A Spectral sequence for polynomially bounded cohomology

A SPECTRAL SEQUENCE F OR POL YNOMIALL Y BOUNDED COHOMOLOGY BOBBY RAMSEY Abstract. W e construct an analogue of the Lyndon-Ho chsc hild- Serre spectral sequence in the con text of polynomially bounded co- homology . F o r G an ex tens ion of Q by H , this sp ectral s equences conv er ges t o the poly nomially bo unded cohomo logy of G , H P ∗ ( G ). If the extension is a po ly nomial extension in the sense of Nos ko v with H and Q iso cohomolo g ical and Q o f type H F ∞ , the sp ectral sequence has E p,q 2 -term H P q ( Q ; H P p ( H )), and G is iso coho mo log- ical for C . By referencing res ults of Connes-Mo s covici a nd Nosko v if H and Q ar e b oth isoco ho mological and hav e the Rapid Deca y prop erty , then G satisfies the Novik ov conjecture. 1. Introduction In [3] Conne s and Mosco vici pro ve the No vik ov conjecture for all finitely generated discrete groups satisfying tw o pro p erties. The first is the Rapid Deca y prop ert y of Jolissain t [11], whic h ensure s the existence of a smo oth dense subalgebra of the reduced gro up C ∗ -algebra. The second prop erty is that eve ry cohomology class can b e represen ted b y a co cycle of p olynomial growth, with res p ect to some (hence any) w ord- length function on the group. The p olynomially b ounded cohomology o f a gro up G , denoted H P ∗ ( G ), obtained b y considering o nly co c hains of p olynomial g r owth, has b een of in terest recen tly . The inclusion of these p olynomially bounded co c hains in to the full co c hain complex yields a homomorphism from the polyno- mially b ounded cohomolo gy to the full cohomology of the gr o up. The second prop ert y of Connes-Mosco vici ab o v e is that this p olynomial comparison homo mo r phism is surjectiv e. A g roup G is iso c ohomo lo g- ic al for M if H P ∗ ( G ; M ) is bor no logically isomorphic t o H ∗ ( G ; M ). The term iso c ohomolo gic al is ta k en f rom Mey er [16], where it describ es a homomorphism betw een t w o b ornological algebras. What is mean t here b y ‘ G is iso coho mo lo gical for C ’, is a we ak ened v ersion of what Mey er refers to as the emb e dding C [ G ] → S G is iso c ohom olo gic al , where S G refers to the F r ´ ec het algebra o f functions G → C of ℓ 1 -rapid deca y . W e call G is o c ohomolo gic al if it is iso cohomolog ical f o r all S G mo dules. 1 2 BOBBY R AMSEY Influenced by Connes and Mosco vici’s approach, in [7] Ji defined p olynomially b ounded cohomolog y and sho w ed that virtually nilp o- ten t groups, are iso cohomological for C . In [13] Mey er sho w ed that p olynomially com bable groups are isocoho mo lo gical for C . By citing a result of Gersten rega r ding classifying spaces for com bable gro ups [4], Ogle indep enden tly show ed that p olynomially combable groups a re iso- cohomological f or C . In [10] it w a s show n that fo r the class of finitely presen ted FP ∞ groups, a g r o up is iso cohomological if and only if it has Dehn functions whic h are p olynomially b ounded in all dimensions. F or a group extension, 0 → H → G → Q → 0 , the Lyndon- Ho c hsc hild-Serre (LHS) sp ectral sequence is a first-quadra nt sp ectral sequence with E 2 -term H ∗ ( Q ; H ∗ ( H )) and con v erg ing to H ∗ ( G ). In [18] Nosk ov generalized the construction of the LHS spectral seque nce to obtain a spectral seque nce in b ounded cohomolo gy for whic h, un- der suitable top ological circumstances, one could iden tify the E 2 -term as H ∗ b ( Q ; H ∗ b ( H )) and whic h con v erges to H ∗ b ( G ). Ogle considered t he LHS sp ectral sequenc e in the con text of P-b o unded cohomology in [19]. There additional tec hnical considerations are also needed to ensure the appropriate E 2 -term. It is not clear for whic h class of extens ions these conditions are satisfied. In this pap er we resolv e this issue for p olynomial extensions, whic h w ere pro p osed b y Nosk o v in [17]. Let ℓ , ℓ H , a nd ℓ Q b e w ord- length functions on G , H , and Q resp ective ly , and let 0 → H → G π → Q → 0 b e a n extension of Q by H . Let q 7→ q be a cross section of π . T o this cross section there is a sso ciated a function [ · , · ] : Q × Q → H b y q 1 q 2 = q 1 q 2 [ q 1 , q 2 ], called the factor set of the extension. The fa c- tor se t has p olynomial grow th if the re exists constan ts C and r suc h that ℓ H ([ q 1 , q 2 ]) ≤ C ((1 + ℓ Q ( q 1 ))(1 + ℓ Q ( q 2 ))) r . The cross section also determines a set-the or etic al action of Q on H . This is a map Q × H → H giv en by ( q , h ) 7→ h q = q − 1 hq . The set-theoretical action is p o lynomially b ounded if there exists constan ts C a nd r suc h that ℓ H ( h q ) ≤ C ℓ H ( h )(1 + ℓ Q ( q )) r . In w hat follo ws, w e adopt the con v en- tion that if Q is the finite generating se t for Q a nd A is the finite generating set for H , then as the generating set for G w e will tak e the set of h ∈ A a nd q for q ∈ Q . Definition 1.1. An extension G of a finitely generated group Q by a finitely generated group H is said to b e a p olynomial extension if there is a cross sec tion yielding a factor set of p olynomial gro wth and inducing a po lynomial set-theoretical a ction of Q on H . A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 3 Our main theorem is as follo ws. Theorem. L et 0 → H → G → Q → 0 b e a p olynomial extension of the H F ∞ gr oup Q , with b oth H and Q iso c oho molo gic al. Ther e is a b ornolo gic al sp e ctr al se quenc e with E p,q 2 ∼ = H P p ( Q ; H P q ( H )) which c onver ges to H P ∗ ( G ) . W e can compare this spectral sequenc e with the LHS sp ectral se- quence. Corollary 1.2. L et 0 → H → G → Q → 0 b e a p olynom ial extension with Q of typ e H F ∞ . If H and Q ar e iso c ohomolo gic al, then G is iso c ohomolo g ic al for C . Applying a result of Nosk o v [17] regarding the p olynomial extension of gr oups with the Ra pid Decay prop erty , as w ell as the results of Connes-Mosco vici, w e obtain the follo wing corollary . Corollary 1.3. L et 0 → H → G → Q → 0 b e a p olynomial gr oup extension with H and Q iso c ohomolo gic al, and Q of typ e H F ∞ . If b oth Q a nd H have the R apid De c ay pr o p erty, G satisfies the Novikov c onje ctur e. It w ould b e con v enien t t o w ork in the category o f F r ´ ec het and DF spaces, how ev er there are many quotien ts in volv ed in the construction yielding spaces whic h need b e neither F r´ ec het nor DF. This is the issue Ogle o v ercomes b y use of a technic al hy p othesis in [19], and it is at the heart o f the top ological consideration in [18], used to iden tif y the E 2 -terms in their spectral sequenc es. T o o v ercome thes e obstacles w e w ork mostly in the b ornological category and utilize an adjo intness relationship of the form Hom( A ˆ ⊗ B , C ) ∼ = Hom( A, Hom( B , C )) whic h is not true in the category of lo cally con v ex top ological v ector spaces. A b o rnology on a space is an analogue of a top ology , in whic h b oundedness replace s openness as t he k ey consideration. In this con- text, w e are also able to b ypass man y of the issues in v olve d in the top ological a na lysis of v ector spaces. Whe n endow ed with the fine b ornology , as defined later, any complex v ector space is a comple te b ornological v ector space. The finest top ology yielding a complete top ological structure on suc h a space is cum b ersome. This borno logy allo ws us to replace analysis of con tinuit y in this t o p ology , to b ound- edness in finite dimensional v ector spaces. In Section 2, w e recall the relev an t concep ts from t he b o r no logi- cal framew ork dev elop ed by Hog b e-Nlend and extende d b y Mey er and 4 BOBBY R AMSEY others. Section 3 consists of translating the usual algebraic sp ectral sequence argumen ts in to this framew ork. This is mainly v erifying that the vec tor space isomorphisms a r e in fact isomorphisms of b ornological spaces. These will b e the main to ols of our cons truction. In Section 4 w e define the relev an t b ornolog ical algebras and define the p olyno- mially b ounded cohomolog y of a discrete group endo w ed with a length function, as w ell as basic materials fo r the construction o f our sp ectral sequence and some of its applications. The a ctual c omputation is the fo cus of the final section. I wish to thank Ron Ji, Crich ton Ogle, Ralf Mey er and the Referee for their in v aluable comme n t s. This article represen ts a revised v ersion of m y thesis. 2. Bornologies Let A and B be subsets of a lo cally con v ex top ological v ector space V . A subset A is cir cle d if λA ⊂ A for all λ ∈ C , | λ | ≤ 1. It is a disk if it is b oth circled and con vex . F or tw o subsets, A absorbs B if there is an α > 0 suc h that B ⊂ λA for all | λ | ≥ α , and A is absorb ent if it absorbs eve ry singleton in V . The cir cle d (diske d, c on vex) hul l of A is the smallest circled (disk ed, con vex ) subse t of V con t a ining A . F or an absorb en t set A there is asso ciated a semi-norm on ρ A on V giv en b y ρ A ( v ) = inf { α > 0; v ∈ αA } . F or an arbitrary subset A , denote b y V A the subspace of V spanned b y A . If A is a disk ed set, ρ A is a semi-norm on V A . A is a c omp letant disk if V A is a Banach space in the top ology induced by ρ A . Let V b e a lo cally conv ex top o logical v ector space. A con v ex v ector space b ornology on V is a collection B of subsets of V suc h that: (1) F or ev ery v ∈ V , { v } ∈ B . (2) If A ⊂ B and B ∈ B then A ∈ B . (3) If A , B ∈ B and λ ∈ C , then A + λB ∈ B . (4) If A ∈ B , and B is the disk ed h ull o f A , then B ∈ B . Elemen ts of B are said to b e the b o unded subsets of V . An importa n t example in what follo ws is the fi ne b ornolo gy . A set is b ounded in the fine b ornolog y on V if it is con tained and b ounded in some finite dimensional subspace of V . Let V and W b e tw o bornolo g ical spaces. A map V → W is b ounde d if the image of ev ery b ounded set in V is b ounded in W . A b ornolo g- ic al isom orphism is a b ounded bijection with b ounded in ve rse. The collection of all b ounded linear maps V → W is denoted bHom( V , W ). There is a canonical complete con ve x b ornolog y on bHom( V , W ), giv en b y the families of equibounded functions; a family U is equib ounded if A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 5 for ev ery b ounded A ⊂ V , U ( A ) = { u ( a ) | u ∈ U, a ∈ A } is b ounded in W . If W is a bo rnological space a nd V ⊂ W , then there is a b ornolo gy on V induced fro m that on W in the obvious w a y . There is also a b ornology on W /V induced fro m W . A subset B ⊂ W /V is b ounded if and only if there is a b ounded C ⊂ W whic h maps to B under the canonical pro jection W → W /V . Let V b e a b ornolo gical v ector space. A sequence ( v i ) in V c onver ges b ornolo gic al ly to 0 if there is a b ounded subse t B ⊂ V and a sequence of scalars λ i tending to 0 suc h that for all i , v i ∈ λ i B . ( v i ) conv erges b ornologically to v if ( v i − v ) con v erges b o rnologically to 0. Let V and W b e b ornological v ector spaces. The complete pro jec- tiv e b ornological tensor pr o duct V ˆ ⊗ W is giv en by the follo wing univ er- sal prop ert y: F or an y complete b ornological v ector sp ace X , a join tly b ounded bilinear map V × W → X extends uniquely to a b ounded ma p V ˆ ⊗ W → X . Unlike the complete to p ological pro jectiv e tensor pro d- uct on the category of locally compact v ector spaces, this b ornolog ical tensor pro duct admits an adjoint. Lemma 2.1 ([16]) . L et A b e a c om p lete b ornolo gi c al algebr a, V and M c omplete b ornolo g ic al A -mo dules, and W a c omp lete b ornolo gi c al s p ac e. Ther e is a b ornolo gic al isomorphi s m bHo m A ( V ˆ ⊗ W , M ) ∼ = bHom( W, bHom A ( V , M )) . W e will be interes ted in b ornologies on F r ´ ec het spaces. F or a F r´ e c het space F , there is a coun table directed family o f seminorms, k·k n yielding the top ology . A set U is said to b e v on Neumann b ounded if k U k n < ∞ for all n . The collection of all v on Neumann b ounded sets forms the v on Neumann Bornolog y on F . This will b e our b ornolog y of c hoice on F r´ ec het spaces , due to the relation with top olog ical constructs. Definition 2.2. A net in a b ornological v ector space F is a f amily of disks, { e i 1 ,i 2 ,...,i k } in F , indexe d b y S k ∈ N I k ( I some c oun table se t ) , whic h satisfies the f ollo wing conditions. (1) F = S i ∈I e i , and e i 1 ,...,i k = S i ∈I e i 1 ,...,i k ,i for k > 1. (2) F or ev ery sequ ence ( i k ) in I , there is a sequence ( ν k ) of p ositiv e reals suc h that for eac h f k ∈ e i 1 ,...,i k and e ac h µ k ∈ [0 , ν k ], the series P ∞ k =1 µ k f k con verges b o rnologically in F , and f or eac h k 0 ∈ N , the serie s P ∞ k = k 0 µ k f k lies in e i 1 ,...,i k 0 . (3) F or ev ery sequ ence ( i k ) in I and ev ery s equence ( λ k ) of p ositiv e reals, S ∞ k =1 λ k e i 1 ,...,i k is b ounded in F . As an example , Hogb e- Nlend sho ws that ev ery bo rnological space with a countable base has a net [6, p58]. 6 BOBBY R AMSEY Lemma 2.3. L et F b e a b ornolo gic al ve ctor sp ac e with a net, and let V b e a subsp ac e of F . Then V has a net. Lemma 2.4. L et F b e a b ornolo gic al ve ctor sp ac e with a net, and let V b e a subsp ac e of F . Then F /V has a n et. Pr o of. Denote the net on F b y R = { e i 1 ,...,i k | i j ∈ I } , and let π : F → F /V b e the pro jection. Set R ′ =  e ′ i 1 ,...,i k = π e i 1 ,...,i k  .  Lemma 2.5. L et U b e a F r` echet sp ac e. Then bHom( U, C N ) has a net. Pr o of. In this case, b o undedness a nd contin uit y are equiv alen t fo r ho- momorphisms, and the equib ounded families are precisely the equicon- tin uo us families. F or each finite sequence of ordered triples of positive in tegers ( n 1 , M 1 , K 1 ), . . . , ( n k , M k , K k ), define b ( n 1 ,M 1 ,K 1 ) ,..., ( n k ,M k ,K k ) to b e the s et o f all f ∈ bHom( U, C N ) suc h that for all i betw een 1 and k , | f ( u ) | < M i for all u ∈ U with k u k U,n i < K i . W e sho w t ha t this giv es a coun table base for the b ornolo gy on bHom( U, C N ). If W is an equib ounded family , then for all neighborho o ds V of zero in C N , there exist n 1 , . . . , n k and K 1 , . . . , K k suc h that { u ∈ U | for all 1 ≤ i ≤ k k u k U,n i < K i } is con tained in W − 1 ( V ). F o r eac h f ∈ W and fo r eac h u in this set, f ( u ) ∈ V . Let V be the op en ball of ra dius 1 in C N , and let n 1 , . . . , n k and K 1 , . . . , K k b e as ab o v e. Then W ⊂ b ( n 1 , 1 ,K 1 ) ,..., ( n k , 1 ,K k ) . Let B = b ( n 1 ,M 1 ,K 1 ) ,..., ( n k ,M k ,K k ) , f ∈ B , M = max M i , and let V be the op en ball of radius R in C . Then V = R M V ′ where V ′ is the op en ball of radius M in C N . f − 1 ( V ) = R M f − 1 ( V ′ ), so if B − 1 ( V ′ ) is a neigh b o rho o d o f zero in U , then so is B − 1 ( V ). Let u ∈ U b e suc h that for a ll 1 ≤ i ≤ k we ha v e k u k U,n i < K i . Then | f ( u ) | < M , so f ( u ) ∈ V ′ . Let S b e the set o f all suc h u ∈ U . It is a neighborho o d of zero. Moreo ver f ( S ) ⊂ V ′ so that S ⊂ f − 1 ( V ′ ), whence S ⊂ B − 1 ( V ′ ). This implie s that the b o rnology on bHom( U, C N ) has a coun table base.  Our in terest in nets is the follo wing ana logue of the op en-mapping theorem. Theorem 2.6 ( [6 , p61] ) . L et E and F b e c onvex b ornolo gi c al sp ac es such that E is c omplete and F has a net. Every b ounde d line ar bije ction v : F → E is a b ornolo gic al isomorphism. A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 7 3. Preliminar y Re sul ts on Bornological Sp ectral Sequences This s ection con tains sev eral results from McCleary’s b o ok [1 2 ] tr a ns- lated in to the b o rnological fra mework. The pro ofs giv en follo w Mc- Cleary , with mo difications to v erify that the vec tor space isomorphisms in volv ed are isomorphisms of b ornological spaces. Let ( A, d ) b e a differential graded b ornological mo dule. That is, A = L ∞ n =0 A n is a gr a ded bo rnological mo dule and d : A → A is a degree 1 b ounded linear map with d 2 = 0 . Let F b e a filtratio n of A whic h is preserv ed by the differen tial, so that for all p , q w e hav e d ( F p A q ) ⊂ F p A q +1 . Assume further that the filtration is decreasing, in that . . . ⊂ F p +1 A q ⊂ F p A q ⊂ F p − 1 A q ⊂ . . . . Such an ( A, F , d ) will b e referred to as a filter e d differ ential gr ade d b ornolo gic al mo dule . Denote b y d p,q : F p A p + q → F p A p + q +1 the restriction of d , so d is the direct sum of d p,q . The filtration F is said to b e b ounded if for eac h n , t here is s = s ( n ) and t = t ( n ) suc h t ha t 0 = F s A n ⊂ F s − 1 A n ⊂ . . . ⊂ F t +1 A n ⊂ F t A n = A n Let Z p,q r = F p A p + q ∩ ( d p + r,q − r ) − 1 ( F p + r A p + q +1 ) B p,q r = F p A p + q ∩ d p − r,q + r − 1 ( F p − r A p + q − 1 ) Z p,q ∞ = F p A p + q ∩ k er d B p,q ∞ = F p A p + q ∩ im d where each of these subspaces are g iv en the subs pace b ornology . Let d n : A n → A n +1 b e the restriction of d . These definitions yield the follo wing ‘to w er’ of submo dules. B p,q 0 ⊂ B p,q 1 ⊂ . . . ⊂ B p,q ∞ ⊂ ‘ Z p,q ∞ ⊂ . . . ⊂ Z p,q 1 ⊂ Z p,q 0 Moreo ver d p − r,q + r − 1 ( Z p − r,q + r − 1 r ) = B p,q r . If the filtration is b o unded and r ≥ max { s ( p + q +1) − p, p − t ( p + q − 1) } then ( d p + r,q − r ) − 1 ( F p + r A p + q +1 ) is the k ernel of d , Z p,q r = Z p,q ∞ , and B p,q r = B p,q ∞ . Lemma 3.1. F or ( A, F , d ) a filter e d differ ential gr ade d b ornolo gic al mo dule, ther e is a sp e ctr al s e quenc e of b orno l o gic al mo dules ( E ∗ , ∗ r , d r ) , r = 1 , 2 , . . . , with d r of bide gr e e ( r, 1 − r ) and E p,q 1 ∼ = H p + q ( F p A/F p +1 A ) . If the filtr ation is b ounde d the sp e ctr al se quenc e c onver ges to H ( A, d ) , E p,q ∞ ∼ = F p H p + q ( A, d ) /F p +1 H p + q ( A, d ) . Pr o of. F or 0 ≤ r ≤ ∞ , let E p,q r = Z p,q r Z p +1 ,q − 1 r − 1 + B p,q r − 1 endo wed with t he quotien t b ornolog y . Thes e a r e the sheets of the spectral sequenc e b eing 8 BOBBY R AMSEY constructed. Con v ergence is guarantee d b y the bo undedness of F . Let η p,q r : Z p,q r → E p,q r b e the pro jection with ke rnel Z p +1 ,q − 1 r − 1 + B p,q r − 1 . The b oundary map d p,q : Z p,q r → Z p + r,q − r +1 r induces a b o unded differ- en tial map d p,q r : E p,q r → E p + r,q − r +1 r yielding the following comm utativ e diagram. Z p,q r d p,q − − − → Z p + r,q − r +1 r η p,q r   y   y η p + r,q − r +1 r E p,q r − − − → d p,q r E p + r,q − r +1 r These definitions ha v e the following consequen ces, used in what fol- lo ws k er d p,q r = η p,q r ( Z p,q r +1 ) ( η p,q r ) − 1 (im d p − r,q + r − 1 r ) = B p,q r + Z p +1 ,q − 1 r − 1 Z p +1 ,q − 1 r − 1 ∩ Z p,q r +1 = Z p +1 ,q − 1 r Z p,q r +1 ∩ ( η p,q r ) − 1 (im d p − r,q + r − 1 r ) = B p,q r + Z p +1 ,q − 1 r This pro of will consist of three steps. The first step consists in v eri- fying that ( E r , d r ) is a b ornological sp ectral sequence. The next ste p is to sho w that it has the appropriate E 1 -term. The final step is ensuring that it has the appropriate E ∞ -term. These steps are carried o ut in the following lemmas.  Lemma 3.2. O n the b orn o lo gic al ve ctor sp ac es E r asso ciate d to ( A, F , d ) , ther e is a b ornolo gic al iso m orphism E p,q r +1 ∼ = H p,q ( E r , d r ) . In p articular ( E r , d r ) is a b ornolo gic al sp e ctr al se quenc e. Pr o of. Let γ : Z p,q r +1 → H p,q ( E r , d r ) b e the b ounded map giv en b y the comp osition Z p,q r +1 η p,q r → k er d p,q r π → H p,q ( E ∗ , ∗ r , d r ) where π is the usual pro jection onto H p,q ( E r , d r ) = k er d p,q r im d p − r,q + r − 1 r . Since k er γ = Z p,q r +1 ∩ ( η p,q r ) − 1 (im d p − r,q + r − 1 r ) = B p,q r + Z p +1 ,q − 1 r , there is an isomorphism of v ector spaces Z p,q r +1 B p,q r + Z p +1 ,q − 1 r = E p,q r +1 ∼ = H p,q ( E r , d r ) giv en b y γ ′ : z + ( B p,q r + Z p +1 ,q − 1 r ) 7→ γ ( z ) + (im d p − r,q + r − 1 r ). W e show that γ ′ is the required b ornological isomorphism. Let U b e a b ounded subset of Z p,q r +1 B p,q r + Z p +1 ,q − 1 r = E p,q r +1 . There is a b ounded subset U ′ of Z p,q r +1 suc h that η p,q r +1 ( U ′ ) = U , so γ ′ ( U ) = η p,q r ( U ′ )+ A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 9 (im d p − r,q + r − 1 r ). As η p,q r is a b ounded map, η p,q r ( U ′ ) is a b ounded set in k er d p,q r and η p,q r ( U ′ ) + (im d p − r,q + r − 1 r ) is b ounded in H p,q ( E r , d r )‘. The b oundedness of γ ′ is v erified. Let φ : k er d p,q r im d p − r,q + r − 1 r → Z p,q r +1 B p,q r + Z p +1 ,q − 1 r b e giv en by z + (im d p − r,q + r − 1 r ) 7→ ( η p,q r ) − 1 ( z ) ∩ Z p,q r +1 + ( B p,q r + Z p +1 ,q − 1 r ). This is the in ve rse of γ ′ . Let U be a b ounded subset of k er d p,q r im d p − r,q + r − 1 r . There exists a b ounded subset U ′ of ke r d p,q r suc h that U ′ + (im d p − r,q + r − 1 r ) contains U in k er d p,q r im d p − r,q + r − 1 r . As k er d p,q r ⊂ E p,q r , U ′ is b ounded in E p,q r , so there is a bo unded subset U ′′ of Z p,q r with U ′ = η p,q r ( U ′′ ). Thus U ′′ + B p,q r − 1 + Z p +1 ,q − 1 r − 1 is the full preimage of U ′ under η p,q r . ( η p,q r ) − 1 ( U ′ ) ∩ Z p,q r +1 = U ′′ ∩ Z p,q r +1 + B p,q r − 1 ∩ Z p,q r +1 + Z p +1 ,q − 1 r − 1 ∩ Z p,q r +1 = U ′′ ∩ Z p,q r +1 + B p,q r − 1 + Z p +1 ,q − 1 r ⊂ U ′′ ∩ Z p,q r +1 + B p,q r + Z p +1 ,q − 1 r Th us φ ( U ) ⊂ U ′′ ∩ Z p,q r +1 + ( B p,q r + Z p +1 ,q − 1 r ) in Z p,q r +1 B p,q r + Z p +1 ,q − 1 r . As U ′′ ∩ Z p,q r +1 is b ounded in Z p,q r +1 , φ ( U ) is b ounded in Z p,q r +1 B p,q r + Z p +1 ,q − 1 r , whence φ is a b o unded map.  Lemma 3.3. The b ornolo gic al sp e ctr al se quenc e ( E r , d r ) asso ciate d to ( A, F , d ) has the pr op erty that E p,q 1 ∼ = H p + q ( F p A/F p +1 A ) as b ornolo gic al ve ctor sp ac es. Pr o of. Since Z p +1 ,q − 1 − 1 = F p +1 A p + q , B p,q − 1 = d ( F p +1 A p + q − 1 ), and Z p,q 0 = F p A p + q ∩ d − 1 ( F p A p + q +1 ), we ha v e E p,q 0 = Z p,q 0 Z p +1 ,q − 1 − 1 + B p,q − 1 = F p A p + q ∩ d − 1 ( F p A p + q +1 ) F p +1 A p + q + d ( F p +1 A p + q − 1 ) = F p A p + q F p +1 A p + q The map d p,q 0 : E p,q 0 → E p,q +1 0 is induced by d p,q : F p A p + q → F p A p + q +1 , fitting into a comm utativ e diag ram F p A p + q d − − − → F p A p + q +1 π   y   y π E p,q 0 = F p A p + q F p +1 A p + q − − − → d 0 F p A p + q + 1 F p +1 A p + q + 1 = E p,q +1 0 10 BOBBY R AMSEY where π are the usu al pro jections. As H p,q ( E 0 , d 0 ) is the homolog y of the complex ( F p A ∗ /F p +1 A ∗ , d 0 ), H p,q ( E 0 , d 0 ) = H p + q ( F p A/F p +1 A ), yielding a bo r no logical isomorphism E p,q 1 ∼ = H p + q ( F p A/F p +1 A )  Lemma 3.4. Assume the filtr ation on ( A, F , d ) is b o und e d. Th e asso ci- ate d b ornolo gic al sp e ctr al se quenc e ( E r , d r ) c onver ges to H ( A, d ) . That is, E p,q ∞ ∼ = F p H p + q ( A, d ) /F p +1 H p + q ( A, d ) . Pr o of. The filtration F on A induces a filtra tion on H ( A, d ), giv en by F p H ( A, d ) = im { H ( inclu sion ) : H ( F p A ) → H ( A ) } . Let η p,q ∞ : Z p,q ∞ → E p,q ∞ and π : k er d → H ( A, d ) denote the pro jections. F p H p + q ( A, d ) = H p + q (im( F p A → A ) , d ) = π ( F p A p + q ∩ k er d ) = π ( Z p,q ∞ ) π (k er η p,q ∞ ) = π ( Z p +1 ,q − 1 ∞ + B p,q ∞ ) = π ( Z p +1 ,q − 1 ∞ ) = F p +1 H p + q ( A, d ) so π induces an isomorphism of v ector spaces d ∞ : E p,q ∞ → F p H p + q ( A, d ) F p +1 H p + q ( A, d ) As π : k er d → H ( A, d ) is b ounded and π ( Z p,q ∞ ) = F p H p + q ( A, d ), the restriction π : Z p,q ∞ → F p H p + q ( A, d ) is a b ounded surjection. Let U b e a b ounded subset of E p,q ∞ . There is a b o unded subset U ′ of Z p,q ∞ suc h that η p,q ∞ ( U ′ ) = U . As π is a b ounded map, π ( U ′ ) is a b o unded subset of F p H p + q ( A, d ). Since d ∞ ( U ) = π ( U ′ ) + F p +1 H p + q ( A, d ) is a b ounded subset of F p H p + q ( A,d ) F p +1 H p + q ( A,d ) , d ∞ is a b ounded map. Consider the map φ : F p H p + q ( A, d ) F p +1 H p + q ( A, d ) → Z p,q ∞ Z p +1 ,q − 1 ∞ + B p,q ∞ giv en by φ : z + ( F p +1 H p + q ( A, d )) 7→ π − 1 ( z ) ∩ Z p,q ∞ + ( Z p +1 ,q − 1 ∞ + B p,q ∞ ). This is the in vers e of d ∞ . It remains to sho w that φ is a b ounded map. Let U b e a bounded subset of F p H p + q ( A,d ) F p +1 H p + q ( A,d ) . There is a b ounded U ′ subset of F p H p + q ( A, d ) whic h pro jects to U . As F p H p + q ( A, d ) is A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 11 con ta ined in H p + q ( A, d ), U ′ is a b ounded subset o f H p + q ( A, d ). There exists a b ounded subset U ′′ in k er d p + q with U ′ = U ′′ + (im d p + q − 1 ). As U ′ is a subset of F p H p + q ( A, d ) we can assume U ′ ⊂ k er d p + q ∩ F p A p + q + (im d p + q − 1 ), so U ′′ ⊂ Z p,q ∞ + B p,q ∞ Therefore U ′′ is b ounded in the subspace Z p,q ∞ + B p,q ∞ , a nd π ( U ′′ ) = U ′′ + (im d p + q − 1 ) ⊃ U ′ = U + F p +1 H p + q ( A, d ). Th us π − 1 ( U ) ⊂ U ′′ + im d p + q − 1 . π − 1 ( U ) ∩ Z p,q ∞ ⊂ U ′′ ∩ Z p,q ∞ + im d p + q − 1 ∩ Z p,q ∞ = U ′′ ∩ Z p,q ∞ + B p,q ∞ So φ ( U ) = π − 1 ( U ) ∩ Z p,q ∞ + ( Z p +1 ,q − 1 ∞ + B p,q ∞ ) ⊂ U ′′ ∩ Z p,q ∞ + ( Z p +1 ,q − 1 ∞ + B p,q ∞ ) As U ′′ is b ounded in Z p,q ∞ + B p,q ∞ , U ′′ ∩ Z p,q ∞ is b ounded in Z p,q ∞ . Th us φ is a b ounded map.  W e now mo ve to the a pplicatio n of Lemma 3.1 in the case whic h will b e of mos t inte rest in the sequel. A double complex of b ornolog- ical mo dules is a bigraded mo dule M = L p ≥ 0 ,q ≥ 0 M p,q , where each M p,q is a b ornological module, alo ng with t w o b ounded linear maps d ′ and d ′′ , of bidegree (1 , 0) and (0 , 1) resp ectiv ely , s atisfying d ′ 2 = d ′′ 2 = d ′ d ′′ + d ′′ d ′ = 0. The tota l complex, ( total ( M ) , d ) of the double com- plex { M ∗ , ∗ , d ′ , d ′′ } is the differen tial graded bornolo gical mo dule with total ( M ) n = L p + q = n M p,q and d = d ′ + d ′′ . There are t wo standard filtrations on the total complex. F p I ( total ( M )) t = M r ≥ p M r,t − r F p I I ( total ( M )) t = M r ≥ p M t − r ,r will be referred to as the column wise filtration and ro wwise filtration re- sp ectiv ely . Bo th are decreasing filtrations, resp ected by the differen tial. As M ∗ , ∗ is first-quadran t, eac h of these filtrations are b ounded a nd b y Lemma 3.1 w e obta in t w o s p ectral s equences of b o rnological mo dules con verging t o H ( total ( M ) , d ) . A t M p,q there are tw o b oundary maps, d ′ and d ′′ , with respect to eac h of whic h w e ma y calculate a bigraded cohomology of M . Sp ecifically , let H p,q I ( M ) = im d ′′ : M p,q − 1 → M p,q k er d ′′ : M p,q → M p,q + 1 and H p,q I I ( M ) = im d ′ : M p − 1 ,q → M p,q k er d ′ : M p,q → M p +1 ,q . In this w a y , H ∗ , ∗ I ( M ) is a double complex with trivial v ertical differential and H ∗ , ∗ I I ( M ) is a double complexes with trivial ho rizon ta l differe n t ia l. W e may then tak e cohomology with resp ect to the nontrivial b oundary map to obta in the iterated coho- mology spaces H ∗ , ∗ I I H I ( M ) and H ∗ , ∗ I H I I ( M ) of M . 12 BOBBY R AMSEY Lemma 3.5. Given a double c ompl e x ( M ∗ , ∗ , d ′ , d ′′ ) of b ornolo gic al mo d- ules and b ounde d maps, ther e ar e two sp e ctr al se quenc es of b ornol o gic al mo dules, ( I E ∗ , ∗ r , I d r ) and { I I E ∗ , ∗ r , I I d r } with I E p,q 2 ∼ = H ∗ , ∗ I H I I ( M ) and I I E p,q 2 ∼ = H ∗ , ∗ I I H I ( M ) . If M ∗ , ∗ is a first-quadr ant double c omplex , b oth sp e ctr al se quenc es c onver ge to H ∗ ( total ( M ) , d ) . Pr o of. The first-quadrant h yp othesis is here to ensure con vergenc e of the sp ectral sequences, and plays no role in the calculation of the E 2 - terms. In the case of F p I w e ha v e I E p,q r = H p + q  F p I ( total ( M )) F p +1 I ( total ( M )) , d  The differen tial on total ( M ) is giv en b y d = d ′ + d ′′ so that d ′ ( F p I ( total ( M )) ) ⊂ F p +1 I ( total ( M )) . There is a b o r nological isomorphism  F p I ( total ( M )) F p +1 I ( total ( M ))  p + q ∼ = M p,q with the induced differen tial d ′′ , th us I E p,q 1 ∼ = H p,q I I ( M ). Consider the follo wing maps i : H n ( F p I ) → H n ( F p − 1 I ) j : H n ( F p I ) → H n ( F p I /F p +1 I ) k : H n ( F p I /F p +1 I ) → H n +1 ( F p +1 I ) d 1 : H p,q I I ( M ) → H p +1 ,q I I ( M ) where i is induced by the inclusion F p − 1 I → F p I , j is induced by the quotien t map F p I → F p I /F p +1 I , k is the connecting homomorphism, and ∂ : F p I /F p +1 I → F p +1 I /F p +2 I is induced by the differen tial d . It is clear that i and j are b ounded. The k map sends [ x + F p +1 I ] ∈ H n ( F p I /F p +1 I ) to [ dx ] ∈ H n +1 ( F p +1 I ). If U is a b ounded subset of H n ( F p I /F p +1 I ) then there is a b ounded subset U ′ in the k ernel of ∂ : F p I /F p +1 I → F p +1 I /F p +2 I with U ′ + (im ∂ ) = U ∈ H n ( F p I /F p +1 I ). There is U ′′ a b ounded subse t of F p I with U ′ = U ′′ + F p +1 I ∈ F p I /F p +1 I . As d is a bounded map, d ( U ′′ ) is a b ounded subset of F p +1 I . It follo ws that [ d ( U ′′ )] is b o unded in H n +1 ( F p +1 I ), and k is a b ounded map. A class in H p + q ( F p I /F p +1 I ) can b e written as [ x + F p +1 I ], where x ∈ F p I and dx ∈ F p +1 I , or it can b e written as a class [ z ] ∈ H p,q I I ( M ), z ∈ M p,q . k sends [ x + F p +1 I ] to [ dx ] ∈ H p + q +1 ( F p +1 I ). T aking z as a represe n t a tiv e this determines [ d ′ z ] ∈ H p + q +1 ( F p +1 I ), since d ′′ ( z ) = 0. Th us w e can consider d ′ z as an elemen t of M p +1 ,q . The map j assigns to a class in H p + q +1 ( F p +1 I ) its represen tativ e in H p + q +1 ( F p +1 I /F p +2 I ). This giv es A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 13 d 1 = j ◦ k as the induced mapping o f d ′ on H p,q I I ( M ), so d 1 = ¯ d ′ . Th us I E p,q 2 ∼ = H p,q I H ∗ , ∗ I I ( M ). Sym metry giv es I I E p,q 2 ∼ = H p,q I I H ∗ , ∗ I ( M ).  In the seq uel, it will b e necessary fo r us to compare sp ectral sequences of b ornolog ical spaces. Definition 3.6. Let ( E r , d r ) and ( E ′ r , d ′ r ) be t w o b ornolo g ical spectral sequence s. A map of b ornolo g ical sp ectral sequenc es is a family of bigraded b ounded linear maps f = ( f r : E r → E ′ r ), eac h of bidegree (0 , 0), suc h that for all r , d ′ r f r = f r d r and f r +1 is the map induced by f r in cohomology . Lemma 3.7. S upp ose f = ( f r : E r → E ′ r ) is a map of b ornolo gic al sp e ctr al se quenc es, e ach E ′ r is c onvex and c omp lete, and e ach E r is c onvex and ha s a net. I f f t is a b ornolo gic al isom orphism for some t , then f r is a b ornolo gic al isomorphism for al l r ≥ t . Mor e over, f induc es an isomorphism E ∞ → E ′ ∞ . Pr o of. It is w ell known that these isomorphisms exis t betw een the v ec- tor spaces. It remains to sho w that these v ector space isomorphisms are b ornological isomorphisms. This follo ws from Lemmas 2 .3, 2.4, and Theorem 2.6.  4. Pol ynomiall y bounded cohomology Let G be a discrete gro up. A length function on G is a function ℓ : G → [0 , ∞ ) suc h that (1) ℓ ( g ) = 0 if and only if g = 1 G is the identit y elemen t of G . (2) F or all g ∈ G , ℓ ( g ) = ℓ ( g − 1 ). (3) F or all g and h ∈ G , ℓ ( g h ) ≤ ℓ ( g ) + ℓ ( h ). T o a finite generating set S of G , w e associate a length function ℓ S defined b y ℓ S ( g ) = min { n | g = s 1 s 2 . . . s n where s i ∈ S ∪ S − 1 } . This length function dep ends on S , but for different c hoices o f S w e obtain linearly equiv alen t length functions. W e refer to an y length function obtained in this w ay as a w ord-length function. Fix some length function ℓ on G . F o r eac h p ositiv e in teger k and for i = 1 and i = 2 define norms on the set of functions φ : G → C b y k φ k i,k = X g ∈ G | φ ( g ) | i (1 + ℓ ( g )) ik ! 1 /i Let S ℓ G b e the set o f all functions f : G → C suc h that for a ll k , k f k 1 ,k < ∞ . S ℓ G is a F r ´ ec het algebra in this family of norms, giving the structure of a bor no logical algebra. In what follo ws we are s olely in terested in the case of a w ord-length function on G . In t his case 14 BOBBY R AMSEY w e denote S ℓ G by S G . P olynomially equiv a lent length functions yield the same S ℓ G algebra, so the particular w ord-length function used is irrelev an t. If R is a subset of G , we also define S R to be the subspace of S G consisting of functions support ed on R . Let A b e a b ornological algebra. A b o rnological A -mo dule is a com- plete con v ex b o rnological space, equipped with a jointly bounded A - mo dule s tructure. A b ornological A -module is b orno lo gically pro jectiv e if it is a direct summand of b o rnological mo dule of the form A ˆ ⊗ E for some bornolo gical v ector space E , with the left-action giv en b y the m ul- tiplication in A . An imp ortant prop ert y of b ornologically free mo dules is that bHom A ( A ˆ ⊗ B , C ) ∼ = bHom( B , C ) . Definition 4.1. The p olynomially b ounded cohomology of G with co- efficien ts in a b orno logical S G -mo dule M is given by H P ∗ ( G ; M ) = bExt ∗ S G ( C ; M ). Here bExt is t he Ext functor in the b ornological category . Notice that eac h of the bExt groups is a complex b ornological v ector space. Mey er shows in [13] that this is equiv alen t to the form ulatio n de scrib ed in the in tro duction, when the co efficien t mo dule is C endo w ed with the trivial S G -action. Using Ext o v er the top ological category one reco vers Ji’s original definition [7], ho wev er from Mey er’s w o rk, for trivial co efficien ts C , the top ological and bo rnological theories coincide. There is a comparison homomorphism H P ∗ ( G ) → H ∗ ( G ) induced b y the inclusion C [ G ] → S G . An imp ortant question with applications to the Novik o v conjecture, as w ell as the ℓ 1 -Bass conjecture (se e [8]) is, ”When is this comparison homomorphism is an isomorphism?” In [13] Mey er shows that this is the case f or an y group equipped with a p oly- nomial length com bing. This wide class includes the w ord-h yp erb olic groups of Gromov [5], the semih yp erb olic g roups of Alonso-Bridson [1], and the automatic gro ups of [2], ho w ev er it do es not include all finitely generated g roups. There are examples of finitely generated groups for whic h it is know n to fail, [9]. Definition 4.2. A group G is isocohomolo gical for M , for M an S G - mo dule, if the comparison homomorphism H P ∗ ( G ; M ) → H ∗ ( G ; M ) is a b ornological isomorphism. It is strongly isocohomolo gical if it is iso cohomological for all S G -mo dule co efficien ts. As w e will not b e in terested in w eak iso cohomologicality , w e drop the adjectiv e “strongly”, a nd refer to a group as b eing iso cohomological if it satisfies this strong iso cohomologicality condition. A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 15 Let H and Q be finitely generated discrete groups with word-length functions ℓ H and ℓ Q resp ectiv ely , and let 0 → H ι → G π → Q → 0 b e an extension of Q by H , w ith w ord-length function ℓ . ( In consid- ering H as a subgroup of G , w e omit the ι when considering h ∈ H as an elemen t of G . ) Let q 7→ q b e a cross section of π . T o this cross section t here is asso ciated a function, called the factor set of the extension, giv en b y [ · , · ] : Q × Q → H by the form ula q 1 q 2 = q 1 q 2 [ q 1 , q 2 ]. The factor set has p olynomial growth if there exist constan ts C and r suc h that ℓ H ([ q 1 , q 2 ]) ≤ C ((1 + ℓ Q ( q 1 ))(1 + ℓ Q ( q 2 ))) r . The cross sec tion also determines a set-theoretic action of Q on H given b y h q = q − 1 hq . The action is p olynomial if there exist constan ts C and r such that ℓ H ( h q ) ≤ C ℓ H ( h )(1 + ℓ Q ( q )) r . Definition 4.3. An extens ion G of a finitely generated gro up Q b y a finitely generated group H is said to b e a p olynomial extension if there is some cross section yielding a f actor set of p olynomial gro wth and inducing a po lynomial action of Q on H . An imp ortant consequence of this definition is that the w ord-length function on H is p olynomially equiv alent to the word-length function on G restricted to H . The f o llo wing follows fro m Lemma 1.4 of [17], and ensures t ha t S ℓ H H = S ℓ | H H . Lemma 4.4. L et G b e a p olynomial extension of the finitely gener ate d gr oup Q by the finitely gener ate d gr oup H . Ther e exists c onstants C and r such that for al l h ∈ H , ℓ ( h ) ≤ ℓ H ( h ) ≤ C (1 + ℓ ( h )) r . Lemma 4.5. As b ornolo gic al S H -mo d ules S G ∼ = S H ˆ ⊗S G/H , wher e H is endowe d with the r estricte d l e ngth function an d G/H is given the minimal leng th function, ℓ ∗ ( g H ) = min h ∈ H ℓ ( g h ) , wher e ℓ is the length function on G . Pr o of. Let R b e a set of minimal length represen tative s fo r righ t cosets. Let r : G → R be the map assigning to g , the represen tative of H g . Eac h g ∈ G has a unique represen tation a s g = h g r ( g ), for h g ∈ H and r ( g ) ∈ R . There is an ob vious equiv alence b etw een S G/H and S R . Consider the map φ : S G → S H ˆ ⊗S R g iven by φ ( g ) = ( h g ) ⊗ ( r ( g )). This is the desire d b ornological isomorphism.  Corollary 4.6. A b ornolo gic al ly pr oje ctive S G -mo dule is a b ornolo gi- c al ly pr oje ctive S H -mo d ule b y r estriction o f t he S G -action. Corollary 4.7. L et M b e an S G -mo d ule. Any b ornolo gic al ly pr oje ctive S G -m o dule r esolution of M is a b ornolo gic al ly pr oje ctive S H -m o dule r esolution of M . 16 BOBBY R AMSEY Consider the follo wing: . . . δ → S G ˆ ⊗ n δ → S G ˆ ⊗ n − 1 δ → . . . δ → S G ˆ ⊗S G δ → S G ǫ → C → 0 where δ : S G ˆ ⊗ n → S G ˆ ⊗ n − 1 is the usual b oundary map giv en by δ ( g 1 , . . . , g n ) = n X i =1 ( − 1) i ( x 1 , . . . , b x i , . . . , x n ) and extend by linearity , where the tuple ( g 1 , . . . , g n ) represen t s the elemen tary tensor g 1 ⊗ . . . ⊗ g n . As defined, δ is a bo unded map and the map s : S G ˆ ⊗ n → S G ˆ ⊗ n +1 giv en on generators b y s ( g 1 , . . . , g n ) = (1 G , g 1 , . . . , g n ) is a b ounded C -linear contracting homotop y for this complex. This is a b o rnologically pro jective r esolution of C ov er S G , which w e call the standar d b ornolo gic al r esolution for the group G . F or groups with additional finiteness conditions, there are resolutions with b etter pro p erties to conside r. By [10], if an is o cohomological group Q is of t yp e H F ∞ , then there is a b ornological pro jectiv e resolution of C o v er S Q of the form . . . → R p → R p − 1 → . . . → R 0 → C → 0 with eac h R p a b ornolog ically free S Q mo dule of finite ra nk. Theorem 4.8. L et 0 → H → G → Q → 0 b e a p olynomial extension of the H F ∞ gr oup Q , with b oth H and Q iso c oho molo gic al. Ther e is a b ornolo gic al sp e ctr al se quenc e with E p,q 2 ∼ = H P p ( Q ; H P q ( H )) which c onver ges to H P ∗ ( G ) . Assuming Theorem 4.8 and Corollary 1.2, w e b egin by v erifying Corollary 1.3. A g roup G acts on ℓ 2 ( G ) via ( g · f ) ( x ) = f ( g − 1 x ). This action ex tends b y linearit y to yield an action b y C G on ℓ 2 ( G ) by b ounded op erators. The comple tion of C G in B ( ℓ 2 ( G )), the space of all b ounded op erators on ℓ 2 ( G ) endo w ed with the op erator norm, is the reduced group C ∗ - algebra, C ∗ r G . Let S 2 G b e the se t of all functions f : G → C suc h that for all k , k f k 2 ,k < ∞ . The group G is said to ha v e the Rapid Deca y prop ert y if S 2 G ⊂ C ∗ r G , [11]. W e use the follo wing result of No sk ov . Theorem 4.9 ([17]) . L et G b e a p olynomial extension of the fi nitely gener ate d gr oup Q by the finitely ge n er ate d gr oup H . If H and Q have the R apid De c ay pr op erty, so do es G . Pr o of of Cor ol lary 1.3. By Corollary 1.2, G is iso cohomological for C . By Nosko v, G has the R apid D eca y prop erty . The result f o llo ws from app ealing to Connes-Mosc o vici.  A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 17 5. Proof of Theorem 4.8 Throughout this section, w e assume the h yp otheses of Theorem 4.8. Let ( P ∗ , d P ) b e the standard b ornolog ical resolution for G , and let T ∗ b e the tensor product of P ∗ b y C ov er S H . The p olynomial extension prop- erties give T q ∼ = S Q ˆ ⊗S G ˆ ⊗ q . As t he P q are b ornolo gical S G -mo dules, they are b y restriction, b ornological S H -mo dules. The quotien t gro up Q acts on bHom S H ( P q , C ) via ( q φ ) ( x ) = q · φ ( q − 1 x ), where · : Q → G is a cross-sec tion giving the p olynomial exten sion prop erties. This ex- tends to a b ornolo g ical S Q -mo dule structure on bHom S H ( P q , C ). Let ( R ∗ , d R ) be a b ornolog ically pro jectiv e resolution of C ov er S Q with eac h R p finite rank. Set C p,q = bHo m S Q ( R p ˆ ⊗ T q , C ) ∼ = bHom S Q ( R p , bHom( T q , C )). The b oundary maps d T and d R induce maps δ T : C p,q → C p,q +1 and δ R : C p,q → C p +1 ,q as follows. ( δ T f ) ( r )( x ) = ( − 1) p f ( r ) ( d T x ) ( δ R f ) ( r )( x ) = f ( d R r )( x ) Filter the do uble complex C ∗ , ∗ b y row s. F or a fix ed q we hav e the complex . . . δ R → C ∗− 1 ,q δ R → C ∗ ,q δ R → C ∗ +1 ,q δ R → . . . The b o unded homotop y for the complex R ∗ induces a con traction on C ∗ ,q , so that E p,q 1 = 0 for p ≥ 1 and E 0 ,q 1 = bHom S Q ( T q , C ). The a d- join tness prop erty giv es a b ornological isomorphism bHom S Q ( T q , C ) ∼ = bHom S G ( P q , C ). This iden tifies E 0 ,q 1 ∼ = bHom S G ( P q , C ). As P ∗ w as a pro jectiv e S G - complex, w e obtain that the E 2 -term is precisely H P ∗ ( G ), and the sp ectral sequence colla pses here. W e no w examine the double complex when filtered by columns. F or a fixed p w e ha ve the complex . . . δ T → C p, ∗− 1 δ T → C p, ∗ δ T → C p, ∗ +1 δ T → . . . By adjoin t ness, C p,q ∼ = bHom S Q ( R p , bHom( T q , C )), and t he bo und- ary map d T induces a map d ∗ T : bHom( T q , C ) → bHom( T q +1 , C ). Lemma 5.1. Ther e ar e identific ations k er δ T = bHo m S Q ( R p , k er d ∗ T ) and im δ T = bHom S Q ( R p , im d ∗ T ) . Pr o of. If ϕ ∈ k er δ T , then fo r all r ∈ R p , ( δ T ϕ )( r )( x ) = 0 for all x ∈ T ∗ . Th us ϕ ( r ) ∈ k er d ∗ T for a ll r and k er δ T ⊂ bHom S Q ( R p , k er d ∗ T ). If ξ ∈ bHom S Q ( R p , k er d ∗ T ) then for all r ∈ R p , d ∗ T ξ ( r ) = 0. That is ξ ( r )( d T x ) = 0 for all x ∈ T ∗ , a nd δ T ξ is the zero map, establishing bHom S Q ( R p , k er d ∗ T ) ⊂ k er δ T . 18 BOBBY R AMSEY F or ϕ ∈ im δ T , there is f ∈ bHom S Q ( R p , bHom( T q , C )) with δ T f = ϕ . Th us, for all r ∈ R p , ϕ ( r ) = ( − 1) p d ∗ T ( f ( r )) ∈ im d ∗ T . In particular, im δ T ⊂ bHom S Q ( R p , im d ∗ T ). That bHom S Q ( R p , im d ∗ T ) ⊂ im δ T follo ws from the finiteness condi- tion on Q . Sp ecifically , w e use the b ornological isomorphism bHom S Q ( R p , M ) ∼ = bHom( R p , M ) for any S Q -mo dule M . Since R p is finite dimensional, a n y linear map R p → M is b ounded. F or each ξ ∈ im d ∗ T , pic k a σ ( ξ ) ∈ bHom( T q , C ) for whic h d ∗ T ( σ ( ξ )) = ξ . W e do not require σ to be a b ounded map. Let R b e a finite basis for R p . Let ϕ ∈ bHom S Q ( R p , im d ∗ T ) ∼ = bHom( R p , im d ∗ T ). Define a map f : R p → bHom( T q , C ) by setting f ( r ) = ( − 1) p σ ( ϕ ( r ) ) for r ∈ R and ex- tending by linearit y . This defines a ma p f ∈ bHom( R p , bHom( T q , C )). F or r ∈ R , δ T f ( r ) = ( − 1) p d ∗ T ( f ( r )) = ϕ ( r ). Th us ϕ ∈ im δ T .  Lemma 5.2. As b ornolo gic al ve ctor sp ac es, bHom S Q ( R p , k er d ∗ T im d ∗ T ) ∼ = bHom S Q ( R p , ke r d ∗ T ) bHom S Q ( R p , im d ∗ T ) Pr o of. Denote b y v the map bHom S Q ( R p , ke r d ∗ T ) bHom S Q ( R p , im d ∗ T ) → bHom S Q ( R p , k er d ∗ T im d ∗ T ), giv en b y v ( f + bHom S Q ( R p , im d ∗ T ))( r ) = f ( r ) + im d ∗ T . F or R p as ab ov e bHom( R p , k er d ∗ T ) ⊂ bHom( R p , bHom( T q , C )) ∼ = bHom( R p ˆ ⊗ T q , C ) . As R p ˆ ⊗ T q is a F r´ ec het space, b y Lemma 2 .5, Lemma 2.3, and Lemma 2.4, bHom( R p , ke r d ∗ T ) bHom( R p , im d ∗ T ) has a net. Moreo ver, bHom( R p , k er d ∗ T ) is a complete b ornological space. The cohomolog y of the complex bHom( T q , C ) ∼ = bHom( S Q ˆ ⊗S G ˆ ⊗ q , C ) ∼ = bHom S H ( S H ˆ ⊗S Q ˆ ⊗S G ˆ ⊗ q , C ) ∼ = bHom S H ( P q , C ) is precis ely H P ∗ ( H ), the p olynomially b ounded cohomology of the sub- group H . As H is iso cohomological for C , b y Theorem 1 1 of [9], for each ∗ ≥ 0 , H P ∗ ( H ) is a finite dimensional complex v ector space equipped with the fine b ornology . Let { γ 1 , . . . , γ k } b e a ba sis for H P ∗ ( H ). F or eac h γ i , tak e an f i ∈ k er d ∗ T with f i + im d ∗ T = γ i . The assignmen t γ i 7→ f i extends to a b ounded linear map H P ∗ ( H ) ∼ = k er d ∗ T im d ∗ T → ke r d ∗ T whic h splits the quotien t map k er d ∗ T → k er d ∗ T im d ∗ T ∼ = H P ∗ ( H ). Let φ ∈ bHom S Q ( R p , k er d ∗ T im d ∗ T ). Since R p has finite rank, t here is a φ ′ ∈ bHom S Q ( R p , k er d ∗ T ) making the follow ing diagram comm ute. A S PECTRAL SEQUENCE F O R POL YNOMIA LL Y BOUNDED COHOMOLOGY 19 (1) R p φ   φ ′ | | ① ① ① ① ① ① ① ① ① ① k er d ∗ T / / k er d ∗ T im d ∗ T This shows the map α : bHom S Q ( R p , k er d ∗ T ) → bHom S Q ( R p , k er d ∗ T im d ∗ T ) is surjectiv e. As k er α = bHom S Q ( R p , im d ∗ T ), v is a b ounded linear bijection. The r esult follow s b y Theorem 2.6.  Pr o of of The or em 4.8. When filtering C ∗ , ∗ b y columns, b y Lemma 5.2 the E p,q 1 ∼ = bHom S Q ( R p , H P q ( H )). Then E 2 ∼ = H P p ( Q ; H P q ( H )). As this sp ectral seque nce con v erges to the same sequence a s that obtained when filtering by ro ws, w e hav e con v ergence to H P p + q ( G ).  Pr o of of Cor ol lary 1.2. W e compare the p olynomial gro wth sp ectral se- quence with the LHS sp ectral seque nce for t he group extens ion. The inclusions C [ H ] → S H and C [ Q ] → S Q induce a mapping of b orno log- ical sp ectral seq uences E r → E ′ r , where E r is the sp ectral sequence resulting from Theorem 4.8, and E ′ r is the usual sp ectral sequence as- so ciated to the group extension. Since Q and H are iso cohomo lo gical, H P p ( Q ; H P q ( H )) ∼ = H p ( Q ; H q ( H )) . The t wo sp ectral sequences hav e b ornologically isomorphic E 2 -terms. Both E r and E ′ r are complete con v ex b ornolog ical spaces . F urthermore, the proo f o f Theorem 4 .8, when com bined with Lemmas 2.4 and 2.3, sho ws that E r has a net. By Lemma 3.7 they ha v e b ornologically isomorphic limits.  Reference s [1] J. Alons o, M. Bridson. Se mihype r bo lic gr o ups. Pro c. London Math. Soc. (3), 70(1):56– 114, 1995 . [2] J. Cannon, D. Epstein, D. Holt, S. Lev y , M. Paterson, W. Th urston. W o rd pro cessing in groups. Jones and Bartlett Publishers, Bosto n, MA, 1992 . [3] A. Connes, and H. Moscovici. Cyclic cohomolog y , the Novik o v conjecture and hyperb olic groups. T opolo g y , 29(3):345– 388, 1990. [4] S. Gersten. Finiteness prop erties o f asynchronously automatic groups. Geo- metric g r oup theory (Co lum bus, O H, 199 2), 121–13 3 , O hio State Univ. Math. Res. Inst. Publ., 3, de Gruyter, Berlin, 1995. [5] M. Gromov. V olume and b ounded cohomolog y . Inst. Hautes Etudes Sci. Publ. Math., (56):5–9 9 (1983 ), 198 2. [6] H. Ho g be- Nlend. Bo r nologies a nd functional analysis. North-Holland P ub- lishing Co., Amsterdam, 197 7. In tro ductory c ourse on the theory of duality top ology-b or no logy and its use in functional analysis, T ransla ted from the 20 BOBBY R AMSEY F r ench by V. B. Moscatelli, North-Holland Ma thema tics Studies, V ol. 2 6 , No- tas de Matematica, No. 62. [Notes on Mathematics, No. 62]. [7] R. Ji. Smo o th dense subalg ebras of r educed g roup C ∗ -algebra s, Sch w artz co - homology of groups, and cyclic cohomology . J. F unct. Anal., 107 (1):1–33,1 992. [8] R. Ji, C. Ogle, and B. Ramsey . Relative hyperb olicity and a generaliza tio n of the Bass conjecture. Jo urnal of Noncomm utative Geometry , 4(1): 83–12 4, 2010. [9] R. Ji, C. Og le, and B. Rams e y . B -b ounded cohomo logy and its applications. arXiv:100 4.4677v4. [10] R. J i, B. Ramsey . The iso co homologica l proper t y , higher Dehn functions, and relatively h yp e rb o lic groups. Adv ances in Mathematics, 22 2(1):255– 2 80, 200 9. [11] P . Joliss aint. Rapidly decrea sing functions in reduced C ∗ -algebra s of groups. T r ans. Amer. Math. Soc., 3 17(1):167 –196, 1990. [12] J . McCle a ry . A user’s guide to sp ectra l se quences, 2nd Edition. Cambridge Univ ersity Pr ess, 2 001. [13] R. Meyer. Com bable groups hav e group cohomolog y o f p olynomial g rowth. Q . J. Math., 57(2):241 –261, 200 6. [14] R. Meyer. Bornolog ical v er sus topolo gical analy sis in metrizable spaces. In Banach algebras a nd their applications, volume 363 of Con temp. Math., pages 249–2 78. Amer. Math. So c., Providence, RI, 200 4. [15] R. Meyer. Analytic cyclic cohomolo gy . arXiv:math/99062 05 v 1. [16] R. Meyer. Embedding der ived categories of borno lo gical mo dules. arXiv:math/04 10596v1. [17] G. Nosko v. Algebras of ra pidly decre asing functions on g roups and co cy cles of po lynomial growth. Sibirs k . Ma t. Zh., 3 3(4):97–1 03, 221, 1992. [18] G. Noskov. The Ho chsc hild-Serre sp ectral sequence for bounded cohomolog y . Pro ceedings of the International Conference on Algebra, Part 1 (Nov o sibirsk, 1989), 613 –629, Contemp. Math., 131, P art 1 , Amer. Math. Soc., Providence, RI, 1992. [19] C. Ogle. Polynomially b ounded co homology and discrete groups. J. Pure Appl. Algebra, 195(2 ):173–20 9, 2005. The Ohio St a te University, Dept. of Ma thema tics, 321 W . 18th A ve., Columbus, OH 43210