Polynomially weighted $\ell^p$-completions and group homology
Alexander Engel Clara Loeh

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Polynomially weighted ℓ^p-completions and group homology
    Alexander Engelmailto:alexander.engel@mathematik.uni-regensburg.dealexander.engel@mathematik.uni-regensburg.de
Clara Löhmailto:clara.loeh@mathematik.uni-regensburg.declara.loeh@mathematik.uni-regensburg.de

    Fakultät für Mathematik
 Universität Regensburg
 93040 Regensburg, Germany
==================================================================================================================================================================================================================




  We introduce polynomially weighted ℓ^p-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of p in the range (1,∞). 






§ INTRODUCTION


Let G be a finitely generated group. In order to state our main result, we quickly introduce the main players in it.

The group homology H_∗(G;) can be computed as the homology of
the bar complex C_*(G;). Chains c ∈ C_k(G;) are of the form
c = ∑_g ∈ G^k a_g · [e,g_1,…, g_k],
where only finitely many of the coefficients a_g are non-zero. We
choose a finite generating set S for G to get a word-metric on
G. For n ∈ and p ∈ [1,∞) we then define a weighted norm on
  C_k(G;) by c_n,p^S := (∑_g∈ G^k |a_g|^p ·_S(g)^n)^1/p. We equip C_k(G;) with the
  family (-_n,p^S + ∂ - _n,p^S)_n∈ of
  norms and denote the corresponding completion to a Fréchet space
  by C_k^p(G). The homology of the resulting chain
  complex C_*^p(G) is denoted by H_*^p(G).

[Theorem <ref>, Proposition <ref>]
  Let G be a finitely generated group of polynomial or exponential growth
  and let p,q ∈ (1,∞) with p < q.
  Then the canonical homomorphism H_*^p(G) ⟶ H_*^q(G)
  is an isomorphism.




  
Relation to the strong Novikov conjecture
Let us explain why we are interested in a theorem like the one above. We first have to recall the following two notions.
Firstly, a group G is called of type F_∞, if it admits a model for its classifying space BG
of finite type (i.e., a CW-complex that in each dimension has only finitely many cells).
Secondly, a group of type F_∞ is called polynomially contractible, if its Dehn function and its higher-dimensional analogues are polynomially bounded. Note that this assumption on G is not very strong: most of the groups that one would call non-positively curved (like hyperbolic groups, CAT(0)-groups, systolic groups or mapping class groups) are polynomially contractible. This follows from the fact that if a group is polynomially combable (i.e., combable with a uniform polynomial bound on the lengths of the combing paths), then it is polynomially contractible <cit.><cit.>.

For hyperbolic groups any choice of geodesics in the Cayley graph will be a suitable combing, for CAT(0)-groups any choice of quasi-geodesics in the group following uniformly closely CAT(0)-geodesics in the underlying space will do the job, for systolic groups one can use the bi-automatic structure found by Januszkiewicz–Świa̧tkowski <cit.> or the combing by Osajda–Przytycki <cit.>, and an automatic structure on mapping class groups was provided by Mosher <cit.>. For a thorough compilation of polynomially contractible groups see the introduction of <cit.>.


  Let G be a group of type F_∞, and of polynomial or exponential growth,
  and let G be polynomially contractible. 
If there exists some p ∈ (1,∞) such that the canonical homomorphism H^1_∗(G) → H^p_∗(G) is injective, then the strong Novikov conjecture holds for G.



The proof relies on the following diagram <cit.>: 

    RK_k(BG) [r] [d]__k    K_k(B_r^p G) @–>[d]
    
    H_k(G;) [r]     H_k^1(G)

Here, B_r^p G denotes the norm completion of G ⊂(ℓ^p G) and the top horizontal map is the analytic assembly map. In the case p=2 we have B_r^2 G = C_r^∗ G, i.e., the reduced group C^∗-algebra, and the strong Novikov conjecture asserts that the analytic assembly map in this case (i.e., for p=2) is rationally injective.

Let x ∈ RK_∗(BG) ⊗ be any non-trivial element. Because the homological Chern character _∗ RK_∗(BG) ⊗→ H_∗(G;) is an isomorphism, there is some k ∈ such that _k(x) ∈ H_k(G;) is non-zero. If G is of type F_∞ and polynomially contractible, then the canonical map H_∗(G;) → H_∗^1(G) is an isomorphism <cit.>. (Analogous statements in the dual situation, i.e., for the corresponding cohomology groups, also hold <cit.>.)
Further,  the right vertical map in the above diagram exists for all p ≤ (k+2) / (k+1). Hence, for such p the element x is not in the kernel of the analytic assembly map.

Since our goal is the strong Novikov conjecture, i.e., to show that the element x is not in the kernel of the assembly map for p=2, we can try to go with the lower horizontal map to H_k^q(G) for
some q >1 instead of to H^1_k(G), i.e, we consider the new diagram

    RK_∗(BG) [rr] [d]__k       K_∗(B_r^p G) @–>[d]
    
    H_k(G;) [r]     H_k^1(G) [r]     H_k^q(G)


Also in this case, we can construct the right vertical map for all p ≤ q · (k+2) / (k+1)
(this can be shown as in previous work of the first named author <cit.>).
In particular, if q is big enough, we can do it for p=2. We have already noted above that polynomial contractibility gives us that the canonical map H_∗(G;) → H_∗^1(G) is an isomorphism. Using the main theorem, we see that if the canonical map H_∗^1(G) → H_∗^p(G) is injective for some p > 1, then H_∗^1(G) → H_∗^q(G) will be injective for every q ∈ (1,∞). Hence our element x is not in the kernel of the assembly map for the case p=2.


Unfortunately, the hypotheses of this corollary are not satisfied for all groups:
For example, for the free group F_2 of rank 2, the canonical homomorphism H^1_1(F_2) ⟶ H^p_1(F_2) is trivial for all p ∈ (1,∞) (Theorem <ref>, Theorem <ref>).
In fact, we expect this vanishing result to hold in far greater generality, and thus this approach to the strong Novikov conjecture is not promising.



  
Questions
Let us collect some open problems arising from the present paper. Since this seems to be the first time that such polynomially weighted ℓ^p-completions of group homology are defined, there are many natural questions left open.
  
  
  * Does Theorem <ref>, i.e., the comparison in the range (1,∞), also hold for groups
    of intermediate growth?
  
  * For which groups of superpolynomial growth does Theorem <ref> also
    hold in the cases “p=1” or “q= ∞” ?
  
  * For which groups G and which p ∈ (1,∞] is the canonical
    map H^1_*(G) ⟶ H^p_*(G), resp. the canonical map H_*(G;) → H^p_*(G), injective?
  
  * For which groups G and which p ∈ [1,∞], k ∈ is H_k^p(G) non-trivial?
    How can such classes be detected? 
  






  
Related work
Though this seems to be the first time that these polynomially weighted ℓ^p-completions of group homology are defined, there are of course similar things already in the literature:



  * Bader, Furman and Sauer <cit.> investigate the comparison maps from ordinary homology and Sobolev homology, respectively, to the ℓ^1-homology of any word hyperbolic group.

  * Nowak and Špakula <cit.> study coarse homology theory with prescribed growth conditions.

  * Weighted simplicial homology was studied by Dawson <cit.> and by Ren, Wu and Wu <cit.>.

  * The dual situation to the one from the present paper, but only in the case of ℓ^1, i.e., group cohomology of polynomial growth, was studied by Connes and Moscovici <cit.> in relation with the strong Novikov conjecture, and further investigated by many others like Ji <cit.>, Meyer <cit.> and Ogle <cit.>.




  
Overview of this article
Section <ref> introduces the polynomially weighted
ℓ^p-versions of group homology in full detail and
discusses the case of groups of polynomial growth. In
Section <ref>, we establish the comparison theorem for
groups of exponential growth.  The vanishing result for the free group
is proved in Section <ref>.



  
Acknowledgements
The authors were supported by the SFB 1085 Higher Invariants of the Deutsche Forschungsgemeinschaft DFG. The first named author was also supported by the Research Fellowship EN 1163/1-1 Mapping Analysis to Homology of the DFG, and the DFG Priority Programme SPP 2026 Geometry at Infinity (EN 1163/3-1 “Duality and the coarse assembly map”). Moreover, we would like to thank the anonymous referee for providing helpful comments.





§ WEIGHTED LP-NORMS ON GROUP HOMOLOGY




 §.§ Definition and basic properties



  Let G be a finitely generated group endowed with a finite generating set S, let k ∈, and let p
  ∈ [1,∞). For n ∈ we define the n-weighted
      ℓ^p-norm (with respect to S) by 
  
    -_n,p^S  C_k(G;)    ⟶_≥ 0
    ∑_g ∈ G^k a_g · [e,g_1,…, g_k]
           ⟼(∑_g∈ G^k |a_g|^p ·_S(g)^n)^1/p,

  where _S(g) := _S{e,g_1,…, g_k} is the diameter with respect
  to the word metric d_S on G.
  
  
  We then equip C_k(G;) with the family (-_n,p^S +
  ∂ - _n,p^S)_n∈ of norms and denote the
  corresponding completion to a Fréchet space by C_k^p(G). By
  construction, the boundary operator of C_k(G;) extends continuously
  to C_k^p(G) and the homology of C_*^p(G) is called
  ℓ^p-polynomially bounded homology of G, denoted
  by H_k^p(G).

  In the case of p = ∞, we proceed in the same manner, using
  the n-weighted ℓ^∞-norms (with respect to S), 
  defined by
  
    -_n,∞^S  C_k(G;)    ⟶_≥ 0
    ∑_g ∈ G^k a_g · [e,g_1,…, g_k]
           ⟼sup_g ∈ G^k |a_g| ·_S(g)^n.




  If G is a finitely generated group, k, n ∈, and p ∈
  [1,∞], then different finite generating sets S,T of G lead
  to equivalent (semi-)norms -_n,p^S and -_n,p^T
  on C_k(G;). Therefore, the completions C_*^p(G) and the
  homology H_*^p(G) are independent of the choice of finite
  generating sets.



  Let G be a finitely generated group and let p,q ∈ [1,∞) with p < q. Then
  the canonical inclusion C_*^p(G) ⟶ C_*^q(G) is
  contractive in the following sense: For every finite generating set S of G
  and every n ∈ the identity map C_*(G;) ⟶ C_*(G;) has norm
  at most 1 with respect to the norms -_n,p^S and -_n,q^S,
  respectively. In particular, we obtain a canonical induced map
  
    H_*^p(G) ⟶ H_*^q(G).

  If p ∈ [1,∞) and n ∈, then
  
    c ∈ C_k(G;)c^S_n, ∞≤c_⌈ n · p⌉,p^S,

  which yields a canonical map H_*^p(G) ⟶ H_*^∞(G).






 §.§ The case p=1


It is already known that the canonical map H_*(G;) ⟶ H_*^1(G) is an isomorphism for a large class of groups.

To state the corresponding theorem, we have to recall two notions.
Firstly, a group G is called of type F_∞, if it admits a model for its classifying space BG
of finite type (i.e., a CW-complex that in each dimension has only finitely many cells).
Secondly, a group of type F_∞ is called polynomially contractible, if its Dehn function and its higher-dimensional analogues are polynomially bounded.

Most of the groups that one calls non-positively curved (like hyperbolic groups, systolic groups, CAT(0)-groups or mapping class groups) are polynomially contractible (see the introduction for references).

The following theorem has been proved (in variations) by different people in different ways <cit.>:


  Let G be a group of type F_∞ that is polynomially contractible. Then
  the canonical map H_*(G;) ⟶ H_*^1(G) is an isomorphism.



Without the assumption of polynomial contractibility, Theorem <ref> is likely false.

Ji, Ogle, and Ramsey provided groups whose comparison maps from bounded cohomology to ordinary cohomology fail to be injective or surjective, respectively <cit.>. Bounded cohomology, as they investigate it, is dual to our H_*^1(-) in the sense that it pairs with it (and this pairing is compatible with the usual pairing of homology with cohomology under the comparison maps). Please be also aware that their bounded cohomology is not the one used by Gromov <cit.>.

It seems therefore plausible that the groups of Ji, Ogle and Ramsey are also examples of groups for which the canonical map H_*(G;) ⟶ H_*^1(G) is not injective or surjective, respectively. 






 §.§ Comparison on groups of polynomial growth




  Let G be a finitely generated group of polynomial growth and let
  p,q ∈ [1,∞] with p < q.
  
    
  * Then the inclusion C_*^p(G) ⟶ C_*^q(G) is bounded
      from below in the following sense: For every finite generating set S
      of G and all k,n ∈ there exist C ∈_>0 and m ∈ with
      
    c ∈ C_k(G;)c_m,q^S ≥ C ·c_n,p^S.

    
  * In particular, the canonical map H_*^p(G) ⟶ H_*^q(G)
      (Remark <ref>) is an isomorphism.
  



  Ad 1.
  We first consider the case q < ∞.
  Let D ∈ be the polynomial growth rate of G, let S be a
  finite generating set of G, and let k, n ∈. Then
  
    m := ⌈ q ·((k · D+2) ·(1/p - 1/q) + n/p)⌉

  has the desired property, as can be seen by the generalized Hölder
  inequality: Because D is the polynomial growth rate of G, there
  is a constant K ∈_>0 with
  
    r ∈_>0β(r) := |B_e^G,S(r)| ≤ K · r^D.

  Moreover, because of q > p, there is q' ∈ [1,∞) with
  
    1/q + 1/q' = 1/p.

  We now consider c ∈ C_k(G;) and the weight functions
  
    w_1,w_2  G^k    ⟶_≥0
    
        w_1 : g    ⟼_S(g)^m/q
    
        w_2 : g    ⟼_S(g)^n/p-m/q.

  By definition, c_n,p^S =  c · w_1 · w_2_p
  and c_m,q^S = c · w_1_q 
  (where
  “·” denotes pointwise multiplication). Applying the
  generalized Hölder inequality, we hence obtain
  
    c_n,p^S   ≤ c · w_1 _q ·w_2_q'
        =  c _m,q^S ·w_2_q'

  and it remains to bound w_2_q' by a constant. The polynomial growth
  condition yields
  
    (w_2_q')^q'    = ∑_g ∈ G^k(g)^q' · (n/p - m/q)
        ≤∑_r =1^∞β(r)^k
        · r^q' · (n/p - m/q)
        ≤ K^k ·∑_r=1^∞ r^k · D· r^q' · (n/p - m/q)
       ≤ K^k ·∑_r=1^∞ r^-2
        = K^k ·ζ(2).


  In the case q = ∞, one can proceed in a similar way
  (with m = ⌈ 1/p · (k · D + n +2)⌉).
  
  Ad 2. 
  By the first part, the identity map on the ordinary chain complex C_*(G;) induces an
  isomorphism C_*^p(G) ⟶ C_*^q(G). 
  Hence, the claim follows.





 §.§ Functoriality of weighted lp-chain complexes



  Let G be a finitely generated group. 
  
  
  * A subgroup H ⊂ G is polynomially controlled, if
    for one (whence every) finite generating set S ⊂ G there
    exist D ∈ and K ∈_>0 such that
    
    r ∈_>0| B_r^G,S(e) ∩ H |
        ≤ K · r^D.

  
  * Let G' be a group. 
    A group homomorphism φ G ⟶ G' has
    polynomially controlled kernel if the subgroup φ of G
    is polynomially controlled in the sense above.



Clearly, all group homomorphisms with finite kernel have polynomially
controlled kernel as well as all group homomorphisms mapping out of
finitely generated groups of polynomial growth. 


If H is a polynomially controlled subgroup of a finitely generated group G, then every finitely generated subgroup K of H has polynomial growth.

The reason for this is that the inclusion K → G does not increase lengths of elements if we choose a finite generating set S of G containing the chosen finite generating set T of K to define the word lengths. More concretely, we have for all r ∈_>0

    |B_r^K,T(e)| ≤ |B_r^G,S(e) ∩ K| ≤ |B_r^G,S(e) ∩ H|.

We see that we even actually have that the growth rates of finitely generated subgroups of H are uniformly bounded from above.



  Let G be a hyperbolic group and H a subgroup of G. 
  Then H is a polynomially controlled subgroup if and only if H is virtually cyclic.



  Let H be a polynomially controlled subgroup. By Remark <ref> finitely
  generated subgroups of H are of polynomial growth. In particular,
  H does not contain a free group of rank 2.  As the ambient
  group G is hyperbolic, this implies that H is virtually
  cyclic <cit.>.
  
  Let H be virtually cyclic. Without loss of generality we can assume that H is isomorphic to . Then H is quasi-isometrically embedded in G <cit.> and hence a polynomially controlled subgroup of G.



  Let p ∈ [1,∞], let G and H be finitely generated
  groups, and let φ G ⟶ H be a group
  homomorphism with polynomially controlled kernel.
  Then the induced chain map C_*(φ;)  C_*(G;)
  ⟶ C_*(H;) is continuous with respect to the
  weighted ℓ^p-Fréchet topologies.


  Let us establish some notation: 
  Let S ⊂ G and T ⊂ H be finite generating sets, and
  without loss of generality we may assume that φ(S) ⊂ T.
  Let D ∈ be the polynomial control rate of φ; hence,
  there is a K ∈_>0 with
  
    r ∈_>0β(r) := | B_e^G, S(r) ∩φ|
         ≤ K · r^D.
   
  Furthermore, let p ∈ [1,∞] and k,n ∈. It then
  suffices to prove that there exist m ∈ and C ∈_>0
  such that
  
    c ∈ C_k(G;) C_k(φ;)(c) ^T_n,p≤ C ·c_m+n,p^S.

  The arguments are similar to the proof of Proposition <ref>:

  Let c ∈ C_k(G;), say c = ∑_g ∈ G^k a_g · [e,g_1,…,g_k].
  By definition of C_k(φ;), we have 
  
    φ(c)
         := C_k(φ;)(c)
         = ∑_h ∈ H^k(∑_g ∈φ^-1(h) a_g) · [e,h_1,…, h_k],

  where φ^-1(h) := { g ∈ G^k |
  j ∈{1,…,k}φ(g_j) = h_j}.    

  We will first consider the case p ∈ (1, ∞). Let 
  
    m := ⌈ (k · D + 2) · (p-1)⌉

  and let p := p / (p-1). As first step, we bound the
  inner sum for a given h ∈ H^k (without loss of generality, we
  may assume h ≠ (e,…,e)): By the Hölder inequality,
  
    | ∑_g ∈φ^-1(h) a_g |^p
           ≤( ∑_g ∈φ^-1(h) |a_g|)^p
        
       ≤∑_g ∈φ^-1(h) |a_g|^p ·_S(g)^m
        ·( ∑_g ∈φ^-1(h)_S(g)^- p· m/p)^p/p
       ≤∑_g ∈φ^-1(h) |a_g|^p ·_S(g)^m
        ·( ∑_g ∈φ^-1(h)_S(g)^- m/p-1)^p-1.

  The first factor is related to c_m,p^S and hence of the
  right type.  We will now take care of the second factor: To this
  end, for j ∈{1,…,k} let g_j(h) ∈ G be a minimiser
  of min{ d_S(e,g) | g ∈ G, φ(g) = h_j}.
  Then we have
  
    κ_j ∈φ d_S(e, g_j(h) ·κ_j)
           ≥1/2· (d_S(e,g_j(h)) + d_S(e, g_j(h) ·κ_j) )
        
       ≥1/2· d_S(g_j(h), g_j(h) ·κ_j)
        = 1/2· d_S(e,κ_j)

  and hence the polynomial control on the kernel yields
  
    ∑_g ∈φ^-1(h)_S(g)^- m/p-1    = ∑_κ∈ (φ)^k_S(g(h) ·κ)^-m/p-1
       ≤∑_κ∈ (φ)^k(max_j ∈{1,…,k} d_S(e, g_j(h) ·κ_j))^-m/p-1
       ≤ 2^m/p-1·∑_κ∈ (φ)^k( max_j ∈{1,…,k} d_S(e,κ_j))^-m/p-1
       ≤ 2^m/p-1·∑_r=1^∞β(r)^k · r^-m/p-1≤ 2^m/p-1· K^k ·∑_r=1^∞ r^k· D· r^-m/p-1
       ≤ 2^m/p-1· K^k ·ζ(2).

  We set C := (2^m/p-1· K^k ·ζ(2))^1/(p-1). 
  Putting it all together, we obtain (because φ(S) ⊂ T)
  
    (φ(c)_n,p^T)^p
            = ∑_h ∈ H^k| ∑_g ∈φ^-1(h) a_g |^p ·_T(h)^n
        
       ≤ C ·∑_h ∈ H^k∑_g ∈φ^-1(h) |a_g|^p ·_S(g)^m ·_T(h)^n
        
       ≤ C ·∑_h ∈ H^k∑_g ∈φ^-1(h) |a_g|^p ·_S(g)^m ·_S(g)^n
        
       ≤ C ·∑_g ∈ G^k |a_g|^p ·_S(g)^m+n
        = C ·(c_m+n,p^S)^p,

  as desired.

  In the case p=1, the estimates above simplify significantly
  because the inner sum can be treated directly with the inherited
  ℓ^1-bound and one obtains
  
    c ∈ C_k(G;)φ(c)_n,1^T ≤c_n,1^S.


  In the case p=∞, we take m := k · D + 2. Then the inner sum
  admits the following estimate for given h ∈ H^k (without loss of generality, we
  may assume h ≠ (e,…,e)):
  
    | ∑_g ∈φ^-1(h) a_g |
           ≤∑_g ∈φ^-1(h) |a_g| ·_S(g)^m ·_S(g)^-m
       ≤sup_g ∈φ^-1(h) |a_g| ·_S(g)^m
        ·∑_g ∈φ^-1(h)_S(g)^-m
       ≤sup_g ∈φ^-1(h) |a_g| ·_S(g)^m · 2^m · K^k ·ζ(2).

  This implies φ(c)_n,∞^T ≤ 2^m · K^k ·ζ(2) ·c_m+n,∞^S.



  Let p ∈ [1,∞], let G and H be finitely generated
  groups, and let φ G ⟶ H be a group
  homomorphism with polynomially controlled kernel.
  
  
  * Then C_*(φ;) admits a well-defined, continuous extension
    
    C_*^p(φ)  C_*^p(G) ⟶ C_*^p(H),

    which is a chain map.
  
  * In particular, we obtain a corresponding
    homomorphism H_*^p(φ)  H_*^p(G) ⟶
    H_*^p(H) that is compatible with H_*(φ;) 
    H_*(G;) ⟶ H_*(H;).
  


  This is a direct consequence of Proposition <ref>.






§ COMPARISON IN THE RANGE [1,INFTY)



  Let G be a finitely generated group of exponential growth
  and p,q ∈ (1,∞) with p < q.
  
  
  * The inclusion C_*^p(G) ⟶ C_*^q(G) is a chain homotopy
    equivalence.
  
  * In particular, the canonical map H_*^p(G) ⟶ H_*^q(G)
      (see Remark <ref>) is an isomorphism.
  


The proof of Theorem <ref> is based on the following
basic chain-level result, which will be proved in Section <ref>.


  Let G be a finitely generated group of exponential growth with finite generating set S
  and let p,q ∈ (1,∞) with p < q.  Then there exists a
  chain map E  C_*(G;) ⟶ C_*(G;) and a chain
  homotopy B between E and the identity with the following
  properties: For all k, n ∈ there exist K ∈_>0 and m ∈
  such that for all c ∈ C_k(G;) we have
  
    E(c) _n,p^S
           ≤ K ·( c_m,q^S + ∂ c _m,q^S )
        
    ∂ E(c) _n,p^S
           ≤ K ·( c_m,q^S + ∂ c _m,q^S )
        
     B(c) _n,p^S
           ≤ K ·( c_m,p^S + ∂ c _m,p^S )
        
    ∂ B(c) _n,p^S
           ≤ K ·( c_m,p^S + ∂ c _m,p^S )



Taking Proposition <ref> for granted, the proof of Theorem <ref> is immediate:


  We write i  C_*^p(G) ⟶ C_*^q(G) for the canonical
  inclusion map. 
  Let E and B be maps as provided by Proposition <ref>.
  Estimates (<ref>) and (<ref>) show that E extends
  to a continuous chain map
  
    E C_*^q(G) ⟶ C_*^p(G).

  Similarly, the Estimates (<ref>) and (<ref>) (for p and
  for q) show that
  B extends to continuous chain homotopies
  
    B(p)  C_*^p(G)    ⟶ C_*^p(G) 
    B(q)  C_*^q(G)    ⟶ C_*^q(G)

  between E ∘ i and the identity on C_*^p(G) and between i
  ∘ E and the identity on C_*^q(G), respectively. Therefore,
  i is a chain homotopy equivalence and thus induces an
  isomorphism H_*^p(G) ⟶ H_*^q(G) on homology.





 §.§ Diffusion


It remains to construct the maps E and B in
Proposition <ref>. The fundamental observation is
that ℓ^p-norms on C_*(G;) can be decreased by diffusing the
coefficients over a large number of simplices. Therefore, we diffuse
the simplices by coning them off with cone points in annuli of suitable
radii (Figure <ref>). 





  Let G be a finitely generated group, k ∈ and let Z  G^*
  ⟶ P_(G) be a map (here P_(G) denotes the collection of finite subsets of G). The diffusion cone
    operator associated with Z is defined by
  
    C_k(G;)    ⟶ C_k+1(G;)
        
    
          [e,g_1,…,g_k]    ⟼1/|Z_(g_1,…, g_k)|·∑_z ∈ Z_(g_1,…,g_k) [z,e,g_1,…,g_k] .



The key parameter of the diffusion cone construction is the
function Z determining the supports of the diffused simplices. We
will use wide enough annuli of large enough radii. More precisely,
we let the radii grow polynomially (of high degree) in terms of
the diameter of the original simplices.


  Let G be a finitely generated group with a chosen finite generating set S
  and let N ∈_>10. We define the diffusion annuli map of degree N for k ∈ by
  
    Z  
        G^k    ⟶ P_(G) 
    
        g    ⟼Z__S(g),

  where 
  
    Z   ⟼ P_(G) 
    
        r    ⟼Z _r := 
        { g ∈ G | r^N - r^N/10 < d_S(e,g) ≤ r^N
        }.

  Moreover, we write 
  
    φ   ⟶
    
        r    ⟼ 2 · r^N.



Before starting with the actual proof of Proposition <ref>,
we first collect some basic estimates concerning this diffusion construction:


  In the situation of Definition <ref>, we have for all k ∈_≥ 1,
  all g ∈ G^k, and all z ∈ Z_g:
  
  
  *  Clearly, _S [z,e,g_1,…,g_k] ≤φ(_S(g)).
  
  * If j ∈{1,…,k}, h ∈ G^k, and w ∈ Z_h satisfy the
    relation 
    [z,e,g_1,…,g_j,…,g_k]
      = [w,e,h_1,…,h_j,…,h_k], then
    
    w = z
             and  _S(g) = _S(h).

  
  * If h ∈ G^k and w ∈ Z_h satisfy [z,g_1,…,g_k] =[w,h_1,…,h_k], then
    
    w = h_1 · g_1^-1· z
             and  _S(g) = _S(h).

  


  Ad 1.
  This is immediate from the construction.

  Ad 2.
  Because both simplices have the same 1-vertex (namely e), all the
  vertices must coincide. Thus, w = z. Because the annuli defined by Z
  are disjoint for different radii and because w = z ∈ Z_h ∩ Z_g, we obtain
  _S(g) = _S(h).

  Ad 3.
  The assumption implies that
  
    j ∈{1,…,k} z^-1· g_j = w^-1· h_j.

  In particular, w = h_1 · g_1^-1· z.
  Using the abbreviations r_g := _S(g) and r_h := _S(h),
  we obtain by the triangle inequality that  d_S(e,z^-1· g_1)
    = d_S(e,w^-1· h_1) is in the intersection 
  
    [r_g^N - r_g^N/10 - r_g, r_g^N + r_g]
        ∩ 
        [r_h^N - r_h^N/10 - r_h, r_h^N + r_h]

  (which is hence non-empty). 
  Therefore, r_g = r_h.



  In the situation of Definition <ref>, let k ∈, and let 
  
    I_k := { (z,g_1,…,g_k) | (g_1,…, g_k) ∈ G^k, z ∈ Z_g }.

  Furthermore, let J_k be a set, let π I_k ⟶ J_k be a
  map, and let β I_k ⟶ be a function controlling the
  size of the fibres of π, i.e.,
  
    i ∈ I_k| π^-1(π(i))| ≤β(i).

  For functions f  I_k ⟶, we define the push-forward
  
    π_* f  J_k    ⟶
    
        j    ⟼∑_i ∈π^-1(j) f(i).

  Finally, let p ∈ (1,∞) and let f  I_k
  ⟶ be a function with finite support.
  
  
  * Then
    
    π_* f_p ≤(
           ∑_i ∈ I_kβ(i)^p ·|f(i) |^p )^1/p.

  
  * Moreover, let q ∈ (1,∞) with p< q, let q' ∈
    (1,∞) with 1/q + 1/q' = 1/p,
    and let w  I_k ⟶
    be a function such that π_* w  J_k ⟶ is q'-summable.
    Then (with respect to pointwise multiplication)
    
    π_*(f · w) _p
             ≤π_* f _q ·π_*w_q'.

  


  The first part is a consequence of the following elementary estimate: For
  all n ∈ and all a_1,…,a_n ∈, we have (by looking at a
  coefficient of maximal modulus) 
  
    |a_1 + … + a_n|^p ≤ n^p ·( |a_1|^p + … + |a_n|^p).

  The second part is just an instance of the generalized Hölder inequality.





 §.§ Completing the proof of the comparison result




  We choose the parameter N := 100 for the construction
  in Definition <ref> (basically any choice will work
  because of the exponential growth of G).
  Let Z  G^* ⟶ P_(G) be the associated
  diffusion annuli map (Definition <ref>) and let B 
  C_*(G;) ⟶ C_*+1(G;) be the diffusion cone operator
  associated with Z (Definition <ref>). We then
  define
  
    E := 𝕀 - ∂∘ B - B ∘∂ C_*(G;) ⟶ C_*(G;).

  It is clear
  that E is a chain map and B a chain homotopy between E
  and the identity on C_*(G;).

  Therefore, it remains to prove the norm estimates. We first replace 
  this zoo of estimates by the following estimates: For all k,n ∈
  there exist K ∈_>0 and m ∈ such that for all c ∈ C_k(G;)
  we have
  
    B(c) _n,p^S
           ≤ K ·c_m,p^S 
        
    
         B(c) _n,p^S
           ≤ K ·c_m,q^S 
        
     c - ∂ B(c) _n,p^S
           ≤ K ·c_m,q^S

  These estimates imply the Estimates (<ref>)–(<ref>) (modulo
  unification of the constants by taking the maximum) as follows:
  
  
  * Estimate (<ref>) follows from Estimate (<ref>).
  
  * Estimate (<ref>) follows from the fact that E = 𝕀 -
    ∂∘ B - B ∘∂ and the Estimates (<ref>) and
    (<ref>) (with modified constants).
  
  * Estimate (<ref>) follows from (<ref>) and the
    fact that E is a chain map.
  
  * Estimate (<ref>) then follows from ∂∘ B = 𝕀 - E - B ∘∂
    and the Estimates (<ref>) (and Remark <ref>), and (<ref>)
    (with modified constants).
  

  In the following, let k, n∈, and let
  
    c= ∑_g ∈ G^k a_g · [e,g_1, …, g_k] ∈ C_k(G;).


  We will first prove (<ref>);
  of course, (<ref>) follows from (<ref>) (with Remark <ref>),
  but we will use this straightforward estimate as warm-up for the other estimates.
  By construction of the
  diffusion cone operator B, we have (using Lemma <ref>.<ref> for the first inequality, and Definition <ref> of φ for the second inequality)
  
    B_k(c) _n,p^S
            = ∑_g ∈ G^k a_g ·1/|Z_g|·∑_z ∈ Z_g [z,e,g_1,…,g_k] _n,p^S
        
        = (
        ∑_g ∈ G^k∑_z∈ Z_g1/|Z_g|^p· |a_g|^p ·(_S[z,e,g_1,…,g_k])^n
        )^1/p
       ≤(
        ∑_g ∈ G^k1/|Z_g|^p-1· |a_g|^p
        ·φ(_S(g))^n
        )^1/p
       ≤(
        ∑_g ∈ G^k |a_g|^p
        · 2^n ·(_S(g))^N · n)^1/p
        = 2^n/p·c_N· n,p.


  Before proving (<ref>) and (<ref>),
  let us first fix some notation:
  Because of q > p, there is some q' such that 1/q +
  1/q' = 1/p; let x := 1/q, let y := 1 - x = 1-1/p+1/q', and let
  ε := y · q' - 1 = q' · (1 - 1/p) > 0.   
  
  Let us establish (<ref>) (with m = 1): 
  The generalized Hölder inequality shows that
  
    B_k(c) _n,p^S
            = (
        ∑_g ∈ G^k∑_z∈ Z_g1/|Z_g|^p· |a_g|^p ·(_S[z,e,g_1,…,g_k])^n
        )^1/p
       ≤(
        ∑_g ∈ G^k∑_z∈ Z_g1/|Z_g|^q· x· |a_g|^q ·_S(g)
        )^1/q
           ·(
        ∑_g ∈ G^k∑_z∈ Z_g1/|Z_g|^q'· y·(_S[z,e,g_1,…,g_k])^q'· n / p/(_S(g))^q'/ q)^1/q'

  We denote the first factor by A_1 and the second factor by A_2.
  As q · x =1, we obtain
  
    A_1     = (
        ∑_g ∈ G^k |Z_g| ·1/|Z_g|^q· x· |a_g|^q ·_S(g)
        )^1/q
        = c_1,q^S.

  The term A_2 can be estimated via
  
    A_2^q'   ≤∑_g ∈ G^k∑_z∈ Z_g1/|Z_g|^q'· y·φ(_S(g))^q'· n / p/(_S(g))^q'/ q
       ≤∑_g ∈ G^k1/|Z_g|^q'· y - 1·φ(_S(g))^q'· n / p/(_S(g))^q' / q
       ≤∑_r ∈| { g∈ G^k |_S{e,g_1,…,g_k} = r}|/|Z_r|^ε·φ(r)^q'· n/p/r^q'/q
       ≤∑_r ∈β_G,S(r)^k/|Z_r|^ε·φ(r)^q'· n/p/r^q'/q,

  where β_G,S⟶ denotes the growth
  function of G with respect to S.
  The second factor in the series above is dominated by a polynomial (in r);
  we will now show that the first factor decreases exponentially in r:
  By definition, we have 
  
    β_G,S(r)^k ≤(4 · |S| )^r · k.

  Because G has exponential growth, there is an α∈_>1
  such that β_G,S(r) ≥α^r holds for all r ∈.
  Therefore, for all r ∈, 
  
    |Z _r|
         ≥|β_G,S(r^N/10/2)| 
         ≥α^r^N/10/2,

  and so
  
    β_G,S(r)^k/|Z_r|^ε   ≤(4 · |S|)^r · k/α^ε/2 · r^N/10,

  which (eventually) decreases exponentially in r. Hence, A_2^q' is dominated by a convergent
  series (whose value is independent of c). This shows Estimate (<ref>).
  
  Finally, we prove the most delicate Estimate (<ref>). By construction,
  
    c - ∂    B_k(c) 
        = ∑_g ∈ G^k a_g ( [e,g_1,…,g_k]
        - 1/|Z_g|·∑_z ∈ Z_g∂ [z,e,g_1,…,g_k])
        
        = ∑_g ∈ G^k a_g ·1/|Z_g|·∑_z ∈ Z_g( -[z,g_1,…,g_k]
            + ∑_j=1^k (-1)^j+1· [z,e,g_1,…,g_j, …, g_k] ).

  Therefore,
  
    c - ∂ B_k(c) _n,p^S
           ≤∑_g∈ G^k1/|Z_g|· a_g ·∑_z ∈ Z_g [z,g_1,…,g_k]_n,p^S
        
            + ∑_j=1^k ∑_g∈ G^k1/|Z_g|· a_g ·∑_z ∈ Z_g [z,e,g_1,…,g_j,…, g_k]_n,p^S.

  We will treat these k+1 sums separately. 
  In order to introduce -_m,q, we again will use the generalized Hölder
  inequality. However, in contrast with the previous estimates, we now have to 
  carefully control accumulations of coefficients
  on k-simplices (using Lemma <ref> and Lemma <ref>).
    
  We will only treat the first sum in detail (the other sums can be handled in
  the same way by modifying J_k accordingly).
  We will apply Lemma <ref> to the following situation:
  
  
  We consider the set
    
    J_k := { [z,g_1,…,g_k] | (z,g_1,…,g_k) ∈ I_k }⊂ C_k(G;),

    together with the canonical projection π I_k ⟶ J_k.    
  
    In view of Lemma <ref>, the projection π has β-controlled
    fibres, where
    
    β I_k    ⟶
    
          (z,g_1,…,g_k)    ⟼β_G,S(_S (g) )^k.

  
    Let δ∈_>0 with δ < y - 1/q' = min(y, ε/q') and 
    
    f  I_k    ⟶
    
          (z,g_1,…,g_k)    ⟼1/|Z_g|^x + δ· a_g ·_S[z,g_1,…,g_k]^1/q ,
          
    
          w  I_k    ⟶
    
          (z,g_1,…,g_k)    ⟼1/|Z_g|^y-δ·_S[z,g_1,…,g_k]^n/p - 1/q .

  
  Then, by construction,
  
    ∑_g ∈ G^k1/|Z_g|· a_g ·∑_z ∈ Z_g [z,g_1,…,g_k] _n,p^S
         = π_* (f · w) _p.

  We will now bound π_*(f · w)_p from above with the help of
  Lemma <ref>: Clearly, f has finite support. Let us show that π_* w is a q'-summable function. By definition of w, we have
  (with Φ(r) := 2 · r^N · (n/p-1/q) · q') 
  
    ∑_i ∈ I_kβ(i)^q'·|w(i)|^q'   ≤∑_(z,g) ∈ I_kβ(z,g)^q'/|Z_g|^(y-δ)· q'·Φ(_S(g))
        
       ≤∑_r∈β_G,S(r)^k · |Z_r| ·β_G,S(r)^q'· k/|Z_r|^(y-δ) · q'·Φ(r)
        
       ≤∑_r∈β_G,S(r)^k+q'· k/|Z_r|^(y-δ) · q' -1·Φ(r).

  Because (y- δ) · q' -1 > 0, the same argument as in the proof
  of Estimate (<ref>) shows that first factor (eventually) decreases at least
  exponentially in r while Φ grows only polynomially in r. Therefore,
  this series is convergent; let A_2 be the value of this series.
  The first part of Lemma <ref>
  shows that π_* w is q'-summable and that
  
    π_* w _q'≤ A_2^1/q^'.

  Therefore, the second part of Lemma <ref> shows that
  
    π_*(f · w)_p
           ≤π_* f _q ·π_* w_q'≤ A_2^1/q^'·π_* f_q.

  It hence remains to provide a suitable estimate for π_*f_q. 
  Using Lemma <ref>, we obtain
  
    π_*f_q^q
           ≤∑_i ∈ I_kβ(i)^q ·|f(i)|^q
        
       ≤∑_(z,g) ∈ I_kβ_G,S(_S (g))^q · k/|Z_g|^q · x + q ·δ· |a_g|^q ·φ(_S g)
        
       ≤ 2 ·∑_g ∈ G^kβ_G,S(_S (g))^q · k/|Z_g|^1 + q ·δ· |Z_g| · |a_g|^q ·_S(g)^N
        
        = 2 ·∑_g ∈ G^kβ_G,S(_S (g))^q · k/|Z_g|^q ·δ· |a_g|^q ·_S(g)^N .

  Again, because q ·δ > 0, we see as in the proof of the Estimate (<ref>)
  that the first factor is bounded, say by A_1. Then,
  
    π_* f_q^q
      ≤ 2 · A_1 ·∑_g ∈ G^k |a_g|^q ·_S(g)^N
      = 2 · A_1 ·c_N,q^q .

  This completes the proof of Proposition <ref>
  and hence of Theorem <ref>.






§ A VANISHING RESULT IN DEGREE 1


We have the following vanishing result for the free group F_2 of rank 2:


  Let p ∈ (1,∞). Then the canonical homomorphism H_1(F_2;)
  ⟶ H_1^p(F_2) is the zero map. 



  Let S := {α,β} be a free generating set of the free group F_2 of
  rank 2. In this proof, all distances, diameters, norms, etc. will be taken with
  respect to this generating set S.

  Before starting with the actual proof, we perform the following
  reductions:
  
  
  * In view of Theorem <ref>, it suffices to prove
    Theorem <ref> for p>2. 
  
  * Because H_1(F_2;) is generated by the homology classes corresponding
    to the cycles [e,α] and [e,β], it suffices to show that
    the classes in H^p_1(F_2) represented by [e,α] and [e,β] are trivial.
  
  * Since the classes represented by [e,α] and [e,β] only differ by an
    isometric automorphism of F_2, it suffices to prove the vanishing for [e,α]. 
  
  To this end, we will construct an explicit chain b in C^p_2(F_2) whose
  boundary is [e,α].
  
  The geometric idea for the construction of such a 2-chain b is
  to start with two 2-simplices with coefficient 1/2 that
  contain [e,α] as an edge; inductively, we then choose two
  2-simplices with halved coefficients that contain the new
  edges … (Figure <ref>).
  The resulting infinite chain will converge in the ℓ^p-setting
  because the coefficients are distributed over enough summands. 
  The main technical difficulty is to ensure that the
  weights are really distributed so that they do not accumulate on
  simplices via accidental cancellations. This will be
  achieved by a careful selection of markers and suffixes that encode
  the induction level and the two different choices at each stage.

  
  
  We will describe the construction of b in a top-down manner, first giving
  the final formula and then explaining all the ingredients: For D∈, we
  set
  
    b(D) := ∑_d=0^D ∑_x ∈ W(d)1/2^d+1·ε(x) ·(s(x) + t(x)) ∈ C_2(F_2;).

  We will then show that the sequence (b(D))_D ∈ converges to a chain
  
    b := lim_D →∞ b(D) ∈ C^p_2(F_2)

  that satisfies ∂ b = [e,α]. But first we have to explain
  the ingredients of b(D): To this end, we define (by mutual recursion)
  the subsets W(d) ⊂ F_2 (keeping track of the set of edges),
  the suffixes s_d, t_d ∈ F_2, the markers m(x) ∈ F_2
  and the 2-simplices s(x) and t(x):
  
  
  * For each d ∈, we set s_d := α^dβ^d and t_d := β^d α^d ∈ F_2.
  
  * We set W(0) := {α} (and m(α) := e)
    and for d ∈_≥ 1, we let
    
    W(d) := ⋃_x∈ W(d-1) W(x,d) ⊂ F_2,

    where (for each x ∈ W(d-1))
    
    W(x,d) := { x m(x) s_d, m(x)s_d, xm(x)t_d, m(x) t_d}⊂ F_2.

  
  * Inductively, we see that |W(d)| ≤ 4^d for all d ∈. We
    can thus choose an injection m  W(d) ⟶{α, β}^2· d and view the words m(x) with x
    ∈ W(d) as elements of F_2.
  
  * For d ∈ and x ∈ W(d), we set
    
    s(x) := [e, x, xm(x)s_d ],   
           t(x) := [e, x, xm(x)t_d ] ∈ C_2(F_2;).

  
  * Finally, the signs ε(…) are defined as follows:
    We set ε(α) :=1; for d ∈_>0
    and x ∈ W(d-1), we set
    
    ε(m(x) s_d)     := - ε(x)
          
    ε(m(x) t_d)     := - ε(x)
          
    ε(xm(x) s_d)     := ε(x)
          
    ε(xm(x) t_d)     := ε(x).

  
  By construction, all elements of W(d) consist of non-negative
  powers of α and β and no cancellations occur in the
  definitions above. Therefore, s, t, and ε are
  well-defined. Moreover, the construction of the edge sets W(d) is justified
  by the following observation: For each d ∈ and each x ∈ W(d),
  we have
  
    ∂( s(x) + t(x) )
            = ∂( [e,x,xm(x)s_d] + [e,x,xm(x)t_d])
        
        = [e,m(x)s_d] - [e,xm(x)s_d] + [e,x]
          + [e,m(x)t_d] - [e,xm(x)t_d] + [e,x].



  
  In order to prove convergence of (b(D))_D ∈ and (∂
  b(D))_D ∈, we need to estimate the diameters of the
  simplices involved: For d ∈ and x ∈ W(d), we have
  
    s(x) ≤ d(e,x) + 4 · d,
         t(x) ≤ d(e,x) + 4 · d;

  inductively, we obtain for x ∈ W(d)

    d(e,x) ∈ O(d^2)
  
  and therefore 
  
    s(x),   t(x) ∈ O(d^2).

  We now give the convergence arguments:
  
  
  * The sequence (b(D))_D ∈ is Cauchy with respect to -_n,p: 
    Let D, D' ∈ with D' > D ≥ 0 and let n ∈. By construction, we
    have
    
    b(D') - b(D)
              = ∑_d=D+1^D'∑_x ∈ W(d)1/2^d+1·ε(x) ·(s(x) + t(x)).

    The markers/suffixes show that all of these 2-simplices are different (so
    no cumulations of coefficients occur). Therefore,
    
    b(D') - b(D) _n,p   ≤∑_d=D+1^D'|W(d)|/(2^d+1)^p· O(d^2n)
          
       ≤∑_d=D+1^D'4^d/(2^d+1)^p· O(d^2n).

    Because p > 2, the corresponding series on the right-hand side
    is convergent. Therefore, these differences between its partial
    sums form a Cauchy sequence.
    
  
  * The sequence (∂ b(D))_D ∈ is Cauchy with respect to -_n,p: 
    Let D ,D' ∈ with D' > D ≥ 0 and let n ∈. By construction, we have
    
    ∂ b(D') - ∂ b(D) 
              = ∑_d = D+1^D'∑_x∈ W(d)1/2^d+1·ε(x)
          ·∂(s(x) + t(x)) 
          
    
    
    
    
              = ∑_x ∈ W(D+1)1/2^D+1·ε(x) · [e,x] - ∑_y ∈ W(D'+1)1/2^D'+1·ε(y) · [e,y].

    The markers/suffixes show that all of these 1-simplices are different (so
    no cumulations of coefficients occur). Therefore,
    
    ∂ b(D') - ∂ b(D) _n,p   ≤|W(D' +1)|/(2^D'+1)^p· O(D'^2n)
          + |W(D+1)|/(2^D+1)^p· O(D^2n)
          
       ≤4^D'+1/(2^D'+1)^p· O(D'^2n)
          + 4^D+1/(2^D+1)^p· O(D^2n).

    Because p > 2, these terms converge to 0 for D,D' →∞.
  

  Thus, we have established that b = lim_D →∞ b(D) ∈
  C^p_2(F_2) is a well-defined chain. By a similar computation as the previous one for ∂ b(D') - ∂ b(0), 
  we have
  
    ∂ b = [e,α],

  as claimed.



amsalpha


