Higher divergence for nilpotent groups
Moritz Gruber









Higher divergence for nilpotent Lie groups
    Moritz Gruber [The author is supported by the German Research Foundation (DFG) grant GR 5203/1-1]
    
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Abstract.  The higher divergence of a metric space describes its isoperimetric behaviour at infinity. It is closely related to the higher-dimensional Dehn functions, but has more requirements to the fillings. We prove that these additional requirements do not have an essential impact for many nilpotent Lie groups. As a corollary, we obtain the higher divergence  of the Heisenberg groups in all dimensions.







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§ INTRODUCTION


Thales' theorem of intersecting lines is one the classical theorems in Euclidean geometry. In the words of a modern geometer, it says that geodesic rays in the Euclidean space diverge from each other linearly. As asymptotic property, the speed of the divergence of geodesics is an interesting object to study in many metric spaces. There is a way to describe it by the needed length to connect two points on the boundary of a ball of radius r by a curve which do not intersect the interior of the ball. This can be considered as an isoperimetric inequality and hence generalised to higher dimensions: fill k-cycles outside a ball with (k+1)-chains also outside the ball.  
So far, this higher divergence has mainly been studied for spaces with non-positive curvature (see <cit.>, <cit.>, <cit.>, <cit.>).

In this article we examine the higher divergence of nilpotent Lie groups with Riemannian metrics. These are metric spaces with all three types of sectional curvature: negative, zero and positive (see <cit.>). We find out that for a class of nilpotent Lie groups the behaviour in low dimensions are the same as for the filling functions. As a corollary, we obtain the higher divergence of the Heisenberg groups H^n_ in all dimensions. As far as we know, this is the first infinite family of metric spaces, not quasi-isometric to non-positively curved spaces, for which the higher divergence is known in all dimensions.

  

Let n be a positive integer and G be stratified nilpotent Lie group with a uniform lattice Γ⊂ G with s_2(Γ) ⊂Γ. We call G n-approximable if there are biLipschitz triangulations (τ, f) of G and (η,g) of G × [0,1] and a n-horizontal, Γ-equivariant map ϕ: τ→ G and a n-horizontal, s_2(Γ)-equivariant map ψ: η→ G such that:


  * ϕ is in finite distance to f, and

  * there are isomorphisms ι_i: g^-1(G × i) → (τ, s_2^i∘ f) such that g ∘ι_i = s_2^i∘ f and ψ∘ι_i = s_2^i∘ϕ for i ∈{0,1}.



A rich class of such n-approximable nilpotent Lie groups is formed by the Jet groups J^m(^n) (compare <cit.>,<cit.>) which include the Heisenberg group H^n_ as J^1(^n).



  

Let G be a stratified nilpotent Lie group equipped with a left-invariant Riemannian metric and let m ∈ be such, that G is (k+1)-approximable. Then:

    Div_G^k(r) ≼ r^k+1

and if there is a number K ∈ such that for all Δ∈η^(k+2) holds (s_t(ψ(Δ))) ≾ t^K+2, then

    Div_G^k+1(r) ≼ r^K+2.


  


As the complex Heisenberg group H^n_ is k-approximable for k ≤ n (see <cit.>), the Main Theorem combined with the results on the higher divergence from <cit.> yields:

  

For the (2n+1)-dimensional complex Heisenberg group H^n_ equipped with a left-invariant Riemannian metric 


    Div_H^n_^k (r)∼ r^k+1 for  1≤ k <n 
     
    											r^n+2 for  k=n 
     
    											r^k(k+2)/k+1 for  n< k <2n



 Using the Main Theorem, the growth of the k-dimensional divergence follows by 
<cit.> for k<n, by <cit.> for k=n and 
by <cit.> for k>n.


Therefore, we now know the higher divergence in all dimensions for the complex Heisenberg groups. 

  

Using the results of <cit.> and <cit.>, we get the corresponding behaviour for the divergence of the quaternionic and octonionic Heisenberg groups H_^n and H_^n in the dimensions ≤ n and codimensions <n. Only for the octonionic Heisenberg groups H_^n we haven't the lower bound for the divergence in dimension n, because of the missing lower bound for the filling function in this dimension.




  

The author is pleased to thank Robert Young for the many enlightening discussions during the development of this article and to thank Enrico Leuzinger for initially drawing the authors attention to this topic.










 §.§ Illustration of our approach


We illustrate the idea behind the proof of our Main Theorem in the case of the Euclidean plane ^2. To be more precise, we show how to construct an ρ r-avoidant 1-chain with a prescribed r-avoidant boundary by using an approximation technique. This technique will generalise to nilpotent Lie groups and higher dimensions (see section <ref>).


Let τ be the grid of squares of side length 1 with vertices in ^2 and let τ_i=2^iτ for i ∈_0 be the scaled grid of squares with side length 2^i, and let r>1. 



0.3

    < g r a p h i c s >

  


0.65
Given a 0-cycle B=a-b with d(B, 0)≥ r, one approximates it by B_0=a_0-b_0 in the 0-skeleton of the grid τ.

Then one approximates B in the grid τ_1 by B_1=a_1-b_1. And further on by 0-cycles B_i=a_i-b_i in the 0-skeleton of the grids τ_i.

After finitely many steps, latest for i_o with  2^i_o-2≤ d(a,b) < 2^i_o-1, the points a and b will be approximated by the same point and therefore the cycle B_i_o is trivial.





0.3

    < g r a p h i c s >




0.65
Having all the above approximations of the 0-cycle B, one can start to construct the filling. First, one connects the original points a and b by straight lines with their approximations a_0 and b_0. These lines are not longer than half the diameter of a square in τ, i.e. at most √(2)/2. This is independent of the cycle.

Then one connects two consecutive approximations B_i and B_i+1 by edges in the 1-skeleton of τ_i. One never needs more than two such edges to connect a_i with a_i+1 and b_i with b_i+1, and hence ends up with a 1-chain F connecting a and b and with 


    (F)≤√(2)/2 + ∑ _i=0^i_o-1 2· 2^i ≤√(2)/2 + 2^i_o+1≤√(2)/2 + 8 · d(a,b) ∼ d(a,b).




  
  


The crux for our purpose is, as seen above, that even when the cycle to fill is far away from the base point 0, the constructed filling may use edges containing the base point. To avoid this, we replace the base point when ever it is used in the approximation by one of its neighbours x_1,x_2,x_3,x_4 in the 0-skeleton of the respective τ_i, say x_1, and replace each edge connecting 0 with x_j,  j∈{1,2,3,4} by a 1-chain, not passing through 0, connecting x_1 with the same x_j.







We have to replace only finitely many edges, here 4, and therefore get a maximal number M of edges used in a replacement chain, here 4, too.
  




0.3

    < g r a p h i c s >




0.65
If we now replace all the edges used in F which contain 0, we obtain a new 1-chain F connecting a and b. As we never substitute an edge by more than M=4 edges, all of the same length as the original edge, we get


    (F) ≤ M ·(F) ∼ d(a,b).


Furthermore, the used simplices of τ_i will always stay in the (2^i√(2))-neighbourhood of B and will never be nearer to 0 than 2^i. Hence 


    d(F, 0) ≥min{max{d(B, 0)-2^i√(2), 2^i}| i ∈_0 }≥1/1+√(2) r.
 












§ PRELIMINARIES





 §.§ Higher Divergence



Let X be a Riemannian manifold or a simplicial complex with base point x_o ∈ X. For r>0 we call a Lipschitz chain b in X r-avoidant if the image of b has trivial intersection with the r-ball around x_o. The higher divergence gives a measure of the difficulty to fill r-avoidant Lipschitz cycles by almost r-avoidant Lipschitz chains. We now give a brief definition of it, for more details see <cit.>, <cit.> and <cit.>. 



  

For ρ∈ (0,1] and α >0 we set


    div^m_X,ρ,α(r) sup{inf{(b) |∂ b=a,  b ρ r-avoidant}| a r-avoidant m-cycle, (a)≤α r^m  }.


The m-dimensional divergence of X is the 2-parameter family


    Div^m_X {div^m_X,ρ,α}_ρ,α


with ρ∈ (0,1] and α >0.

Usually one wants to see the higher divergence as a quasi-isometry invariant. For this purpose one considers its equivalence class by the relation defined in the following.

  

Let F={f_s,t} and H={h_s,t} be two 2-parameter families indexed over s ∈ (0,1] and t>0 with increasing functions f_s,t, h_s,t: ^+ →^+∪{∞}. For fixed m ∈ we write F ≼_m H if there are thresholds s_o and t_o, and constants L,M ≥ 1 such that for all s≤ s_o and all t≥ t_o there is a constant A=A(s,t) ≥ 1 with


    f_s,t(x) ≤ Ah_Ls,Mt(Ax+A) + O(x^m)   ∀ x ∈^+.


If F ≼_m H and H ≼_m F, we write F ∼_m H. This defines an equivalence relation.


In the case of the m-dimensional divergence we will use the relations ≼_m and ∼_m. For brevity we will omit the index and only write ≼ and ∼. For the same reason we denote the 2-parameter family F={f_s,t} consisting of the same function f for all indices, i.e. f_s,t=f for all s,t, shortly by this function f. Further we will use for functions f,h : ^+ →^+∪{∞} the notation f≾ h when there is a constant C>0 such that f(x) ≤ Ch(x)+Cx+C for all x∈_+.



  

The relation ≼ (and consequently ∼) only captures the asymptotic behaviour of the functions for r →∞. 
Let h:_+ →_+∪{∞} be an increasing function. If  r_o≥1 and div^m_X,ρ,α(r_o)≤ h(r_o) then div^m_X,ρ,α(r)≤ r_o· h(r_o r+ r_o) for all r ≤ r_o, because both sides are increasing. So we need to examine the relation ≼ (and consequently ∼)  only for r larger than an arbitrary constant r_o=r_o(ρ,α)≥ 1.






 §.§ Stratified nilpotent Lie groups



We call a simply connected, d-step nilpotent Lie group G with Lie algebra g stratified if there are subspaces V_i ⊂ g with


    g = V_1 ⊕ V_2 ⊕ ... ⊕ V_d   and   [V_j, V_1]=V_j+1


where V_k={0} for all k>d. For every stratified nilpotent Lie group G there is a family {ŝ_t}_t>0 of Lie algebra automorphisms ŝ_t:  g → g, (v_1,v_2,...,v_d) ↦ (tv_1,t^2v_2,...,t^dv_d). These induce (uniquely determined) Lie group automorphisms s_t : G → G with differential ŝ_t at the neutral element 1 ∈ G, the so called scaling automorphisms or dilations. A special role in the geometry of stratified nilpotent Lie groups plays the horizontal distribution


    H ⋃_g ∈ GdL_g V_1 ⊂ TG.


Let G be equipped with a left-invariant Riemannian metric. A Lipschitz map f:M → G from a Riemannian manifold M to G is horizontal if its image is almost everywhere tangent to H. By this, one can define the Carnot-Carathéodory metric d_c(·,·) as the length metric of horizontal curves with the Riemannian length. This is a left-invariant metric on G with 


    d_c(s_t(x),s_t(y))=t · d_c(x,y)   ∀ x,y ∈ G ∀ t>0.


If we denote the Riemannian distance by d(·,·), one has d(x,y)≤ d_c(x,y) for all x,y ∈ G. Further, both metrics induce the same topology on G. This leads to the following useful estimate.

  

Let G be a stratified nilpotent Lie group equipped with a left-invariant Riemannian metric and let d(·,·) be the induced length metric and let r>0. Then there is a constant C_r>0 such that:


    d(s_t(x),s_t(y))≤ t · C_r · r   ∀ t >0 ∀ x,y ∈ G  with  d(x,y)≤ r.


In particular, for all x,y ∈ G  with d(x,y)=r this implies


    d(s_t(x),s_t(y))≤ t · C_r · d(x,y)   ∀ t >0.




Let d_c(·,·) be the corresponding Carnot-Carathéodory metric. 

We have x∈ B_r(y) {z ∈ G | d(z,y)≤ r}. 
As B_r(y) is compact and z ↦ d_c(z,y) is continuous, we have 


    C_rmax{d_c(z,y)/r| z ∈ B_r(y)} < ∞.


This implies

    d(s_t(x),s_t(y))≤ d_c(s_t(x),s_t(y))=t · d_c(x,y)=t ·r/r· d_c(x,y)=t ·d_c(x,y)/r· r≤ t · C_r · r

for all y ∈ G and all x ∈ B_r(y).








§ FILLING BY APPROXIMATION IN NILPOTENT GROUPS 


Young developed in <cit.> a technique for nilpotent Lie groups to construct fillings by approximation. In this section we give a reminder of this construction. In the second half of the section we will introduce a modification of it which allows us to avoid simplices near the base point.


A main ingredient for Young's construction is the following approximation theorem. 



Let X be a polyhedral complex with finitely many isometry types of cells. Then there is a constant C_X>0 depending only on X, such that for every Lipschitz k-chain b, whose boundary ∂ b is a polyhedral (k-1)-chain, there is a polyhedral k-chain P_X(b) and a Lipschitz (k+1)-chain Q_X(b) with:

	
  * (P_X(b))≤ C_X ·(b),				

	
  * (Q_X(b))≤ C_X ·(b),				

	
  * ∂ Q_X(b)=b-P_X(b).				

Furthermore, P_X(b) and Q_X(b) are contained in the smallest subcomplex of X which contains b.




In the following, we start with a recapitulation of the notation and the filling technique from <cit.>. For a Lipschitz map φ: X → Y between to metric spaces X and Y, we denote by φ_# the map that sends the Lipschitz chain a=∑_i z_i α_i in X to the Lipschitz chain φ_#(a)=∑_i z_i (φ∘α_i) in Y. 

Let G always denote an n-dimensional, stratified nilpotent Lie group equipped with a left-invariant Riemannian metric and let Γ≤ G be a (uniform) lattice in G with s_2(Γ) ⊂Γ.
Let (τ, f:τ→ G) be a biLipschitz triangulation of G and ϕ: τ→ G be a (k+1)-horizontal, Γ-equivariant Lipschitz map in bounded distance c_ϕ> 0 to f. Hence, there is a Lipschitz homotopy h:G × [0,1] → G from Id_G to ϕ∘ f^-1.

If a is a Lipschitz k-cycle in G, then f^-1_#(a) is a Lipschitz k-cycle in τ and one can define in G the Lipschitz k-cycle

    P_ϕ(τ)(a)ϕ_#(P_τ(f^-1_#(a))).

(<cit.>)

There is a constant c_Q, only depending on τ, ϕ and k, such that if


	Q_ϕ(τ)(a) h(a × [0,1]) + ϕ_#(Q_τ(f^-1_#(a)))

then


  * ∂ Q_ϕ(τ)(a)=a - P_ϕ(τ)(a), 


  * (Q_ϕ(τ)(a)) ≤ c_Q ·(a).


By defining P_i(a) s_2^i(P_ϕ(τ)(s_2^-i(a)))
one obtains a sequence of rougher and rougher approximations of a and can prove that there is a constant c_P, only depending on τ, ϕ and k, such that (P_i(a)) ≤ c_P ·(a) for all i ∈_0 (<cit.>).

In the following crucial step one constructs chains interpolating between two consecutive approximations P_i and P_i+1. For this, denote for i∈{0,1} by (τ_i,f_i) the triangulation (τ, s_2^i∘ f) of G and let (η, g:η→ G×[0,1]) be a biLipschitz triangulation of G×[0,1] such that  the subcomplexes g^-1(G ×{i}), i ∈{0,1}, are isomorphic to τ_i by isomorphisms ι_i with g∘ι_i=f_i. Further let ψ:η→ G be a (k+1)-horizontal, s_2(Γ)-equivariant Lipschitz map and define ϕ_i: τ_i → G by ϕ_iψ∘ι_i.
By <cit.> such a triangulation (η,g) always exists, when (τ,f) is chosen Γ-equivariant for a  lattice Γ of G with s_2(Γ) ⊂Γ.

(<cit.>)

There is a constant c_R, only depending on η, ψ and k, such that if

    X(a) Q_τ_1((f_1^-1)_#(a))+g^-1_#(a×[0,1]) - Q_τ_0((f_0^-1)_#(a))

and R_ψ(η)(a)ψ_#(P_η(X(a))) then


  * ∂ R_ψ(η)(a)= P_ϕ_1(τ_1)(a)-P_ϕ_0(τ_0)(a), 


  * (R_ψ(η)(a)) ≤ c_R ·(a).



By scaling the chain R_ψ(η) in the same way as the cycle P_ϕ(τ) one gets	R_i(a) s_2^i(R_ψ(η)(s_2^-i(a))) and the following holds for all i∈_0 (<cit.>):


  1) ∂ R_i(a)= P_i+1(a)-P_i(a) 


  2) (R_i(a)) ≤ c_R · 2^i ·(a)

Using all these constructions, Young shows (<cit.>) that for any Lipschitz k-cycle a and  i_o ∈ such that 
2^(i_o-1)k≤ c_Δ(a) ≤ 2^i_ok with c_Δ c_τLip(f^-1)^k 1/(Δ^(k)), the Lipschitz (k+1)-chain 

    b - (Q_ϕ(τ)(a)+ ∑_i=0^i_o-1 R_i(a) )

is a filling of a with (b) ≾(a)^k+1/k.


Furthermore, the same construction yields upper bounds on the filling functions if one replaces the horizontality condition on the map ψ by a bound on the scaling behaviour of ψ-images of simplices (compare <cit.>). For this let f:^+→^+ be a function so that for every (d+1)-simplex Δ∈η^(d+1) one has (s_t(ψ(Δ))) ≤ f(t). For every Lipschitz d-cycle a in G we have (P_η(X(a))) ≤ c_τ( 2 c_τLip(g^-1)^d + Lip(g^-1)^d+1) (a) and therefore P_η(X(a)) consists of not more than c_τ( 2 c_τLip(g^-1)^d + Lip(g^-1)^d+1)/(Δ^(d+1))(a) many (d+1)-simplices. For this reason and with c_X c_τ( 2 c_τLip(g^-1)^d + Lip(g^-1)^d+1)/(Δ^(d+1)) one has

    (R_i(a)) = (s_2^i(ψ_#(P_η(X(s_2^-i(a)))))) ≤ f(2^i)c_X (s_2^-i(a))≤ 2^-dic_Xf(2^i) (a).

If there is a D∈ such that one has the above situation with a function f(t) ∼ t^D, this leads to a filling b -Q_ϕ(τ)(a)- ∑_i=0^i_o-1 R_i(a) with (b) ≾(a)^D/d.




We would like to use these constructions to fill r-avoidant cycles by ρ r-avoidant chains for some ρ∈ (0,1]. Unfortunately, one can't exclude the occurrence of chains R_i containing the base point 1∈ G or getting arbitrarily near to it. 


To fill r-avoidant cycles, we modify this construction to get a control on the distance of the chains R_i to the neutral element 1∈ G of the group G. To do this, we replace simplices too near to 1 by chains in larger distance.


Let ε >0 and let {w_1,...,w_m} be the set of all (n+1)-simplices of η which are simplices  Δ^(n+1) or share a vertex with a simplex Δ^(n+1) with 


    ψ(Δ^(n+1))∩ B_ε(1) ∅.


Let {v_1,...,v_m_τ} be the set of all simplices of τ which have, as simplices of τ_0 or τ_1, non-empty intersection with one of the simplices of {w_1,...,w_m}. Both of these sets are finite as ψ is s_2(Γ)-equivariant. As τ is homeomorphic to ^n, we find a finite subcomplex V_τ,ε of it which contains all the simplices v_i and is homeomorphic to an n-ball and therefore has an (n-2)-connected boundary.

As the first step for our modification, we choose a vertex x_o ∈ (∂ V_τ,ε)^(0) and define the map


    π_0: V_τ,ε^(0)→ (∂ V_τ,ε)^(0), x ↦π_0(x)


with


  * π_0(x)=x     ∀ x ∈ (∂ V_τ,ε)^(0),

  * π_0(x)=x_o   ∀ x ∈ V_τ,ε^(0)∖ (∂ V_τ,ε)^(0).

We denote by C^(l)(T), T⊂τ, the integral l-chains in the subcomplex T. Then we inductively define for 1≤ l ≤ n-1 the maps


    π_l: V_τ,ε^(l)→ C^(l)(∂ V_τ,ε), y ↦π_l(y)


with


  * π_l(y)=y     ∀ y ∈ (∂ V_τ,ε)^(l),

  * if ∂ y=∑_j=0^l (-1)^j y_j, so is ∂π_l(y)=∑_j=0^l (-1)^j π_l-1(y_j).

We can construct such maps by filling the boundaries of the mapped simplices by l-chains in ∂ V_τ,ε. This is possible as ∂ V_τ,ε is (n-2)-connected and l≤ n-1.

Each of the skeletons V_τ,ε^(l) is finite because V_τ,ε is finite. For this reason, the maximum


    M_l max{(π_l(y))/(Δ^(l))| y ∈ V_τ,ε^(l)} < ∞


exists for all l ∈{0,...,n-1}.

Using property (π_l1), we can extend the maps π_l to the whole l-skeleton of τ by the identity. By property (π_l2), we can extend this map to arbitrary l-chains:


    π_l: C^(l)(τ) → C^(l)((τ∖ V_τ,ε) ∪∂ V_τ,ε), b=∑_jβ_j y_j ↦π_l(b) ∑_j β_j π_l(y_j).


As no l-simplex is mapped to a l-chain of mass more than M_l ·(Δ^(l)), we get


    (π_l(b)) ≤ M_l ·(b)   ∀ b ∈ C^(l)(τ)  ∀ l ∈{0,...,n-1}


We defined the maps π_l on the l-chains of the simplicial complex τ. As τ_0 and τ_1 are isomorphic to τ, we can consider the maps π_l as maps on l-chains of these subcomplexes of η, too.

As next step, we define similar maps on the skeletons of the simplicial complex η. For this let W_η,ε be a finite subcomplex of η which contains all the simplices w_j,  j ∈{1,...,m}, is homeomorphic to an (n+1)-ball and such that W_η,ε∩τ_i = V_τ_i,ε, i ∈{0,1}, where V_τ_i,ε is the copy of V_τ,ε in τ_i.
Then the subcomplexes V_τ_i,ε, i ∈{0,1}, are contained in the boundary of W_η,ε. We denote W(∂ W_η,ε∖ (V_τ_0,ε∪ V_τ_1,ε))∪ (∂ V_τ_0,ε∪∂ V_τ_1,ε) and choose a vertex x_1 ∈W^(0). Then we define


    p_0: W_η,ε^(0)→W^(0), x ↦ p_0(x)


with 


  * p_0(x)=π_0(x) 	  ∀ x ∈ V_τ_0,ε^(0)∪ V_τ_1,ε^(0),

  * p_0(x)=x ∀ x ∈ (∂ W_η,ε)^(0)∖ (V_τ_0,ε^(0)∪ V_τ_1,ε^(0)),

  * p_0(x)=x_1 ∀ x ∈ W_η,ε(0)∖ (∂ W_η,ε)^(0).

We denote by C^(l)(E), E⊂η, the integral l-chains in the subcomplex E. Then we inductively define for 1≤ l ≤ n-1 the maps


    p_l: W_η,ε^(l)→ C^(l)(W), y ↦ p_l(y)


with


  * p_l(y)=π_l(y)   ∀ y ∈ V_τ_0,ε^(l)∪ V_τ_1,ε^(l),

  * p_l(y)=y ∀ y ∈ (∂ W_η,ε)^(l)∖ (V_τ_0,ε^(l)∪ V_τ_1,ε^(l)),

  * if ∂ y = ∑_j=0^l (-1)^jy_j, so is ∂ p_l(y)=∑_j=0^l (-1)^jp_l-1(y_j).

We are able to construct such maps by filling the boundaries of the mapped simplices by l-chains in W. This is possible as W is homeomorphic to an n-dimensional cylinder and therefore (n-2)-connected and because π_l satisfies property (p_l3) by property (π_l2).

As every skeleton W_η,ε^(l) is finite, the maximum


    M'_l max{(p_l(y))/(Δ^(k))| y ∈ W_η,ε^(l)} < ∞


exists. 

By property (p_l2) and by property (p_l1), combined with property (π_l1), we can extend the maps p_l to the whole l-skeleton of η by the identity. Using further property (p_l3), we can extend them to maps


    p_l: C^(l)(η) → C^(l)((η∖ W_η,ε)∪W),  b=∑_j β_j y_j ↦p_l(b) ∑_j β_j p_l(y_j)


of l-chains.

Again, no l-simplex is mapped to an l-chain of mass more than M'_l ·(Δ^(l)) and so we obtain


    (p_l(b)) ≤ M'_l ·(b)   ∀ b ∈ C^(l)(η) ∀ l ∈{0,...,n-1}.



We use the maps p_l to adapt the chains P_i and R_i (compare above) for our purpose to fill avoidant cycles with avoidant chains. Let r>0 and let a be an r-avoidant Lipschitz k-cycle in G. We define

    P_ϕ(τ)(a)ϕ_#( p_k (P_τ (f_#^-1(a))))

and

    P_i(a)  s_2^i(P_ϕ(τ)(s_2^-i(a)))

as well as

    R_ψ(η)(a) ψ_#(p_k+1(P_η(X(a))) )

and

    R_i(a)  s_2^i(R_ψ(η)(s_2^-i(a))).



  

For the above defined chains holds


    ∂R_i(a)=P_i+1(a) -P_i(a) .


Further there is a constant c_R, only depending on the triangulation (η,g), the map ψ and the map p_k+1, such that for all k-cycles a as above 


    (R_i(a))≤ 2^i c_R(a).




By equation (<ref>) we have ∂ X(a)= P_τ_1(f_#^-1(a)) - P_τ_0(f_#^-1(a)). Therefore

    ∂R_ψ(η)(a)=   ψ_#(∂p_k+1(P_η(X(a))) (p_k+13)=ψ_#(p_k(∂ P_η(X(a))))=ψ_#(p_k(P_τ_1(f_#^-1(a)) - P_τ_0(f_#^-1(a))))
    
    =   P_1(a)- P_0(a)

So we obtain

    ∂R_i(a)(<ref>)=   s_2^i(∂ R_ψ(η)(s_2^-i(a)))=s_2^i( P_1(s_2^-i(a))- P_0(s_2^-i(a))) 
    (<ref>)=   s_2^i( s_2(P_0(s_2^-1(s_2^-i(a)))))- s_2^i(P_0(s_2^-i(a)))(<ref>)=P_i+1(a)-P_i(a)

For the estimate of the mass we use (X(a)) ≤(2c_η·Lip(g^-1)^k +  Lip(g^-1)^k+1) (a) from the proof of <cit.>, where c_η denotes the constant in the Federer-Fleming Deformation Theorem for the complex η. With this we get

    (R_ψ(η)(a))    ≤Lip(ψ)^k+1(p_k+1(P_η(X(a)))
       ≤Lip(ψ)^k+1 M'_k+1(P_η(X(a)))
       ≤Lip(ψ)^k+1 M'_k+1c_η(X(a)) 
       ≤Lip(ψ)^k+1 M'_k+1c_η(2c_η·Lip(g^-1)^k +  Lip(g^-1)^k+1) (a) .

As the chain R_ψ(η)(a) is horizontal, it follows with 


    c_RLip(ψ)^k+1 M'_k+1c_η(2c_η·Lip(g^-1)^k +  Lip(g^-1)^k+1)


that


    (R_i(a)) = 2^i(k+1)(R_ψ(η)(s_2^-i(a))) ≤ 2^i(k+1) c_R 2^-ik(a) = 2^ic_R(a).




We close our modification by constructing a filling of a cycle a consisting of the old chain Q_ϕ(τ)(a) and the new chains R_i(a). Let r>Lip(ψ)(diam(W_η, ε)+ diam(Δ^(n+1)))>0, so that for every r-avoidant k-cycle a in G the preimage f_#^-1(a) does not intersect V_τ,ε. Let a be an r-avoidant k-cycle. Therefore P_τ(f_#^-1(a)) uses no simplices of V_τ,ε∖∂ V_τ,ε as it is contained in the smallest subcomplex of τ that contains f_#^-1(a). In particular, we have P_0(a)=P_0(a). 

  

Let a be as above, c_Δ c_τLip(f^-1)^k 1/(Δ^(k)) and i_o ∈ such that 
2^(i_o-1)k≤ c_Δ(a) ≤ 2^i_ok. Then the (k+1)-chain


    b -(Q_ϕ(τ)(a) + ∑_i=0^i_o-1R_i(a))


is a filling of a with (b) ≾(a)^k+1/k.


By the proof of <cit.> we know P_τ(s_2^-i_o(a))=0 and therefore 

P_i_o(a) = s_2^i_0(ϕ_#(p_k(P_τ(s_2^-i_o(a)))))=0. As P_0(a)=P_0(a), we have

    ∂b=    - (∂ Q_ϕ(τ)(a) + ∑_i=0^i_o-1∂ R_i(a)) =-((P_0(a)-a) + ∑_i=0^i_o-1 (P_i+1(a)-P_i(a)))
    
    =   -((P_0(a)-a) + (P_i_o(a)-P_0(a))) = a.

With Lemma <ref> and Lemma <ref> we get

    (b) ≤    c_Q(a) + ∑_i=0^i_0-1(2^ic_R(a)) ≤  c_Q(a) + 2^i_oc_R(a)
    def  i_o≤    (c_Q + 2 c_R(c_Δ(a))^1/k) (a).

For (a) ≥ 1 this implies (b) ≤ (c_Q + 2 c_R(c_Δ)^1/k ) (a)^k+1/k≾(a)^k+1/k.






Again, if one replaces the horizontality condition on the map ψ by a bound on the scaling behaviour of ψ-images of simplices, one can still bound the mass of the filling b. 
  

Let f:^+→^+ be a function so that for every (d+1)-simplex Δ∈η^(d+1) one has (s_t(ψ(Δ))) ≤ f(t). Then there is a constant c_X such that for every Lipschitz d-cycle a in G

    (R_i(a)) ≤ 2^-dic_Xf(2^i) (a).



We have 


    (p_d+1(P_η(X(a)))) ≤ M'_d+1c_τ( 2 c_τLip(g^-1)^d + Lip(g^-1)^d+1) (a)


and therefore p_d+1(P_η(X(a))) consists of not more than 


    1/(Δ^(d+1))M'_d+1c_τ( 2 c_τLip(g^-1)^d + Lip(g^-1)^d+1) (a)


many (d+1)-simplices. For this reason and with c_XM'_d+1c_τ( 2 c_τLip(g^-1)^d + Lip(g^-1)^d+1)/(Δ^(d+1)) we get

    (R_i(a))    = (s_2^i(ψ_#(p_d+1(P_η(X(s_2^-i(a)))))))
       ≤ f(2^i)c_X(s_2^-i(a))
       ≤ 2^-dic_Xf(2^i) (a).



If one can use a polynomial function f this leads to the following.

  

Let D ∈, D>d≥ 1, and C >0 be so that for every (d+1)-simplex Δ∈η^(d+1) one has (s_t(ψ(Δ))) ≤ Ct^D+Ct+C. Then for every r-avoidant Lipschitz d-cycle a the Lipschitz (d+1)-chain


    b -(Q_ϕ(τ)(a) + ∑_i=0^i_o-1R_i(a))


is a filling of a with (b) ≾(a)^D/d.


The fact that b is a filling of a we have already seen in Proposition <ref>. 
As we now can use Lemma <ref> with f(t)=Ct^D+Ct+C we obtain for the mass

    (b)    ≤ c_Q(a) + ∑_i=0^i_0-1(2^-di(C(2^i)^D+C2^i+C)c_X(a)) 
       ≤ c_Q(a) + ∑_i=0^i_0-1C((2^(D-d)i+2^(1-d)i+2^-di)c_X(a))
       ≤ c_Q(a) + ∑_i=0^i_0-1C((2^(D-d)i+2·2^(1-d)i)c_X(a))
       ≤ c_Q(a) + (2^(D-d)i_o+2^(1-d)i_o)Cc_X(a)
       ≤ (c_Q + 2 Cc_X((c_Δ(a))^D-d/d+(c_Δ(a))^1-d/d)) (a).
 
For (a) ≥ 1 this yields (b)≤ (c_Q + 2 C c_X((c_Δ )^D-d/d+ c_Δ)) (a)^D/d≾(a)^D/d.







§ THE PROOF OF THE MAIN RESULT



In this section we use all the notations as in section <ref>. To prove our Main Theorem we will show that in early approximation steps the chains R_i(a) stay near the cycle a and that in late approximation steps they stay far away from the base point.


By construction we have that the  (k+1)-chains p_k+1(P_η(X(a))) only use simplices Δ of η with ψ(Δ) ∩ B_ε(1)= ∅. 

Set D_ηdiam( Δ^(n+1)) and let r>Lip(ψ)(diam(W_η, ε)+ D_η)>0 as before.



  

Let a be an r-avoidant Lipschitz k-cycle in G. Then holds 


    d(R_i(a),1) ≥ 2^i ·ε.




By the defining equation (<ref>) we have R_ψ(η)(a) ψ_#(p_k+1(P_η(X(a))) ) and therefore 
d(R_ψ(η)(a), 1) > ε.
By equation (<ref>) we have R_i(a)=s_2^i(R_ψ(η)(s_2^-i(a))) and so
d(R_i(a),1)≥  2^i  d(R_ψ(η)(s_2^-i(a)),1) ≥ 2^i ·ε.




  

Let a be an r-avoidant Lipschitz k-cycle in G. Then the chain R_i(a) is contained in the (2^i· C_D·D)-neighbourhood of a, where  D only depends on the triangulation (η, g) and the maps ϕ and ψ and the choice of the subcomplex W_η,ε and C_D is the constant from Lemma <ref>.


We prove that R_ψ(η)(a) is contained in the D-neighbourhood of a for D c_ϕ + (ψ)·((g^-1) + D_η+ diam(W_η,ε)). Then the claim follows by Lemma <ref> and equation (<ref>).

Remember the definition R_ψ(η)(a)=ψ_#(p_k+1(P_η(X(a)))) with 


    X(a)= Q_τ_1((f_1^-1)_#(a))+g^-1_#(a×[0,1]) - Q_τ_0((f_0^-1)_#(a)).


As f_0 and f_1 are restrictions of g and as Q_τ_i((f_i^-1)_#(a)) is contained in the smallest subcomplex of τ_i that contains (f_i^-1)_#(a), we have that P_η(X(a)) is contained in the smallest subcomplex of η that contains g^-1_#(a×[0,1]). As further g^-1_#(a×[0,1]) lies in the (g^-1)-neighbourhood of (f_0^-1)_#(a), this subcomplex is contained in the ((g^-1)+D_η)-neighbourhood of (f_0^-1)_#(a). Further the map p_k+1 moves chains at most by diam(W_η,ε).
 Therefore R_ψ(η)(a) is contained in the ((ψ)·((g^-1)+D_η+diam(W_η,ε)))-neighbourhood of ψ_#((f_0^-1)_#(a))=ϕ_#((f^-1)_#(a)). Finally, ϕ and f are in distance at most c_ϕ and so ϕ_#((f^-1)_#(a)) is in the c_ϕ-neighbourhood of a.



  

Let a be a Lipschitz k-cycle in G. Then there is a constant D_Q>0, only depending on the triangulations (τ,f), (η,g), the map ϕ and the homotopy h, such that Q_ϕ(τ)(a) is contained in the D_Q-neighbourhood of a.


Remember the definition Q_ϕ(τ)(a)=h_#(a× [0,1]) + ϕ_#(Q_τ((f^-1)_ #(a))). As Q_τ((f^-1)_ #(a)) is contained in the smallest subcomplex of τ which contains (f^-1)_ #(a), and as τ is a subcomplex of η, it is contained in the D_η-neighbourhood of (f^-1)_ #(a). So ϕ_#(Q_τ((f^-1)_#(a))) is contained in the ((ϕ) · D_η)-neighbourhood of ϕ_#((f^-1)_ #(a)). As ϕ and f are in distance at most c_ϕ, ϕ_#((f^-1)_#(a)) is in the c_ϕ-neighbourhood of a. We get that Q_ϕ(τ)(a) is contained in the D_Q-neighbourhood of a for D_Q(h)+c_ϕ+ (ϕ)· D_η.



Now we are prepared for the proof of our Main Theorem.


the Main Theorem  

Let r>0, α >0. By Remark <ref> we can assume r> 10 ·max{ε,D_Q,C_D·D}. Let further 


    ρ≤min{9/10 , ε/ε+C_D·D}.


We will show that for every r-avoidant Lipschitz k-cycle a in G with (a) ≤α· r^k
the filling b=-Q_ϕ(τ)(a)- ∑_i=0^i_o-1R_i(a)  is ρ r-avoidant.
For this we treat the Q-part and the R-part separately. 

By the lower bound on r we have in particular r > 10 · D_Q and therefore D_Q < 1/10 r. As a is r-avoidant, Q_ϕ(τ)(a) is 9/10 r-avoidant by Lemma <ref> and therefore ρ r-avoidant.

For the R-part we use Lemma <ref> and Lemma <ref>. By these we have for every i∈{0,...,i_o-1} that R_i(a) is simultaneously (2^i ·ε)-avoidant and (r-2^i· C_D·D)-avoidant. For every single i we can use the bigger of these two numbers, and in the whole we get:


    d(1,∑_i=0^i_o-1R_i(a))    ≥min{max{2^i ·ε , r-2^i· C_D·D}| i∈{0,...,i_o-1}}
       ≥min{max{2^i ·ε , r-2^i· C_D·D}| i∈_0 }
       ≥min{max{2^i ·ε , r-2^i· C_D·D}| i∈ [0,∞) }


We set M_1(i) 2^i ·ε and M_2(i) r-2^i· C_D·D for i ≥ 0. The function M_1 is strictly increasing in i and the function M_2 is strictly decreasing in i. So we find the minimal maximum of {M_1(i),M_2(i)} in the point where M_1 and M_2 coincide. This is for


    M_1(i)=M_2(i) ⇔ 2^i ·ε=r-2^i· C_D·D ⇔ i=log_2(r/ε+C_D·D)


and hence the minimal maximum is


    M_1(log_2(r/ε+C_D·D))= 2^log_2(r/ε+C_D·D)·ε=r/ε+C_D·D·ε=ε/ε+C_D·D· r ≥ρ r .


Therefore ∑_i=0^i_o-1R_i(a) is ρ r-avoidant. Altogether, the filling b is ρ r-avoidant and satisfies


    (b) ≾(a)^k+1/k≾ r^k+1.



If we start with na r-avoidant (k+1)-cycle a' with (a')≤α· r^k+1, the same computations show, that the filling b'=-Q_ϕ(τ)(a')- ∑_i=0^i'_o-1 R_i(a') is ρ r-avoidant. 
As we have (s_t(ψ(Δ))) ≾ t^K+2 for all Δ∈η^(k+2) the filling b' satisfies by Proposition <ref>


    (b') ≾(a')^K+2/k+1≾ r^K+2.


This completes the proof.

















plain

  

Courant Institute of Mathematical Sciences, New York University, New York, USA

E-mail address: moritz.gruber@cims.nyu.edu










