A Subgroup Theorem for Homological Filling Functions
aa r X i v : . [ m a t h . G R ] A ug A SUBGROUP THEOREM FOR HOMOLOGICAL FILLING FUNCTIONS
RICHARD GAELAN HANLON AND EDUARDO MART´INEZ-PEDROZAA bstract . We use algebraic techniques to study homological filling functions of groupsand their subgroups. If G is a group admitting a finite ( n + K ( G ,
1) and H ≤ G is of type F n + , then the n th –homological filling function of H is bounded above bythat of G . This contrasts with known examples where such inequality does not hold underweaker conditions on the ambient group G or the subgroup H . We include applications tohyperbolic groups and homotopical filling functions.
1. I ntroduction
The n th homological and homotopical filling functions of a space are generalized isoperi-metric functions describing the minimal volume required to fill an n –cycle or n –sphere withan ( n + n + n th homological filling functions of a finitely presentedgroup and its subgroups. Our main result provides su ffi cient conditions for the n th -fillingfunction of a subgroup to be bounded from above by the n th -filling function of the ambientgroup. The hypotheses of our theorem are in terms of finiteness properties of the ambientgroup and the subgroup. Our result contrasts with known examples illustrating that thisrelation does not hold under weaker conditions [4, 22, 21].1.1. Statement of Main Result. A K ( G ,
1) for a group G is a cell complex X with con-tractible universal cover e X and fundamental group isomorphic to G . If G admits a K ( G , n -skeleton, then G is said to be of type F n . Such finiteness properties are natural(topological) generalizations of being finitely generated (type F ) and finitely presented(type F ).If X is a K ( G ,
1) with finite ( n + n th –homological filling function of G is an optimal function FV n + G : N → N such that FV n + G ( k ) bounds the minimal volumerequired to fill an n –cycle γ of e X of volume at most k , with an ( n + µ of e X havingboundary ∂ ( µ ) = γ . See Section 3 for precise definitions.It can be shown that the growth rate of FV n + G is independent of the choice of X up toan equivalence relation ∼ , hence FV n + G is an invariant of the group G , see [9, 20]. Therelation f ∼ g between functions is defined as f (cid:22) g and g (cid:22) f , where f (cid:22) g means thatthere is C > n ∈ N , f ( n ) ≤ Cg ( Cn + C ) + Cn + C . Our main result is ageneralization of a result of Gersten [12, Thm C] to higher dimensions. Theorem 1.1.
Let n ≥ . Let G be a group admitting a finite ( n + -dimensional K ( G , and let H ≤ G be a subgroup of type F n + . ThenFV n + H (cid:22) FV n + G . Mathematics Subject Classification.
Key words and phrases.
Filling Functions, Isoperimetric Functions, Dehn functions, Hyperbolic Groups,Finiteness Properties.
Some examples that contrast with Theorem 1.1 are the following. In [4], Noel Bradyconstructed a group G admitting a finite 3–dimensional K ( G ,
1) such that FV G is linear,and G contains a subgroup H ≤ G of type F with FV H at least quadratic. Anothersource of examples are the generalized Heisenberg groups H n + , for which Robert Youngcomputed the homological filling invariants in [22, 21]. For instance, H admits a finite5–dimensional K ( H ,
1) and has quadratic FV H . On the other hand, H can be embeddedin H , admits a 3–dimensional K ( H , FV H . Likewise, H has quadratic FV H and can be embedded in H which has FV H polynomial of degree 3 / H with decidable word problem in nondeterministicpolynomial time, Birget, Ol’shanskii, Rips and Sapir produce an embedding of H into afinitely presented group G with polynomial Dehn function [3]. For this construction, The-orem 1.1 implies that if H has a finite 2-dimensional K ( H ,
1) and FV H is not bounded bya polynomial function, then G does not admit a finite 2-dimensional K ( G , H is the Baumslag-Solitar group B ( m , n ) with | m | , | n | , for whichthe embedding constraint is known [12, Thm A].We discuss some applications of Theorem 1.1 to hyperbolic groups and homotopicalfilling functions below. Recall that a group G is hyperbolic if it has a linear Dehn function.In [13], Gersten proved the following: Theorem 1.2. [13, Thm 4.6]
Let G be a hyperbolic group of cohomological dimension .Then every finitely presented subgroup H ≤ G is hyperbolic.
Gersten’s result does not hold in higher dimensions as Brady has exhibited a hyperbolicgroup G of cohomological dimension 3 containing a non–hyperbolic finitely presented sub-group H ≤ G [4]. We can however, obtain a result similar to Theorem 1.2 by consideringhomotopical filling functions of higher dimensions. The n th –homotopical filling function δ nG of a group G is defined analogously to FV n + G but restricts to filling n –spheres with( n + K ( G ,
1) with finite ( n + δ nG ( k ) bounds the minimum volume required to fill an n –sphere of volume atmost k , with an ( n + δ nG can be found in [2, 5]. Corollary 1.3.
Let G be a hyperbolic group of geometric dimension n + , where n ≥ .Let H ≤ G be of type F n + . Then δ nH is linear. Recall that the geometric dimension of a group G is the minimum dimension among K ( G , G are equal for dimensions greater or equal than 3. This justifiesour use of geometric dimension in the corollary above. In addition to Corollary 1.3, wehave the following homotopical version of Theorem 1.1 for su ffi ciently large n . Corollary 1.4.
Let n ≥ . Let G be a group admitting a finite ( n + –dimensional K ( G , .Let H ≤ G be of type F n + . Then δ nH (cid:22) δ nG . Corollaries 1.3 and 1.4 follow from Theorem 1.1 and the following results:
Theorem 1.5. [1, pg. 1 and references therein]
For n ≥ , the n th –homotopical andhomological filling functions δ nG and FV n + G are equivalent. For n = , δ G (cid:22) FV G . Theorem 1.6. [17]
Let G be a hyperbolic group. Then FV n + G is linear for all n ≥ .Proof of Corollary 1.3. A theorem of Rips imples that G admits a compact K ( G , G admits a compact ( n + SUBGROUP THEOREM FOR HOMOLOGICAL FILLING FUNCTIONS 3 K ( G , FV n + H is linear. It then follows fromTheorem 1.5 that δ nH is also linear. (cid:3) Proof of Corollary 1.4.
Apply Theorems 1.5 and 1.1. (cid:3)
Remark 1.7.
Corollary 1.3 does not apply to Brady’s example H ≤ G mentioned abovesince H is not of type F . It is an open question whether or not the subgroups H inCorollary 1.3 are in fact hyperbolic. Remark 1.8.
It is an open question whether or not the statement of Corollary 1.4 holdsfor n = or . In general δ G / FV G and δ G / FV G , examples of such groups are givenin [1, 20] . Outline of the Paper.
The rest of the paper is organized into three sections. Sec-tion 2 contains the definition of a filling norm on a finitely generated Z G -module and lem-mas required for the proof of Theorem 1.1. Section 3 contains algebraic and topologicaldefinitions for FV n + G . Section 4 contains the proof of Theorem 1.1.1.3. Acknowledgments.
Thanks to Noel Brady and Mark Sapir for comments on an ear-lier version of the article. We especially thank the referee for a list of useful comments andcorrections. We acknowledge funding by the Natural Sciences and Engineering ResearchCouncil of Canada, NSERC.2. F illing N orms on Z G -M odules In this section we define the notion of a filling-norm on a finitely generated Z G –module.Most ideas in this section are based on the work of Gersten in [13]. The section containsfour lemmas on which the proof of the main result of the paper relies on. Definition 2.1 (Norm on Abelian Groups) . A norm on an Abelian group A is a function k · k : A → R satisfying the following conditions: • k a k ≥ with equality if and only if a = , and • k a k + k a ′ k ≥ k a + a ′ k .If, in addition, the norm satisfies • k na k = | n | · k a k , for n ∈ Z ,then it is called a regular norm . If A is free Abelian with basis X , then X induces a regular ℓ –norm on A given by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈ X n x x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X x ∈ X | n x | , where n x ∈ Z . Definition 2.2 (Linearly Equivalent Norms) . Two norms k · k and k · k ′ on a Z –module Mare linearly equivalent if there exists a fixed constant C > such thatC − k m k ≤ k m k ′ ≤ C k m k for all m ∈ M. This is an equivalence relation and the equivalence class of a norm k · k iscalled the linear equivalence class of k · k . Definition 2.3 (Based Free Z G –modules and Induced ℓ -norms) . Suppose G is a groupand F is a free Z G–module with Z G − basis { α , . . . , α n } . Then { g α i : g ∈ G , ≤ i ≤ n } is afree Z -basis for F as a (free) Z -module. This free Z –basis induces a G-equivariant ℓ -norm k · k on F. We call a free Z G − module based if it is understood to have a fixed basis, andwe use this basis for the induced ℓ –norm k · k . R.G. HANLON AND E. MART´INEZ-PEDROZA
Definition 2.4 (Filling Norms on Z G -modules) . Let η : F → M be a surjective homomor-phism of Z G-modules and suppose that F is free, finitely generated, and based. The fillingnorm on M induced by η and the free Z G -basis of F is defined as k m k η = min { k x k : x ∈ F , η ( x ) = m } . Observe that this norm is G-equivariant.
Remark 2.5.
Gersten observed that filling norms are not in general regular norms. Heillustrated this fact with the following example [11] . Let X be the universal cover of thestandard complex of the group presentation h x | x , x k i , where k ≥ . The filling normon the integral cellular -cycles Z ( X ) induced by C ( X ) ∂ → Z ( X ) is not regular since k x k ∂ = k kx k ∂ = . Remark 2.6 (Induced ℓ -norms are Filling Norms) . If F is a finitely generated based free Z G-module, then the ℓ -norm induced by a free Z G-basis is a filling norm.
The following lemma is reminiscent of the fact that linear operators on finite dimen-sional normed spaces are bounded.
Lemma 2.7 ( Z G -Morphisms between Free Modules are Bounded) . [13, Proof of Propo-sition 4.4] Let ϕ : F → F ′ be a homomorphism between finitely generated, free, based Z G − modules. Let k · k and k · k ′ denote the induced ℓ –norms of F and F ′ . Then thereexists a constant C > such that for all x ∈ F k ϕ ( x ) k ′ ≤ C · k x k . Proof.
Let A = { α , . . . , α n } be the Z G − basis of F inducing the norm k · k . Then ϕ isgiven by a finite n × m matrix whose entries are elements of Z G . Observe that for any g ∈ G , x ∈ F , we have k x k = k gx k . Define C = max ≤ i ≤ n {k ϕ ( α i ) k} and let x ∈ F be arbitrary.Then k ϕ ( x ) k ′ = k ϕ ( λ α + · · · + λ n α n ) k ′ , where λ i ∈ Z G ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X g ∈ G λ , g g ϕ ( α ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ′ + · · · + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X g ∈ G λ n , g g ϕ ( α n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ′ , where λ j = X g ∈ G λ j , g g and λ j , g ∈ Z ≤ X g ∈ G | λ , g | k ϕ ( α ) k ′ + · · · + X g ∈ G | λ n , g | k ϕ ( α n ) k ′ ≤ C m X i = X g ∈ G | λ i , g | = C k x k . (cid:3) Lemma 2.8 ( Z G -Morphisms with Projective Domain are Bounded) . Let ϕ : P → Q be ahomomorphism between finitely generated Z G − modules. Let k · k P and k · k Q denote fillingnorms on P and Q respectively. If P is projective then there exists a constant C > suchthat for all p ∈ P k ϕ ( p ) k Q ≤ C · k p k P . Proof.
Consider the commutative diagram A ˜ ϕ / / ρ (cid:15) (cid:15) B (cid:15) (cid:15) P ϕ / / ψ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ Q SUBGROUP THEOREM FOR HOMOLOGICAL FILLING FUNCTIONS 5 constructed as follows. Let A and B be finitely generated and based free Z G -modules, andlet A → P and B → Q be surjective morphisms inducing the filling norms k · k P and k · k Q .Since P is projective and B → Q is surjective, there is a lifting ψ : P → B of ϕ ; then let ˜ ϕ be the composition A ρ → P ψ → B . Let C be the constant provided by Lemma 2.7 for ˜ ϕ . Let p ∈ P and let a ∈ A that maps to p . It follows that k ϕ ( p ) k Q ≤ k ψ ( p ) k = k ˜ ϕ ( a ) k ≤ C k a k . Since the above inequality holds for any a ∈ A with ρ ( a ) = p , it follows that k ϕ ( p ) k Q ≤ C · min ρ ( a ) = p {k a k } = C · k p k P . (cid:3) Lemma 2.9 (Equivalence of Filling Norms for Z G -modules) . [13, Lemma 4.1] Any twofilling norms on a finitely generated Z G–module M are linearly equivalent.Proof.
Consider a pair of surjective homomorphisms of Z G − modules η : F → M and η ′ : F ′ → M such that F and F ′ are finitely generated, free, based modules inducing thefilling norms k · k η and k · k η ′ on M . Since η ′ is surjective, the universal property of F provides a homomorphism ϕ such that η = η ′ ◦ ϕ . Let m ∈ M be arbitrary and take x ∈ F such that η ( x ) = m . Since η ′ ◦ ϕ ( x ) = m , by Lemma 2.7 there exists C > k m k η ′ = min η ′ ( x ′ ) = m k x ′ k ′ ≤ k ϕ ( x ) k ′ ≤ C · k x k . As this inequality holds for all x ∈ F satisfying η ( x ) = m , we have k m k η ′ ≤ C · min η ( x ) = m {k x k } = C · k m k η . The other inequality proceeds in a similar manner. (cid:3)
Lemma 2.10 (Retraction Lemma) . [13, Prop. 4.4] Let → M ι → N → P → be a shortexact sequence of Z G − modules where M is finitely generated and equipped with a filling-norm k · k M . N is free, based, and equipped with the induced ℓ -norm k · k . P is projective.Then there exists a retraction ρ : N → M for the inclusion ι : M → N and a fixed constantC > such that k ρ ( x ) k M ≤ C k x k for all x ∈ N.Proof.
Since P is projective there is a retraction ρ ′ for ι . Since M is finitely generated, N is isomorphic to a product I ⊕ Q of free modules where I is finitely generated and containsthe image of M . Define ρ : N → M by ρ | I = ρ ′ | I and ρ | Q =
0. Then ρ is a retraction for ι with support contained in I .Each x ∈ N has a unique decomposition x = y + q where y ∈ I , q ∈ Q such that ρ ( x ) = ρ ( y ) and k y k ≤ k x k . Apply Lemma 2.8 to the restriction ρ : I → M to obtain C > k ρ ( x ) k M = k ρ ( y ) k M ≤ C k y k ≤ C k x k . (cid:3) R.G. HANLON AND E. MART´INEZ-PEDROZA
3. H omological F illing F unctions of G roups In this section, given a group G of type FP n + , where n ≥
1, we define the groupinvariant FV n + G . In the first part of the section we provide an algebraic definition of FV n + G and prove that it is well defined. This algebraic approach, while naturally inspired by thetopological approach, provides a convenient algebraic framework suitable for some of thearguments in this paper. This algebraic approach has been also explored in [16]. In thesecond part, we recall the topological approach to FV n + G and show that the topological andalgebraic approaches are equivalent for finitely presented groups of type FP n + . The finalsubsection discusses why FV n + G ( k ) is a finite number.3.1. Algebraic Definition of FV n + G .Definition 3.1 (Linearly Equivalent Functions) . Let f and g be functions from N to N .Define f (cid:22) g if there exists C > such that for all n ∈ N f ( n ) ≤ Cg ( Cn + C ) + Cn + C . The functions f and g are linearly equivalent , f ∼ g, if both f (cid:22) g and g (cid:22) f hold. Thisis an equivalence relation and the equivalence class containing a function f is called the linear equivalence class of f . Definition 3.2 ( FP n group) . [6] A group G is of type FP n if there is a resolution of Z G–modules P n ∂ n −→ P n − ∂ n − −→ . . . ∂ −→ P ∂ −→ P −→ Z → , such that for each i ∈ { , . . . , n } the module P i is a finitely generated projective Z G–module. In this case, such a resolution is called an FP n –resolution . Definition 3.3 (Algebraic definition of FV n + G ) . Let G be a group of type FP n + . Thealgebraic n th –filling function is the (linear equivalence class of the) functionFV n + G : N → N defined as follows. LetP n + ∂ n + −→ P n ∂ n −→ . . . ∂ −→ P ∂ −→ P −→ Z → , be a resolution of Z G–modules for Z of type FP n + . Choose filling norms for P n and P n + ,denoted by k · k P n and k · k P n + respectively. ThenFV n + G ( k ) = max (cid:8) k γ k ∂ n + : γ ∈ ker( ∂ n ) , k γ k P n ≤ k (cid:9) , where k γ k ∂ n + = min (cid:8) k µ k P n + : µ ∈ P n + , ∂ n + ( µ ) = γ (cid:9) . Remark 3.4 (Finiteness of FV n + G ) . It is not immediately clear that the maximum in Defi-nition 3.3 is a finite number. In Section 3.3 we recall some results from the literature which,under the assumption G is finitely presented, imply that FV n + G is a finite valued functionfor n = and n ≥ . The authors are not aware of a proof for the case n = .For n = , all results in this paper regarding FV G hold under the following naturalmodifications. First, work with the standard extensions of addition, multiplication, andorder, of the positive integers N to N ∪{∞} . Definition 3.1 is extended to functions N → N ∪{∞} , but we emphasize that the constant C remains a finite positive integer. In Definition 3.3the function FV G is defined as an N → N ∪ {∞} function. We observe that no argument inthis paper relies on FV n + G ( k ) being finite. SUBGROUP THEOREM FOR HOMOLOGICAL FILLING FUNCTIONS 7
Theorem 3.5 ( FV n + G is a Well-defined Group Invariant) . Let G be a group of type FP n + .Then the algebraic n th –filling function FV n + G of G is well defined up to linear equivalence.Proof. Let ( F ∗ , ∂ ∗ ) and ( P ∗ , δ ∗ ) be a pair of resolutions of Z G -modules of type FP n + withchoices of filling-norms for their n th and ( n + th modules denoted by k·k F n and k·k F n + , and k · k P n and k · k P n + respectively. Let FV n + F ∗ and FV n + P ∗ be the induced functions accordingto Definition 3.3. By symmetry, it is enough to show that FV n + F ∗ (cid:22) FV n + P ∗ .It is well known that any two projective resolutions of a Z G -module are chain homotopyequivalent, see for example [6, pg.24, Thm 7.5], and hence the resolutions F ∗ and P ∗ arechain homotopy equivalent. Therefore there exists chain maps f i : F i → P i , g i : P i → F i ,and a map h i : F i → F i + such that ∂ i + ◦ h i + h i − ◦ ∂ i = g i ◦ f i − id . Let C denote the maximum of the constants for the maps g n + , h n , and f n and the chosenfilling-norms provided by Lemma 2.8. We claim that for every k ∈ N , FV n + F ∗ ( k ) ≤ C · FV n + P ∗ ( Ck + C ) + Ck + C . Fix k . Let α ∈ ker( ∂ n ) be such that k α k F n ≤ k . Choose β ∈ P n + such that δ n + ( β ) = f n ( α )and k f n ( α ) k δ n + = k β k P n + . By commutativity of the chain maps and the chain homotopyequivalence, ∂ n + ◦ h n ( α ) + h n − ◦ ∂ n ( α ) = g n ◦ f n ( α ) − α = g n ◦ δ n + ( β ) − α = ∂ n + ◦ g n + ( β ) − α. Since α ∈ ker( ∂ n ), we have that h n − ◦ ∂ ( α ) =
0. Rearranging the above equation, we obtain α = ∂ n + ◦ g n + ( β ) − ∂ n + ◦ h n ( α ) = ∂ n + ( g n + ( β ) − h n ( α )) . Hence g n + ( β ) − h n ( α ) has boundary α . Observe that k α k ∂ n + ≤ k g n + ( β ) − h n ( α ) k F n + since ∂ n + ( g n + ( β ) − h n ( α )) = α ≤ k g n + ( β ) k F n + + k h n ( α ) k F n + by the triangle inequality ≤ C · k β k P n + + C · k α k F n by Lemma 2.8 = C · k f n ( α ) k δ n + + C · k α k F n by definition of β ≤ C · FV n + P ∗ ( k f n ( α ) k P n ) + C k α k F n by definition of FV n + P ∗ ≤ C · FV n + P ∗ (cid:0) C k α k F n (cid:1) + C k α k F n by Lemma 2.8 ≤ C · FV n + P ∗ ( Ck + C ) + Ck + C since k α k F n ≤ k .Since α was arbitrary, FV n + F ∗ ( k ) ≤ C · FV n + P ∗ ( Ck + C ) + Ck + C for all k ∈ N . This showsthat FV n + F ∗ (cid:22) FV n + F ′∗ completing the proof. (cid:3) Topological Definition of FV n + G . For a cell complex X , the cellular chain group C i ( X ) is a free Abelian group with basis the collection of all i -cells of X . This naturalbasis induces an ℓ -norm on C i ( X ) that we denote by k · k . Recall that a complex X is n -connected if its first n -homotopy groups are trivial. Definition 3.6 ( F n group) . A group G is of type F n if there is a K ( G , -complex with afinite n-skeleton, i.e., with only finitely many cells in dimensions ≤ n. R.G. HANLON AND E. MART´INEZ-PEDROZA
Definition 3.7 (Topological Definition of FV n + G ) . [9, 20] Let G be a group acting properly,cocompactly, by cellular automorphisms on an n–connected cell complex X. The topologi-cal n th –filling function of G is the (linear equivalence class of the) function FV n + G : N → N defined as FV n + G ( k ) = max { k γ k ∂ : γ ∈ Z n ( X ) , k γ k ≤ k } , where k γ k ∂ = min { k µ k : µ ∈ C n + ( X ) , ∂ ( µ ) = γ } . J. Fletcher and R. Young have independently provided geometric proofs that the topo-logical n th –filling function FV n + G is well defined as an invariant of the group, see [9, Theo-rem 2.1] and [20, Lemma 1] respectively. In the work of Fletcher, the topological definitionof FV n + G requires X to be the universal cover of a K ( G , Theorem 3.8. [20]
Let G be a group admitting a proper and cocompact action by cellularautomorphisms on an n–connected cell complex. Then the topological n th –filling invariantFV n + G of G is well defined up to linear equivalence. Even in the topological definition, it is not trivial that FV n + G is a finite valued functionand Remark 3.4 also applies in this case. For the rest of the section, we show that thetopological and algebraic approaches to FV n + G are equivalent for finitely presented groupsof type FP n + . Proposition 3.9.
Let n ≥ and let G be a group of type F n + . Then G is of type FP n + andthe algebraic and topological n th –filling functions of G are linearly equivalent.Proof. Let X be a K ( G ,
1) with finite ( n + G -action on the universal cover e X of X induces the structure of a Z G -module to the group of cellular chains C i ( e X ) and eachboundary map ∂ i is a morphism of Z G -modules. Since the G -action on e X is cellular andfree, each C i ( e X ) is a free Z G -module with basis any collection of representatives of the G -orbits of i -cells. Since the action is cocompact on the ( n + C i ( e X ) is afinitely generated free Z G -module for i ∈ { , , . . . , n + } . Since e X is a contractible space,all its homology groups are trivial and therefore we have a resolution of Z G -modules · · · −→ C n + ( X ) ∂ n + −→ C n ( X ) ∂ n −→ . . . ∂ −→ C ( X ) ∂ −→ C ( X ) −→ Z → , of type FP n + . Under our assumptions, the induced topological n th –filling function of G isa particular instance of an algebraic n th –filling function of G . The conclusion then followsfrom Theorems 3.5 and 3.8. (cid:3) Proposition 3.10. [6, pg 205, proof of Thm. 7.1]
Let G be finitely presented and of typeFP n where n ≥ . Then G is of type F n . Propositions 3.9 and 3.10 imply the following statement.
Corollary 3.11.
Let G be a finitely presented group of type FP n + . Then the topologicaland algebraic definitions of FV n + G are equivalent. Finiteness of FV n + G ( k ) . Let G be a finitely presented group of type FP n + , or equiv-alently assume that G is of type F n + ; see Proposition 3.10. We will sketch why FV n + G isa finite valued function for n = n ≥ SUBGROUP THEOREM FOR HOMOLOGICAL FILLING FUNCTIONS 9
Case n = . Finiteness of FV G follows from that of the Dehn function δ G . Wesummarize the argument from Gersten’s article [10, Prop 2 . X be a K ( G ,
1) withfinite 2–skeleton and let z ∈ Z (cid:16)e X (cid:17) be a 1–cycle with k γ k ≤ k . Then z = z + . . . z m forsome m ≤ k where each z i is the 1–cycle induced by a simple edge circuit γ i in e X and m X i = ℓ ( γ i ) = k z k . Then k z k ∂ ≤ m X i = Area ( γ i ) ≤ m X i = δ G ( ℓ ( γ i )) ≤ k · δ G ( k ) < ∞ . Case n ≥ . A group G of type F n + has a well defined invariant called the n th –homotopical filling function δ nG : N → N . There are multiple approaches to define δ nG , wesketch the approach found in [1, 5]. Roughly speaking, if X is a K ( G ,
1) with finite ( n + δ nG ( k ) measures the number of ( n + S n → e X comprised of at most k n –balls. Here the maps f : S n → e X and fillings ˜ f : D n + → e X arerequired to be in a particular class of maps called admissible maps . This allows one todefine the volumes, vol ( f ) and vol ( ˜ f ), as the number of n -balls and ( n + S n and D n + respectively, mapping homeomorphically to open cells of e X . The filling volume of f is given by FVol ( f ) = sup { vol ( ˜ f ) | ˜ f : D n + → e X , ˜ f | ∂ D n + = f } and δ nG by δ nG ( k ) = max { FVol ( f ) | f : S n → e X , vol ( f ) ≤ k } . Alonso et al. use higher homotopy groups as π ( X )–modules to provide a more algebraicapproach to δ nG , in particular they show that δ nG is a finite valued function [2, Corollary 1].It is observed in [5, Remark 2 . FV n + G then follows from the inequality FV n + G (cid:22) δ nG which holds for all n ≥
3. We outline the argument for this inequality described in theintroduction of [1]. Let X be a K ( G ,
1) with finite ( n + γ ∈ Z n (cid:16)e X (cid:17) with k γ k ≤ k . Using the Hurewicz Theorem, one can show (see [15, 19]) that γ is the imageof the fundamental class of an n –sphere for a map f : S n → e X such that vol ( f ) = k γ k .If ˜ f : D n + → e X is an extension of f to the ( n + D n + , then the image of thefundamental class of D n + is an ( n + µ with ∂ ( µ ) = γ and vol (cid:16) ˜ f (cid:17) ≥ k µ k . Thereforethe filling volume FVol ( f ) = sup { vol ( ˜ f ) | ˜ f : D n + → e X , ˜ f | ∂ D n + = f } is greater than or equal to the filling norm k γ k ∂ n + . It follows that FV n + G ( k ) ≤ δ nG ( k )4. M ain R esult As we will be working with cell complexes, all relevant computations in this section areunderstood to occur within cellular chain complexes.
Definition 4.1 (Stably free) . A Z G-module P is stably free if there exists finitely generatedfree Z G module F such that P ⊕ F is free.
Lemma 4.2 (The Eilenberg Trick) . [6, pg.207] Let G = π ( X , x ) , where X is a cellcomplex. Then X is a subcomplex of a complex Y such that the inclusion X ֒ → Y is ahomotopy equivalence, and the cellular n-cycles of the universal covers e Y and e X satisfyZ n (cid:16)e Y (cid:17) ≃ Z n (cid:16)e X (cid:17) ⊕ Z G as Z G-modules.Proof.
Let x be a 0-cell of X , and glue an n -cell D n to ( X , x ) by mapping its boundaryto x . The resulting space is the wedge sum of X and an n -sphere S n . To obtain Y , attachan ( n + D n + by the attaching map that identifies ∂ D n + with the n -sphere S n . Then Z n (cid:16)e Y (cid:17) ≃ Z n (cid:16)e X (cid:17) ⊕ Z G where the Z G factor is generated by a lifting of the n -cell D n to e Y . Itis clear that X ֒ → Y is a homotopy equivalence. (cid:3) Lemma 4.3 (Schanuel’s Lemma) . [6, pg.193, Lemma 4.4] Let → P n → P n − → · · · → P → M → and → P ′ n → P ′ n − → · · · → P ′ → M → be exact sequences of R–modules with P i and P ′ i projective for i ≤ n − . ThenP ⊕ P ′ ⊕ P ⊕ P ′ ⊕ . . . ≃ P ′ ⊕ P ⊕ P ′ ⊕ P ⊕ . . . We are now ready to prove our main result which is a generalization of [12, Thm C]. Theproof is based on Gersten’s proof of [13, Thm 4.6] and is adjusted for higher dimensions:
Theorem 4.4.
Let G be a group admitting a finite ( n + -dimensional K ( G , and letH ≤ G be a subgroup of type F n + . Then FV n + H (cid:22) FV n + G . Proof.
Let W be a finite ( n + K ( G , X be the ( n + K ( H , H is of type F n + , we may assume that X is a finite cell complex. Then,after subdivisions, there exists a cellular map f : X → W inducing the inclusion H ֒ → G at the level of fundamental groups. Let M f be the mapping cylinder of f and consider theexact sequences of Z G -modules(4.1) 0 → Z n (cid:16) e M f (cid:17) → C n (cid:16) e M f (cid:17) → · · · → C (cid:16) e M f (cid:17) → Z → → C n + (cid:16) e W (cid:17) → C n (cid:16) e W (cid:17) → · · · → C (cid:16) e W (cid:17) → Z → , where e W and e M f denote the universal covers of W and M f respectively.Applying Schanuel’s lemma to the above sequences shows that the Z G − module Z n (cid:16) e M f (cid:17) is finitely generated and stably free. Let Y be the space obtained by attaching a finitenumber of ( n + M f as in Lemma 4.2 such that Z n (cid:16)e Y (cid:17) is finitelygenerated and free as a Z G -module.From here on, we are only concerned with the inclusion map X → Y realizing theinclusion H → G at the level of fundamental groups with the property that Z n (cid:16)e Y (cid:17) is finitelygenerated and free as a Z G -module. Since the inclusion X → Y is injective at the level offundamental groups, any lifting e X → e Y is an embedding. Moreover, we can choose thelifting to be equivariant with respect to the inclusion H → G . Without loss of generality,assume that e X is an H -equivariant subcomplex of e Y .Since the ring Z G is free as a Z H –module, it follows that C i (cid:16)e Y (cid:17) is a free Z H -module.Since e X is an H -equivariant subcomplex of e Y , the Z H − module C i (cid:16)e X (cid:17) is a free factor of C i (cid:16)e Y (cid:17) . Hence the quotient C i (cid:16)e Y ( n ) , e X ( n ) (cid:17) = C i (cid:16)e Y (cid:17) / C i (cid:16)e X (cid:17) is a free Z H − module. SUBGROUP THEOREM FOR HOMOLOGICAL FILLING FUNCTIONS 11
Restricting our attention to n -skeleta, the following short exact sequence of chain com-plexes of Z H -modules arises(4.3) 0 → C ∗ (cid:16)e X ( n ) (cid:17) → C ∗ (cid:16)e Y ( n ) (cid:17) → C ∗ (cid:16)e Y ( n ) , e X ( n ) (cid:17) → . Consider the induced long exact homology sequence(4.4) 0 → e H n (cid:16)e X ( n ) (cid:17) → e H n (cid:16)e Y ( n ) (cid:17) → e H n (cid:16)e Y ( n ) , e X ( n ) (cid:17) → e H n − (cid:16)e X ( n ) (cid:17) → · · · . Since X is the ( n + K ( H , e H n − ( e X ( n ) ) is trivial.Now the exact sequence (4.4) can be truncated, obtaining the short exact sequence(4.5) 0 → Z n (cid:16)e X (cid:17) ι → Z n (cid:16)e Y (cid:17) → Z n (cid:16)e Y , e X (cid:17) → , where ι is induced by the inclusion e X ⊆ e Y . We claim that the short exact sequence (4.5)satisfies the three hypothesis of Lemma 2.10.First, since X is a finite cell complex, C n + (cid:16)e X (cid:17) is finitely generated as a Z H -module.Therefore Z n (cid:16)e X (cid:17) is also finitely generated as a Z H -module.Second, the construction of Y guarantees that Z n (cid:16)e Y (cid:17) is a free Z G –module, hence Z n (cid:16)e Y (cid:17) is a free Z H -module.Third, we need to verify that Z n (cid:16)e Y , e X (cid:17) is a projective Z H -module; in fact we show thatit is stably free. Indeed, since X ( n ) and Y ( n ) are the ( n + K ( H ,
1) and a K ( G ,
1) respectively, the reduced homology groups e H k (cid:16)e X ( n ) (cid:17) and e H k (cid:16)e Y ( n ) (cid:17) are trivial for1 ≤ k < n . Then, considering the exact sequence (4.4), we have that(4.6) 0 → Z n (cid:16)e Y ( n ) , e X ( n ) (cid:17) → C n (cid:16)e Y ( n ) , e X ( n ) (cid:17) → · · · → C (cid:16)e Y ( n ) , e X ( n ) (cid:17) → Z H -modules C i (cid:16)e Y ( n ) , e X ( n ) (cid:17) are free, and application of Schanuel’sLemma to (4.6) and a trivial resolution of C (cid:16)e Y ( n ) , e X ( n ) (cid:17) shows that Z n (cid:16)e Y ( n ) , e X ( n ) (cid:17) is a stablyfree Z H -module.Thus we have shown that the short exact sequence (4.5) satisfies the three hypothesis ofLemma 2.10. Before invoking this lemma and concluding the proof, we set up notation forthe norms required to specify representatives of FV n + G and FV n + H .Let k · k denote the ℓ -norm on C i ( e Y ) induced by the basis consisting on all i -cells of e Y .Let k · k Z n ( e Y ) denote the ℓ -norm on Z n ( e Y ) induced by a free Z G -basis; by definition this isalso filling norm on Z n ( e Y ). Then (a representative of) FV n + G is given by(4.7) FV n + G ( k ) = max n k γ k Z n ( e Y ) : γ ∈ Z ( e Y ) , k γ k ≤ k o . Since C n + ( e X ) ⊆ C n + ( e Y ) is a free factor, the ℓ -norm on C n + ( e X ) induced by the ( n + e X equals the restriction of k · k to C n + ( e X ). Let k · k Z n ( e X ) denote the filling-norm on Z n ( e X ) as a Z H -module induced by the boundary map C n + ( e X ) ∂ n + → Z n ( e X ). Then(4.8) FV n + H ( k ) = max n k γ k Z n ( e X ) : γ ∈ Z ( e X ) , k γ k ≤ k o . By Lemma 2.10 applied to the short exact sequence (4.5), there exists a constant C > Z H -modules ρ : Z n ( e Y ) → Z n ( e X ) such that(4.9) k ρ ( α ) k Z n ( e X ) ≤ C · k α k Z n ( e Y ) , for every α ∈ Z n ( e Y ), and ρ ◦ ı is the identity on Z n ( e X ).Let k ∈ N and let γ ∈ Z n ( e X ) such that k γ k ≤ k . Then (4.9) implies that(4.10) k γ k Z n ( e X ) = k ρ ◦ ι ( γ ) k Z n ( e X ) ≤ C · k ι ( γ ) k Z n ( e Y ) ≤ C · FV n + G ( k ) . Since γ was arbitrary, we have FV n + H ( k ) ≤ C · FV n + G ( k ). (cid:3) Remark 4.5.
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The Journal of Geometric Analysis , pages1–21, 2015.M emorial U niversity , S t . J ohn ’ s , N ewfoundland , C anada E-mail address : [email protected] E-mail address ::