pansu’s result, Theorem 2.1, the unique asymptotic cone of X : = ( G , d ) is Y : = ( G , d c ), where d c is the associated Carnot-Carath´eodory distance. Since Y is geodesicand is not a metric tree, it follows for example from Proposition 3.1 in [26] that there exists T ∈ I ( Y ) with ∂ T = T ,
0. Suppose there exists S ∈ I ( Y ) with ∂ S = T . Then k S k is concentrated on a countably H -rectifiable subset A ⊂ Y and is absolutely continuouswith respect to H . Since Y is purely 2-unrectifiable by [15] (see also [2] for the case of thefirst Heisenberg group) it follows that H ( A ) = S =
0. As a consequence,we obtain that T = ∂ S =
0, a contradiction. It thus follows that there exists no S with ∂ S = T and, in particular, Y cannot admit a quadratic isoperimetric inequality for I ( Y ).Theorem 3.1 therefore shows that X does not admit a quadratic isoperimetric inequality for I ( X ). By Lemma 2.3 in [25], the filling area function FA( r ) of X cannot be bounded by Cr for any C . (cid:3) Simple examples of Carnot groups satisfying the hypotheses of Theorem 5.1 are the firstHeisenberg group and its generalizations using quaternions and octonions. It is knownthat the first Heisenberg group and the quaternionic Heisenberg group have cubic Dehnfunction, see [20] for the quaternionic case. In [20] it is also claimed that the octonionicHeisenberg group has cubic Dehn function. As was pointed out in [13] there is a signerror in the proof of the octonionic case, so that the best previously known lower bound isquadratic. From Theorem 5.1 we obtain a super-quadratic lower bound:
Corollary 5.2.
The filling area function
FA( r ) of the octonionic Heisenberg group en-dowed with a left-invariant Riemannian metric grows strictly faster than quadratically. For completeness, we recall the definition of the quaternionic and octonionic Heisenberggroup. Denote by C the complex numbers, by H the quaternions, and by O the octonions.Recall that using the Cayley-Dickson construction, one obtains K : = C from K : = R , K : = H from K , and K : = O from K as follows: for a ∈ K define the conjugate a ∗ of a by a ∗ : = a and the imaginary part of a by Im( a ) : =
0. Suppose that for some i ≥ K i − has been defined together with multiplication, the conjugate a ∗ of a , and the imaginary partIm( a ). Set K i : = { ( a , b ) : a , b ∈ K i − } and define multiplication on K i by( a , b )( c , d ) : = ( ac − db ∗ , a ∗ d + cb ) . For ( a , b ) ∈ K i , define the conjugate by( a , b ) ∗ : = ( a ∗ , − b )and the imaginary part by Im( a , b ) : = (Im( a ) , b ) . Now, set L i : = Im( K i ) and define a stratified nilpotent Lie algebra g i over R of step 2 by g i : = K i ⊕ L i , where the Lie bracket on g i is defined by[ z ⊕ z ′ , w ⊕ w ′ ] : = Im( zw ∗ ) ∈ L i . It is not di ffi cult to check that for z , w ∈ K i we have Im( zw ∗ ) = λ ∈ R such that w = λ z . In particular, the first layer K i of g i does not contain a2-dimensional subalgebra. R eferences [1] D. Allcock: An isoperimetric inequality for the Heisenberg groups , Geom. Funct. Anal. 8 (1998), no. 2,219–233.[2] L. Ambrosio, B. Kirchheim:
Rectifiable sets in metric and Banach spaces , Math. Ann. 318 (2000), 527–555.[3] L. Ambrosio, B. Kirchheim:
Currents in metric spaces , Acta Math. 185 (2000), no. 1, 1–80.[4] G. Baumslag, A. Myasnikov, V. Shpilrain:
Open problems in combinatorial group theory. Second edition ,Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002.[5] G. Baumslag, C. F. Miller III, H. Short:
Isoperimetric inequalities and the homology of groups , Invent.Math. 113 (1993), no. 3, 531–560.[6] N. Brady, T. Riley, H. Short:
The geometry of the word problem for finitely generated groups , AdvancedCourses in Mathematics CRM Barcelona, Birkh¨auser-Verlag, 2007.[7] M. R. Bridson, A. Haefliger:
Metric Spaces of Non-Positive Curvature , Grundlehren der mathematischenWissenschaften 319, Springer, 1999.[8] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, W. P. Thurston:
Word processingin groups , Jones and Bartlett Publishers, Boston, MA, 1992.[9] S. M. Gersten, D. F. Holt, T. R. Riley:
Isoperimetric inequalities for nilpotent groups , Geom. Funct. Anal.13 (2003), 795–814.[10] M. Gromov:
Groups of polynomial growth and expanding maps , Inst. Hautes Etudes Sci. Publ. Math. No.53 (1981), 53–73.[11] M. Gromov:
Filling Riemannian manifolds , J. Di ff . Geom. 18 (1983), 1–147.[12] M. Gromov: Asymptotic invariants of infinite groups , in Geometric group theory, Vol. 2 (Sussex, 1991),1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993.[13] E. Leuzinger, Ch. Pittet:
On quadratic Dehn functions , Math. Z. 248 (2004), no. 4, 725–755.[14] V. Magnani: Di ff erentiability and Area formula on stratified Lie groups , Houston J. Math., vol. 27 (2001),no. 2, 297–323.[15] V. Magnani: Unrectifiability and rigidity in stratified groups , Arch. Math. 83 (2004), 568–576. by KluwerAcademic, Dordrecht, 1991.[16] A. Yu. Ol’shanskii, M. Sapir:
Quadratic isoperimetric functions of the Heisenberg groups. A combinato-rial proof , Jour. of Math. Sciences, 93 (1999), no. 6, 921–927.[17] P. Pansu:
Croissance des boules et des g´eodesiques ferm´ees dans les nilvari´et´es , Ergod. Th. & Dynam.Sys. (1983), no. 3, 415–445.[18] P. Pansu:
M´etriques de Carnot-Carath´eodory et quasiisom´etries des espaces symmetriques des rang un ,Annals of Math., 129 (1989), 1–60.[19] P. Papasoglu:
On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality , J. Di ff .Geom. 44 (1996), no. 4, 789–806.[20] Ch. Pittet: Isoperimetric inequalities for homogeneous nilpotent groups , Geometric Group Theory, 159 –164, Ohio State Univ. Math. Res. Inst. Publ. 3, de Gruyter, Berlin 1995.[21] Ch. Pittet:
Isoperimetric inequalities in nilpotent groups , J. London Math. Soc (2) 55 (1997), 588–600.[22] S. K. Vodopyanov, A. D. Ukhlov:
Approximately di ff erentiable transformations and change of variableson nilpotent groups Sib. Math. Journal, 37 (1996), 62–78.[23] S. Wenger:
Isoperimetric inequalities of Euclidean type in metric spaces , Geom. Funct. Anal. 15 (2005),no. 2, 534–554.[24] S. Wenger:
Flat convergence for integral currents in metric spaces , Calc. Var. Partial Di ff erential Equa-tions 28 (2007), no. 2, 139–160.[25] S. Wenger: Gromov hyperbolic spaces and the sharp isoperimetric constant
Invent. Math. 171 (2008), no.1, 227–255.[26] S. Wenger:
Characterizations of metric trees and Gromov hyperbolic spaces , Math. Res. Lett. 15 (2008),no. 5, 1017–1026.[27] R. Young:
Scaled relators and Dehn functions for nilpotent groups , preprint 2006.D epartment of M athematics , U niversity of I llinois at C hicago , 851 S. M organ S treet , C hicago , IL 60607–7045 E-mail address ::