Secondary Higher Invariants and Cyclic Cohomology for Groups of Polynomial Growth
aa r X i v : . [ m a t h . K T ] J u l SECONDARY HIGHER INVARIANTS AND CYCLIC COHOMOLOGY FOR GROUPSOF POLYNOMIAL GROWTH
SHEAGAN A. K. A. JOHN
Abstract
We prove that if Γ is a group of polynomial growth then each delocalized cycliccocycle on the group algebra has a representative of polynomial growth. For eachdelocalized cocyle we thus define a higher analogue of Lotts delocalized eta invariantand prove its convergence for invertible differential operators. We also use a determinantmap construction of Xie and Yu to prove that if Γ is of polynomial growth then thereis a well defined pairing between delocalized cyclic cocyles and K -theory classes of C ∗ -algebraic secondary higher invariants. When this K -theory class is that of a higherrho invariant of an invertible differential operator we show this pairing is precisely theaforementioned higher analogue of Lotts delocalized eta invariant. As an application ofthis equivalence we provide a delocalized higher Atiyah-Patodi-Singer index theoremgiven M is a compact spin manifold with boundary, equipped with a positive scalarmetric g and having fundamental group Γ = π ( M ) which is finitely generated and ofpolynomial growth. Table of Contents C ∗ -algebras, and Smooth Dense Sub-algebras . . . . . . . . . . . 62.2 Cyclic and Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 10 The author was partially supported by NSF 1800737, NSF 1952693 § Given a Fredholm operator T : X −→ Y between two Banach spaces the classic indextheory for Fredholm operators provides an integer valued analytic indexind( T ) = dim ker( T ) − dim coker( T )which is invariant under perturbations of T by compact operators. The non-vanishing ofind( T ) is thus an obstruction to invertibility of a Fredholm operator T . When T is anelliptic differential operator with X and Y smooth vector bundles over a smooth closedmanifold M the work of Atiyah and Singer [3] showed the equivalence between ind( T ) andthe often more tractable topological index (see (4.4.2) of Section 4.4).Let M be a complete n -dimensional Riemannian manifold with a discrete group G acting on it properly and cocompactly by isometries. Each G -equivariant elliptic differen-tial operator D on M gives rise to a higher index class Ind G ( D ) in the K -theory group K n ( C ∗ r ( G )) of the reduced group C ∗ -algebra C ∗ r ( G ). Higher index classes are invariantunder homotopy, and being an obstruction to the invertibility of D , are often referred toas primary invariants. Higher index theory provides a far-reaching generalization of theFredholm index by taking into consideration the symmetries of the underlying spaces;in particular, if M is a complete compact Riemannian manifold with an associated Dirac-type operator D , a higher index theory intrinsically involves the fundamental group π ( M ).The higher index theory plays a fundamental role in the studies of many important openproblems having relations to geometry and topology, such as the Novikov conjecture, theoperator K -theoretic Baum-Connes conjecture, and the Gromov-Lawson-Rosenberg con-jecture.A secondary higher invariant– so called due to its natural appearance upon the van-ishing of a primary invariant such as Ind G ( D )– was developed by Lott [28] within theframework of noncommutative differential forms, for manifolds with fundamental groupsof polynomial growth and D invertible. Lott’s work was heavily inspired by the work ofBismut and Cheeger on eta forms [6], which naturally arise in the index theory for familiesof manifolds with boundary [1]. Lott’s higher eta invariant, despite being defined by anexplicit integral formula of noncommutative differential forms is unfortunately difficult tocompute in general. To reduce the computability difficulty and make this higher invariantmore applicable to problems in geometry and topology it is useful to introduce (periodic)cyclic cohomology. The delocalized eta invariant of Lott [29] can be formally thought of asprecisely such a pairing with respect to traces (see the formula (4.1.2)).Given a delocalized cyclic cocycle class [ ϕ γ ] ∈ HC ∗ ( C π ( M ) , cl( γ )) of any degree, where γ ∈ π ( M ) and the conjugacy class cl( γ ) is not trivial, a higher analogue of Lott’s delo-calized eta invariant η [ ϕ γ ] ( e D ) is given in Definition 4.2. The precise definition of thesedelocalized cyclic cocyles is found in Definition 2.14, and the explicit formula for η [ ϕ γ ] ( e D )is described in terms of the transgression formula for Connes-Chern character [9, 12]. Thenatural problem which arises is in determining when this delocalized higher eta invariant can actually be rigorously well-defined and involves some subtle convergence issues. Im-portantly, we prove that with respect to a group Γ of polynomial growth every delocalizedcyclic cocyle class on the group algebra has a representative of polynomial growth. It isalso essential to prove that the pairing is independent of the choice of representative of2HEAGAN A. K. A JOHNany given cocycle class. We are thus able to show that whenever M possesses a finitelygenerated fundamental group of polynomial growth this higher analogue pairing of Lottshigher eta invariant with (delocalized) cyclic cocycles is well-defined, under the conditionsthat the Dirac operator on f M is invertible– or more generally has a spectral gap at zero. Theorem 1.1.
Let M be a closed odd-dimensional spin manifold equipped with a positivescalar metric g , and fundamental group of polynomial growth. Supposing the Dirac operator D has an associated lift e D to the universal cover f M , the higher delocalized eta invariant η [ ϕ γ ] ( e D ) converges absolutely for every [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) . Moreover, if S ∗ γ : HC m ( C π ( M ) , cl( γ )) −→ HC m +2 ( C π ( M ) , cl( γ )) denotes the delocalized Connes periodicity operator, then η [ S γ ϕ γ ] ( e D ) = η [ ϕ γ ] ( e D ) . When the higher index class of an operator is trivial– given a specific trivialization– asecondary index theoretic invariant naturally arises through a C ∗ -algebraic approach. Forexample, consider the associated Dirac operator on the universal covering f M of a closed, n -dimensional spin manifold M equipped with a positive scalar curvature metric g . TheLichnerowicz formula (see (4.1.1) of Section 4.1) asserts that the Dirac operator on f M isinvertible [26], and so Ind G ( D ) must necessarily be trivial. In this case, there is a natural C ∗ -algebraic secondary invariant ρ ( e D, e g ) introduced by Higson and Roe [17, 18, 19], calledthe higher rho invariant, which is an obstruction to the inverse of the Dirac operator beinglocal. This higher rho invariant describes a class belonging to the group K n ( C ∗ L, ( f M ) π ( M ) ),where π ( M ) is the fundamental group of M . As mentioned before, such a secondary indextheoretic invariant often plays an important role in problems in geometry and topology(cf. [40, 41, 43]). The precise description of the geometric C ∗ -algebra C ∗ L, ( f M ) π ( M ) isprovided in Definition 2.6, and the particular construction of the higher rho invariant isgiven at the beginning of Section 4.2. In the case that π ( M ) is of polynomial growthwe provide– using the construction of the determinant map of [44]– in Section 4.2 anexplicit formula (see Definition 4.11)for a pairing of C ∗ -algebraic secondary invariants anddelocalized cyclic cocycles of the group algebra is realized. Moreover, in the particularinstance that [ u ] ∈ K ( C ∗ L, ( f M ) π ( M ) ) is the K -theory class of the higher rho invariant ρ ( e D, e g ), then the pairing is given explicitly in terms of the higher delocalized eta invariant η [ ϕ γ ] ( e D ). Theorem 1.2.
Let M be a closed odd-dimensional spin manifold with fundamental groupof polynomial growth, then every delocalized cyclic cocycle [ ϕ γ ] ∈ HC m ( C π ( M ) , cl( γ )) induces a natural map τ [ ϕ γ ] : K ( C L, ( f M ) π ( M ) ) −→ C If M has positive scalar curvature metric g then τ [ ϕ γ ] ( u ) converges absolutely. When [ u ] = ρ ( e D, e g ) is the K -theory class of the higher rho invariant there is an equivalence τ [ ϕ γ ] ( ρ ( e D, e g )) = ( − m η [ ϕ γ ] ( e D )The above theorem holds in the more general case that the Dirac operator D on M hasan associated lift e D to the universal cover f M which is invertible. Showing that the map τ [ ϕ γ ] ϕ γ fromthe group algebra to the localization algebra requires the existence of a certain smooth densesubalgebra of C ∗ r ( π ( M )) introduced by Connes and Moscovici [13]. In [44], Xie and Yuestablished such a pairing between delocalized cyclic cocycles of degree m = 0– delocalizedtraces– and classes [ u ] belonging to K ( C ∗ L, ( f M ) π ( M ) ), under the assumption that therelevant conjugacy class has polynomial growth. Later, in [8], under the assumption that π ( M ) is a hyperbolic group, this construction was extended to allow for a pairing betweendelocalized cyclic cocycles of all degrees and the K-theory classes [ u ] ∈ K n ( C ∗ L, ( f M ) π ( M ) ).In the hyperbolic case, convergence of τ [ ϕ γ ] relies on the properties of Puschnigg’s [35]smooth dense subalgebra in an essential way.The C ∗ -algebraic map τ [ ϕ γ ] allows for a constructive and explicit approach to a higherdelocalized Atiyah-Patodi-Singer index theorem. In Section 4.4 we prove a direct relation-ship between pairings of K -theory classes [ u ] ∈ K n ( C ∗ L, ( f M ) π ( M ) ) with τ [ ϕ γ ] , and thepairing of classes ∂ [ u ] with respect to the delocalized Chern-Connes character map [9, 12](see (4.4.3) for the explicit expression used here), where ∂ : K n ( C ∗ ( f M ) π ( M ) ) −→ K n − ( C ∗ L, ( f M ) π ( M ) )is the usual K -theory connecting map. Combined with Theorem 1.2 this provides thefollowing version of a higher delocalized Atiyah-Patodi-Singer index theorem. Theorem 1.3.
Let W be a compact spin manifold with boundary, equipped with a scalarcurvature metric g which is positive on ∂W , and fundamental group which is of polyno-mial growth. Denote by e D W and e D ∂W the lifted Dirac operators on W and its boundary,respectively. ch [ ϕ γ ] (cid:16) Ind π ( W ) ( e D W ) (cid:17) = ( − m +1 η [ ϕ γ ] ( e D ∂W ) for any [ ϕ γ ] ∈ HC m ( C π ( M ) , cl( γ )) , where ch [ ϕ γ ] is the delocalized Chern-Connes char-acter map which pairs cyclic cocyles with the K -theoretic index class. There have been various versions of a higher Atiyah-Patodi-Singer theorem in the lit-erature, such as [23, 24, 14] and [38]. The form of the this result strongly mirrors thatconjectured by Lott [28, Conjecture 1], and is essentially a general case of that provenby Xie and Yu [44, Proposition 5.3] for zero dimensional cyclic cocyles. In Section 4.4 weprovide the basic background of the original APS index theorem, and show how the abovetheorem is specifically related to it. See the discussion following [8, Theorem 7.3] for moredetails on the relationships and differences of the above theorem with other existing resultsof higher APS index theorems.This paper is organized as follows. In Section 2.1 we provide the properties of the ge-ometric C ∗ -algebras which shall be used throughout, as well as detail the construction ofimportant smooth dense sub-algebras. Section 2.2 is concerned primarily with providingthe definition of cyclic cohomology and detailing the relationship between this and coho-mology for groups; in addition we recall an essential construction for explicit representativeof cyclic cocyle classes. In Section 3.1 we review the long exact sequence of periodic cycliccohomology involving the (delocalized) Connes periodicity operator (see Definition 3.2 and4HEAGAN A. K. A JOHN(3.1.3)); combining this with a cohomomological dimension result we prove a necessary tor-sion argument. We are thus able to construct a rational isomorphism between cohomologygroups of a certain complex of cyclic cocyles and group cohomology of a particular sub-group of π ( M ). Using the universal classifying description of group cohomology and theprevious rational isomorphism, the entirety of Section 3.2 is devoted to proving that everydelocalized cyclic cocyle has a representative of polynomial growth. In Section 4.1, given adelocalized cyclic cocyle of polynomial growth we define a higher analogue of Lott’s delocal-ized eta invariant and prove it converges for invertible elliptic operators. In Section 4.2, wefirst review a construction of Higson and Roe’s higher rho invariant as an explicit K -theoryclass. We provide an explicit formula for the pairing between C ∗ -algebraic secondary invari-ants and delocalized cyclic cocycles of the group algebra for groups of polynomial growth,and prove it is well-defined. In particular, in the case that the secondary invariant is a K -theoretic higher rho invariant of an invertible elliptic differential operator, we show inSection 4.3 that this pairing is precisely the higher delocalized eta invariant of the givenoperator. In Section 4.4, we use the determinant map of the previous section to determinea pairing between delocalized cyclic cocycles and C ∗ -algebraic Atiyah-Patodi-Singer indexclasses for manifolds with boundary, when the fundamental group of the given manifold isof polynomial growth. Acknowledgements:
The author would like to thank Zhizhang Xie for his invaluablesupport and advice. § In all that follows we will take M to be a closed odd-dimensional spin manifold, which isequipped with a positive scalar metric g . By D we denote the Dirac operator associated to M , and analogously by e D the associated lift to the universal cover f M . By Γ = π ( M ) werefer to a countable discrete finitely generated group which is also the fundamental groupof M . Given γ ∈ Γ the centralizer of γ will be denoted Z Γ ( γ ), or if there is no confusion asto the group Γ, by Z γ ; likewise, if γ Z is the cyclic group generated by γ , then the quotientgroup Z γ /γ Z will be denoted by N γ . By C Γ and Z Γ we mean the group algebra withcomplex coefficients and the group ring with integer coefficients, respectively.We recall that a finitely generated discrete group Γ comes equipped with a lengthfunction l S – with respect to some given symmetric generating set S ⊂ Γ. l S ( g ) = min { c ∈ N : ∃ s , . . . , s c ∈ S, s · · · s c = g } (2.0.1)There exists an associated word metric d S ( g, h ) = || g − h || ; = l S ( gh ) which is left-invariantwith respect to the group action. More importantly, since the metric spaces (Γ , S ) and(Γ , T ) are quasi-isomorphic for any choice of generating sets S and T , we are able to ignorethis choice when dealing with the word metric (or length function). Henceforth we willmerely refer to the length function l Γ or the word metric d Γ . Unless otherwise stated wewill assume throughout that Γ is of polynomial growth, which is defined as there existingpositive integer constants C and m such that |{ g ∈ Γ : || g || ≤ n }| ≤ C ( n + 1) m ∀ n ∈ N (2.0.2)5IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPS § C ∗ -algebras, and Smooth Dense Sub-algebras Let X be a proper metric space, and C ( X ) the algebra of continuous functions on X which vanish at infinity. An X -module H X is a separable Hilbert space equipped with a ∗ -representation π : C ( X ) −→ B ( H X ) into the algebra of bounded operators on H X , andis called non-degenerate if the ∗ -representation of C ( X ) is non-degenerate. If no non-zerofunction f ∈ C ( X ) acts a compact operator under this ∗ -representation, then we call H X a standard X -module. Definition 2.1.
Recall that an operator T acting on a Hilbert space H belongs to thealgebra of compact operators K ⊂ B ( H ) if the image under T of every bounded subset hascompact closure. (i) Let T ∈ B ( H X ) be a bounded linear operator acting on H X , then T is locally compactif for all f ∈ C ( X ) both f T and T f are compact operators. We similarly call T pseudo-local if the weaker condition, [ T, f ] =
T F − f T is a compact operator for all f ∈ C ( X ), is satisfied. (ii) Again assume that T belongs to B ( H X ); the propagation of T is defined to besup { d ( x, y ) : ( x, y ) ∈ Supp ( T ) } where Supp ( T ) denotes the support of T , which is the set { ( x, y ) ∈ X × X : ∃ f, g ∈ C ( X ) such that gT f = 0 and f ( x ) = 0 , g ( y ) = 0 } c If we further impose that H X is a standard and non-degenerate X -module, then thereexist important constructions of certain geometric C ∗ -algebras. The first two of these, de-scribed in Definition 2.2 were introduced by Roe in [36], and the coarse homotopy invarianceof their K -theory was subsequently proven by Higson and Roe [16]. Definition 2.2.
The C ∗ -algebra generated by all locally compact operators with finitepropagation in B ( H X ), is the Roe algebra of X and is denoted by C ∗ ( X ). If we insteadconsider the C ∗ -algebra generated by all pseudo-local operators with finite propagation in B ( H X ), then we obtain a related algebra D ∗ ( X ). In fact, D ∗ ( X ) is a subalgebra of themultiplier algebra M ( C ∗ ( X ))– which is the largest unital C ∗ -algebra containing C ∗ ( X ) asan ideal. Definition 2.3.
Let prop ( T ) denote the propagation of an operator T ∈ B ( H X ). Thelocalization algebras C ∗ L ( X ) and D ∗ L ( X ) introduced by Yu [45] are defined as the C ∗ -algebras generated by S and S respectively, where f is bounded and uniformly norm-continuous S = n f : [0 , ∞ ) −→ C ∗ ( X ) | lim t −→∞ prop ( f ( t )) = 0 o S = n f : [0 , ∞ ) −→ D ∗ ( X ) | lim t −→∞ prop ( f ( t )) = 0 o Once again D ∗ L ( X ) is a subalgebra of the multiplier algebra M ( C ∗ L ( X )). The kernel of theevaluation map ev : C ∗ L ( X ) −→ C ∗ ( X ) defined by ev ( f ) = f (0) is an ideal of C ∗ L ( X ), and isitself a C ∗ -algebra which we denote by C ∗ L, ( X ). Analogously we also define the C ∗ -algebra D ∗ L, ( X ) as the kernel of ev : D ∗ L ( X ) −→ D ∗ ( X ).6HEAGAN A. K. A JOHNIt follows that the Roe algebra and its localization fit into a short exact sequence–analogously for D ∗ ( X )– which give rise to a six term K -theoretic long exact sequence withconnecting map ∂ , and for which i = 0 , C ∗ L, ( X ) C ∗ L ( X ) C ∗ ( X ) 0 ev (2.1.1) K i ( C ∗ L, ( X )) K i ( C ∗ L ( X )) K i ( C ∗ ( X )) K i − ( C ∗ ( X )) K i − ( C ∗ L ( X )) K i − ( C ∗ L, ( X )) ∂∂ (2.1.2)Assuming that a group G acts properly and cocompactly on X by isometries, we can equip H X with a covariant unitary representation of G , which we will denote by ̟ . Explicitly, if g ∈ G, f ∈ C ( X ) and v ∈ H X ̟ ( g )( π ( f ) v ) = π ( f g )( ̟ ( g ) v )where f g ( x ) = f ( g − x ). We call the system ( H X , π, ̟ ) a covariant system. Definition 2.4.
Suppose that H X is a standard and non-degenerate X -module, and G actson X properly and cocompactly. Moreover, for each x ∈ X the action of the stabilizer group G x on H X is isomorphic to the action of G x on l ( G x ) ⊗ H for some infinite dimensionalHilbert space H , where G x acts trivially on H and by translations on l ( G x ). Under theseconditions a covariant system ( H X , π, ̟ ) is called admissible .If it is not necessary to emphasize the representations we shall simply refer to theadmissible system ( H X , π, ̟ ) by H X , and describe it as an admissible ( X, G )-module.
Remark 2.5.
For every locally compact metric space X which admits a proper and co-compact isometric action of G , there exists an admissible covariant system ( H X , π, ̟ ) . Definition 2.6.
Consider a locally compact metric space X which admits a proper andcocompact isometric action of G , and fix some admissible ( X, G )-module H X . The G -equivariant Roe algebra C ∗ ( X ) G is the completion in B ( H X ) of the ∗ -algebra C [ X ] G ofall G -invariant locally compact operators with finite propagation in B ( H X ). Replacing G -invariant locally compact operators with G -invariant pseudo-local operators we similarlyobtain D ∗ ( X ) G . The G -equivariant localization algebras C ∗ L ( X ) G and D ∗ L ( X ) G are definedas the C ∗ -algebras generated by S and S respectively, where f is bounded and uniformlynorm-continuous S = n f : [0 , ∞ ) −→ C ∗ ( X ) G | lim t −→∞ prop ( f ( t )) = 0 o S = n f : [0 , ∞ ) −→ D ∗ ( X ) G | lim t −→∞ prop ( f ( t )) = 0 o Analogous to Definition 2.3 we can also define the ideals C ∗ L, ( X ) G and D ∗ L, ( X ) G as thekernels of the evaluation map. 7IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSThe equivariant Roe algebra– analogously for D ∗ ( X ) G – fits into similar short exactsequence as did the original Roe algebra0 C ∗ L, ( X ) G C ∗ L ( X ) G C ∗ ( X ) G ev An especially useful consequence of the cocompact action of G on X is that there existsa ∗ -isomorphism between C ∗ r ( G ) ⊗ K and C ∗ ( X ) G , where C ∗ r ( G ) is the reduced group C ∗ -algebra of G . Remark 2.7.
The geometric C ∗ -algebras defined in Definition 2.3 and Definition 2.6 areall unique up to isomorphism, independent of the choice of H X is a standard and non-degenerate X -module. Likewise the G -equivariant versions are also, up to isomorphism,independent of the choice of admissible ( X, G ) -module H X . Let Γ and M be as described above; we turn our attention to construction of two im-portant smooth dense subalgebras of C ∗ r (Γ) ⊗ K ∼ = C ∗ ( f M ) Γ , the first of which is essentiallya slight modification of Connes and Moscovici’s [13]. Definition 2.8.
Fixing a basis of L ( M ), the algebra R of smooth operators on M canbe identified with the algebra of matrices ( a ij ) i,j ∈ N satisfyingsup i,j ∈ N i k j l | a ij | < ∞ ∀ k, l ∈ N Consider the unbounded operators ∆ : ℓ ( N ) −→ ℓ ( N ) and ∆ : ℓ (Γ) −→ ℓ (Γ) definedon basis elements according to∆ ( δ j ) = jδ j , j ∈ N and ∆ ( g ) = || g || · g, g ∈ ΓDenoting by I the identity operator and with [ · , · ] being the usual commutator bracket,we have unbounded derivations ∂ ( T ) = [∆ , T ] of operators T ∈ B ( ℓ (Γ)) and unboundedderivations e ∂ ( T ) = [∆ ⊗ I, T ] of operators T ∈ B ( ℓ (Γ) ⊗ ℓ ( N )). Define an algebra B ( f M ) Γ = { A ∈ C ∗ r (Γ) ⊗ K : e ∂ k ( A ) ◦ ( I ⊗ ∆ ) is bounded ∀ k ∈ N } The crucial property of B ( f M ) Γ is that it contains C Γ ⊗ R as a dense subalgebra, isitself a smooth dense subalgebra of C ∗ ( f M ) Γ , and it is closed under holomorphic functionalcalculus. Moreover, B ( f M ) Γ is a Fr´echet algebra under the sequence of seminorms {||·|| B ,k : k ∈ N } , where || A || B ,k = || e ∂ k ( A ) ◦ ( I ⊗ ∆ ) || op is the operator norm of e ∂ k ( A ) ◦ ( I ⊗ ∆ ) . Definition 2.9.
We define a kind of localization algebra B L ( f M ) Γ associated to B ( f M ) Γ ,which by construction is a smooth dense subalgebra of C ∗ L ( f M ) Γ and thus is closed underholomorphic functional calculus. B L ( f M ) Γ = { f ∈ C ∗ L ( f M ) Γ : f is piecewise smooth w . r . t t , f ( t ) ∈ B ( f M ) Γ ∀ t ∈ [0 , ∞ ) } and also define B L, ( f M ) Γ to be the kernel of the usual evaluation map ev : B L ( f M ) Γ −→ B ( f M ) Γ defined by ev ( f ) = f (0). 8HEAGAN A. K. A JOHN Proposition 2.10.
The inclusions B L ( f M ) Γ ֒ → C ∗ L ( f M ) Γ and B L, ( f M ) Γ ֒ → C ∗ L, ( f M ) Γ induce isomorphisms on K -theory K i ( B L ( f M ) Γ ) ∼ = K i ( C ∗ L ( f M ) Γ ) K i ( B L, ( f M ) Γ ) ∼ = K i ( C ∗ L, ( f M ) Γ ) Proof.
This follows immediately from the definitions of the smooth dense subalgebras.We now look at the second kind of smooth dense subalgebra of C ∗ ( f M ) Γ , this timeworking more directly with f M . Let A belong to the algebra C ∞ ( f M × f M ) of smooth functionson f M × f M , and assume that A is both Γ-invariant and of finite propagation. Explicitly, wemean that A ( gx, gy ) = A ( x, y ) ∀ g ∈ Γ ∃ R > A ( x, y ) = 0 , ∀ ( x, y ) ∈ f M × f M satisfying d f M ( x, y ) > R Definition 2.11.
Denote by L ( f M ) Γ the convolution algebra of A ∈ C ∞ ( f M × f M ) whichare both Γ-invariant and of finite propagation. The action of L ( f M ) Γ on L ( f M ) is accordingto ( Af )( x ) = Z f M A ( x, y ) f ( y ) dy for A ∈ L ( f M ) Γ , f ∈ L ( f M )Denote by ˆ ρ : f M −→ [0 , ∞ ) the distance function ˆ ρ ( x ) = ˆ ρ ( x, y ) for some fixed point y ∈ f M , with ρ being the modification of ˆ ρ near y to ensure smoothness. The multiplication op-erator T ρ thus acts as an unbounded operator on L ( f M ), according to ( T ρ f )( x ) = ρ ( x ) f ( x ).Using the commutator bracket we can define a derivation e ∂ = [ T ρ , · ] : L ( f M ) Γ −→ L ( f M ) Γ . A ( f M ) Γ = { A ∈ C ∗ ( f M ) Γ : e ∂ k ( A ) ◦ (∆ + 1) n is bounded ∀ k ∈ N } where ∆ is the Laplace operator on f M , and n is a fixed integer greater than dim( M ). Theassociated norm is given by || A || A ,k = || e ∂ k ( A ) ◦ (∆ + 1) n || op , which is the operator normof e ∂ k ( A ) ◦ (∆ + 1) n .The same proof of Connes and Moscovici [13, Lemma 6.4] shows that A ( f M ) Γ is closedunder holomorphic functional calculus, and contains L ( f M ) Γ as a subalgebra. Before pro-ceeding to define the generalized higher eta invariant in Section 4.1 we first recall a necessaryextension of A ( f M ) Γ , by introducing bundles. Consider the bundle on f M × f M given by End ( S ) = p ∗ ( S ) ⊗ p ∗ ( S ∗ ), where p i : f M × f M −→ f M are the obvious projection maps,with S and S ∗ being the spinor bundle on f M and its dual bundle, respectively. Considerthe set C ∞ ( f M × f M ,
End ( S )) of all smooth sections of the bundle End ( S ) on f M × f M ,and note that there exists a natural diagonal action of Γ on End ( S ). Thus, we can con-struct L ( f M , S ) Γ as the convolution algebra of all Γ-invariant finite propagation elementsof C ∞ ( f M × f M ,
End ( S )). Let L ( f M , S ) denote the the space of L -sections of S over f M ;there is an action of L ( f M , S ) Γ on L ( f M , S )( Af )( x ) = Z f M A ( x, y ) f ( y ) dy A ∈ L ( f M , S ) Γ f ∈ L ( f M , S ) (2.1.3)9IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSNow since L ( f M , S ) is an admissible ( f M , S )-module we can construct the Γ-equivariantRoe algebra C ∗ ( f M , S ) Γ associated to it; however by Remark 2.7 Roe algebras are up toisomorphism independent of the choice of admissible module. Thus, we will also denote by C ∗ ( f M ) Γ the Γ-equivariant Roe algebra constructed with respect to L ( f M , S ). Definition 2.12.
Let e D be the Dirac operator on f M , and fix some integer n > dim M ,then A ( f M , S ) Γ = { A ∈ C ∗ ( f M ) Γ : e ∂ k ( A ) ◦ ( e D n + 1) is bounded ∀ k ∈ N } where e ∂ = [ T ρ , · ] is the derivation on L ( f M , S ) Γ if we take T ρ to be the multiplicationoperator on L ( f M , S ). The algebras A L ( f M , S ) Γ and A L, ( f M , S ) Γ are defined analogouslyto those in Definition 2.9. The associated norm is given by || A || A , S ,k = || e ∂ k ( A ) ◦ ( e D n +1) || op , which is the operator norm of e ∂ ( A ) k ◦ ( e D n + 1).If there is no cause for confusion, we shall remove the explicit spinor notation andsimply denote the above norm on A ( f M , S ) Γ by || A || A ,k . We end this section with a briefreminder of the notion of projective tensor product A ˆ ⊗ mπ with respect to any of the ∗ -algebras constructed above. If A ⊗ B is the algebraic tensor product, then recall that theprojective tensor product A ˆ ⊗ π B is the completion of A ⊗ B with respect to the projectivecross norm π ( x ) = inf ( n x X i =1 || A i || A || B i || B : x = n x X i =1 A i ⊗ B i ) (2.1.4)where || · || A denotes the norm on A . We will denote the norm on A ˆ ⊗ mπ by || · || A ˆ ⊗ m andusually write elements of A ˆ ⊗ mπ as A ˆ ⊗ · · · ˆ ⊗ A m . § Definition 2.13.
Denote by C n ( C Γ) the cyclic module consisting of all ( n + 1)-functionals f : ( C Γ) ⊗ n +1 −→ C together with maps d i : ( C Γ) ⊗ n +1 −→ ( C Γ) ⊗ n defined according to d i ( a ⊗ · · · ⊗ a n ) = a ⊗ · · · ⊗ a i − ⊗ ( a i a i +1 ) ⊗ a i +2 ⊗ a n for 0 ≤ i < nd n ( a ⊗ · · · ⊗ a n ) = ( a n a ) ⊗ a ⊗ · · · ⊗ a n − and a cyclic operator t , where t f ( a ⊗ · · · ⊗ a n ) = ( − n f ( a n ⊗ a · · · ⊗ a n − ). Define thecoboundary differential b : C n ( C Γ) −→ C n ( C Γ) by b = P n +1 i =0 ( − i δ i , where δ i is the dualto d i ; that is h δ i f, a i = h f, d i ( a ) i . Hence we have( bf )( a ⊗· · ·⊗ a n +1 ) = n X i =0 ( − i f ( a ⊗· · ·⊗ ( a i a i +1 ) ⊗ a n )+( − n +1 f ( a n +1 a ⊗ a ⊗· · ·⊗ a n )The cohomology of the complex ( C n ( C Γ) , b ) is the cyclic cohomology HC ∗ ( C Γ).
Definition 2.14.
Fix γ ∈ Γ and denote by ( C Γ , cl( γ )) ⊗ n +1 the subcomplex of ( C Γ) ⊗ n +1 spanned by all elements ( g , . . . , g n ) ∈ Γ n +1 satisfying g · · · g n ∈ cl( γ ), where cl( γ ) is theconjugacy class of γ . This gives rise to a cyclic submodule C n ( C Γ , cl( γ )) of C n ( C Γ) whichcomprises the collection of functionals which vanish on ( g , . . . , g n ) if g · · · g n / ∈ cl( γ ).The coboundary differential b preserves this cyclic subcomplex, and we thus denote thecohomology of ( C n ( C Γ , cl( γ )) , b ) by HC ∗ ( C Γ , cl( γ )).10HEAGAN A. K. A JOHN Definition 2.15. By H ∗ ( N γ , C ) we are referring to the groups Ext ∗ Z N γ ( Z , C ) defined overthe projective Z N γ -resolution of Z . Namely, consider the resolution · · · Z N k +1 γ Z N kγ · · · Z N γ Z ∂ k ∂ k − ∂ ∂ ǫ where, if ˆ h i denotes a deleted entry, ∂ k acts on the basis elements according to ∂ k ( h , . . . , h k ) = k X i =0 ( − k ( h , . . . , b h i , . . . , h k )Dropping the Z term and applying the contravariant functor Hom N γ ( − , C ) to this resolutionproduces a cochain complex with coboundary differential ˆ b · · · Hom N γ ( Z N kγ , C ) · · · Hom N γ ( Z N γ , C ) 0 ˆ b ˆ b ˆ b ˆ b (ˆ bφ )( h , · · · , h k +1 ) = k +1 X i =0 ( − i φ ( h , . . . , b h i , . . . , h k +1 )The cohomology of this complex is defined to be the group cohomology H ∗ ( N γ , C ).Note that every cochain φ ∈ H n ( N γ , C ) satisfies the ”homogeneous” condition: that is,for every h ∈ N γ , hφ ( h , . . . , h n ) = φ ( hh , . . . , hh n ). It will be extremely useful to replacethe standard cochain complex with the sub-complex of homogeneous skew-cochains ϕ ( σ ( h , h , . . . , h n )) = ϕ ( h σ (0) , h σ (1) , . . . , h σ ( n ) ) = sgn( σ ) ϕ ( h , h , . . . , h n ) ∀ σ ∈ S n +1 (2.2.1)where S n +1 is the symmetric group on n + 1 letters. It is an immediate consequence of thisdefinition that ϕ ( h , . . . , h n ) vanishes whenever h i = h j for i = j ; just take σ to be thepermutation satisfying σ ( i ) = j, σ ( j ) = i , and which fixes all other indices. Define the map F : Hom N γ ( Z N nγ , C ) −→ Hom N γ ( Z N nγ , C ) according to( F φ )( h , . . . , h n ) = 1( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h , . . . , h n )) (2.2.2) Proposition 2.16.
For every φ ∈ Hom N γ ( Z N nγ , C ) the cochain F φ is a skew cochain ϕ .Proof. Let σ be any fixed even permutation– that is σ is decomposable as an even numberof 2-cycles, hence sgn( σ ) = 1. Since left multiplication of any group on itself is a free andtransitive action, it follows that for each σ there exists a unique τ σ such that σ τ σ = σ .( F φ )( σ ( h , . . . , h n )) = 1( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ σ ( h , . . . , h n ))= 1( n + 1)! X σ τ σ ∈ S n +1 sgn( σ τ σ ) φ ( σ σ ( h , . . . , h n ))11IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPS= 1( n + 1)! X τ σ ∈ S n +1 sgn( τ σ ) φ ( σ ( h , . . . , h n ))Since sign( σ )sign( τ σ ) = sign( σ τ σ ) = sgn( σ ) and σ is an even permutation then τ σ musthave the same parity as σ . It follows that( F φ )( σ ( h , . . . , h n )) = 1( n + 1)! X τ σ ∈ S n +1 sgn( σ ) φ ( σ ( h , . . . , h n )) = ( F φ )( h , . . . , h n )Follow the same argument, if σ is an odd permutation then again for each σ there exists aunique τ σ such that σ τ σ = σ . However, since sgn( σ ) = − τ σ must possessopposite parity to σ , hence( F φ )( σ ( h , . . . , h n )) = 1( n + 1)! X τ σ ∈ S n +1 − sgn( σ ) φ ( σ ( h , . . . , h n )) = − ( F φ )( h , . . . , h n ) Lemma 2.17.
The induced map F ∗ : H ∗ ( N γ , C ) −→ H ∗ ( N γ , C ) is an isomorphism.Proof. That F induces an isomorphism on cohomology (with real or complex coefficients)follows if we can show F ≃ Id as chain complex maps. First a straightforward calculationproves that F is a chain complex map, in the sense that the following diagram commutesfor all n . Hom N γ ( Z N n +1 γ , C ) Hom N γ ( Z N nγ , C ) Hom N γ ( Z N n +1 γ , C ) Hom N γ ( Z N nγ , C ) F ˆ b F ˆ b (ˆ b ◦ F φ )( h , . . . , h n +1 ) = n +1 X i =0 ( − i ( F φ )( h , . . . , b h i , . . . , h n +1 )= 1( n + 1)! X σ ∈ S n +1 sgn( σ ) n +1 X i =0 ( − i φ ( σ ( h , . . . , b h i , . . . , h n +1 ))= 1( n + 1)! X σ ∈ S n +1 sgn( σ )(ˆ bφ )( σ ( h , . . . , h n +1 )) = ( F ◦ ˆ bφ )( h , . . . , h n +1 )Now, F is chain homotopic to Id on each Hom N γ ( Z N nγ , C ) if there exists a sequence of maps { p k | p k : Hom N γ ( Z N kγ , C ) −→ Hom N γ ( Z N k − γ , C ) } such that F − Id = b ◦ p n + p n +1 ◦ b . Forease of notation denote h n − = ( h , . . . , h n − ); we will define( p n φ )( h n − ) = ( − n ( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h n − , eh n − )) − ( − n φ ( h n − , eh n − )12HEAGAN A. K. A JOHNwhere eh n − = h n − denotes a copy of h n − inserted into the n ’th position. For furtherease of notation we will denote ( h , . . . , b h i , . . . , h n , eh n ) by ( h n, ˆ i , eh n ) for i ≤ n .( b ◦ p n φ )( h n ) = n X i =0 ( − i ( p n φ )( h , . . . , b h i , . . . , h n )= n X i =0 ( − i ( − n ( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h n, ˆ i , eh n )) − ( − n φ ( h n, ˆ i , eh n ) ( p n +1 ◦ bφ )( h n ) = ( − n +1 ( n + 2)! X σ ∈ S n +1 sgn( σ )( bφ )( σ ( h n , eh n )) − ( − n +1 ( bφ )( h n , eh n )= n +1 X i =0 ( − i ( − n +1 ( n + 2)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h n, ˆ i , eh n )) − ( − n +1 φ ( h n, ˆ i , eh n ) Using the fact that by Proposition 2.16 the expressions( − n ( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( h n, ˆ i , eh n ) and ( − n +1 ( n + 2)! X σ ∈ S n +1 sgn( σ ) φ ( h n, ˆ i , eh n )vanish for all i ≤ n − h n = eh n , we thus have the reduced identities( b ◦ p n φ )( h n ) = ( − n ( − n ( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h n, ˆ n , eh n )) − n X i =0 ( − i + n φ ( h n, ˆ i , eh n )( p n +1 ◦ bφ )( h n ) = 1( n + 2)! X σ ∈ S n +1 sgn( σ ) (cid:16) ( − n +2 φ ( σ ( h n , e c h n )) + ( − n +1 φ ( σ ( h n, ˆ n , eh n )) (cid:17) − n +1 X i =0 ( − i + n φ ( h n, ˆ i , eh n )= n +1 X i =0 ( − i + n φ ( h n, ˆ i , eh n ) + 1( n + 2)! X σ ∈ S n +1 n +1 X i =0 ( − i + n φ ( h n, ˆ i , eh n )where we have used the fact ( h n, ˆ n , eh n ) = ( h n , e c h n ). Moreover, it is readily apparent thatboth these tuples are also equal to h n ; it follows that ( b ◦ p n φ + p n +1 ◦ bφ )( h n ) simplifiesto exactly the expression for ( F − Id) φ ( h n )( − n ( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h n, ˆ n , eh n )) + n +1 X i =0 ( − i + n φ ( h n, ˆ i , eh n ) − n X i =0 ( − i + n φ ( h n, ˆ i , eh n )= 1( n + 1)! X σ ∈ S n +1 sgn( σ ) φ ( σ ( h n )) − φ ( h n ) = ( F − Id) φ ( h n )13IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSFor the remainder of this paper, when referring to group cohomology it will be withrespect to the subcomplex of skew cochains. The following splitting of cyclic (co)-homologywas proven by Burghelea [7] using topological arguments, and Nistor [32] provided a lateralgebraic proof. Theorem 2.18. HC ∗ ( C Γ) ∼ = Y cl( γ ) HC ∗ ( C Γ , cl( γ )) Moreover there exist isomorphisms with group (co)-homology HC ∗ ( C Γ , cl( γ )) ∼ = (cid:26) H ∗ ( N γ , C ) γ is of infinite order H ∗ ( N γ , C ) ⊗ C HC ∗ ( C ) γ is of finite order Definition 2.19.
Fix a group element γ with conjugacy class cl( γ ), and let C k (Γ , Z γ , γ )be the collection of all multilinear forms α : Γ k +1 −→ C satisfying α ( g σ (0) , g σ (1) , . . . , g σ ( n ) ) = sgn( σ ) α ( g , g , . . . , g k ) ∀ σ ∈ S k +1 α ( zg , zg , . . . , zg k ) = α ( g , g , . . . , g k ) ∀ z ∈ Z γ α ( γg , g , . . . , g k ) = α ( g , g , . . . , g k )The coboundary map ˆ b : C k (Γ , Z γ , γ ) −→ C k +1 (Γ , Z γ , γ ) gives rise to a cochain complex( C n (Γ , Z γ , γ ) , ˆ b ), the cohomology of which we will denote by H ∗ ( C , C ) · · · C k +1 (Γ , Z γ , γ ) C k (Γ , Z γ , γ ) · · · C (Γ , Z γ , γ ) 0 ˆ b ˆ b ˆ b ˆ b ˆ b (ˆ bφ )( g , · · · , g k +1 ) = k +1 X i =0 ( − i φ ( g , . . . , b g i , . . . , g k +1 )Recall that a cyclic cocyle of ( C n ( C Γ) , b ) is a functional ϕ which belongs to the kernel ZC n ( C Γ) of the coboundary differential. If cl( γ ) is non-trivial we call the cyclic cocyles ϕ γ of ( C n ( C Γ , cl( γ )) , b ) delocalized cyclic cocycles. Following the example of Lott [28, Section4.1] we can construct explicit representations of any delocalized cyclic cocyle as follows:associated to each α ∈ H ∗ ( C , C ) define ϕ α,γ ( g , g , . . . , g n ) = (cid:26) g g · · · g n / ∈ cl( γ ) α ( h, hg , . . . , hg g · · · g n − ) if g g · · · g n = h − γh (2.2.3)By multilinearity of α , it is immediate that given a i = P g i ∈ Γ c g i · g i in the group algebra ϕ α,γ ( a ⊗ · · · ⊗ a n ) = X g g ··· g n ∈ cl( γ ) c g · · · c g n ϕ α,γ ( g , g , . . . , g n )It is also apparent that the property ϕ α,γ ( γg , g , . . . , g k ) = α ( g , g , . . . , g k ) generalizes toany element of γ Z , that is for any r ∈ Z – it suffices to consider r ≥
0– we have ϕ α,γ ( γ r g , g , . . . , g k ) = ϕ α,γ ( γ r − g , g , . . . , g k ) = · · · = ϕ α,γ ( g , g , . . . , g k )14HEAGAN A. K. A JOHNIf A is a unital algebra such that ϕ γ admits an extension to A then we define a unitizedversion of the cyclic cocyle. Let A + be the algebra formed from adjoining a unit to A ,then the homomorphism ( A, λ ) ( A + λ A , λ ) provides an isomorphism between A + and A ⊕ C
1. For any ϕ ∈ ZC n ( C Γ , cl( γ )) we define ϕ γ ( A ˆ ⊗ · · · ˆ ⊗ A n ) = ϕ γ ( A ˆ ⊗ · · · ˆ ⊗ A n ) where A i = ( A i , λ i ) ∈ A + (2.2.4)and as shown in [12, Chapter 3.3] the condition bϕ γ = 0 still holds. Remark 2.20.
With respect to the delocalized cyclic cocycle representations ϕ α,γ there isan elementary way to move between α ( g , g , . . . , g n ) and the normalized form α ( h, hg , . . . , hg g · · · g n − ) which clearly vanishes if g i = e for any ≤ i ≤ n − . For each y ∈ cl( γ ) fix some h y ∈ Γ such that ( h y ) − γh y = y . In particular, the elements y = g g · · · g n and y i = g i · · · g n g · · · g i − all belong to cl( γ ) for ≤ i ≤ n , since by hypothesis y ∈ cl( γ ) , anddirect computation shows that y i = ( g · · · g i − ) − y ( g · · · g i − ) .The map F defined by F ( g i ) = h y ( g i · · · g n ) − y i induces a map on H ∗ ( C , C ) accordingto F ∗ [ α ] = [ α ◦ F ] where since g g · · · g n = y = ( h y ) − γh y ( α ◦ F )( g , g , . . . , g n ) = α ( F ( g ) , F ( g ) , . . . , F ( g n ))= α ( h y ( g · · · g n ) − y , h y ( g · · · g n ) − y , . . . , h y g − n y n )= α ( h y ( g · · · g n ) − ( g · · · g n ) , h y ( g · · · g n ) − g · · · g n g , . . . , h y g − n g n g · · · g n − )= α ( h y , h y g , . . . , h y g · · · g n − ) This property of F carries over to ϕ α,γ acting on the group algebra C Γ by extending F linearly. § The convergence properties of the integrals defining the pairing of delocalized cyclic cocyleswith higher invariants depend crucially on the growth conditions of the cyclic cocyles. Thisin turn is linked to the growth properties of conjugacy classes of Γ, in particular it isproven in [20] that polynomial growth groups are of polynomial cohomology– with respectto coefficients in C . Definition 3.1.
The group G is of polynomial cohomology if for any [ φ ] ∈ H ∗ ( G, C ) thereexists (a skew cocyle) ϕ ∈ Z ( Hom Γ ( Z G ∗ , C ) , ˆ b ) such that [ ϕ ] = [ φ ], and ϕ is of polynomialgrowth. That is ϕ satisfies the following bound for positive integer constants R ϕ and k | ϕ ( g , g , . . . , g n ) | ≤ R ϕ (1 + || g || ) k (1 + || g || ) k · · · (1 + || g n || ) k (3.0.1)By Remark 2.20 it follows that any normalized group cocyle α also has polynomialgrowth if the non-normalized version does, since | α ( h, hg , . . . , hg g · · · g n − ) | = | α ( F ( g ) , F ( g ) , . . . , F ( g n )) | γ is atorsion element. § In the next section our results depend crucially on H ∗ ( N γ , C ) not contributing to thedelocalized cyclic cohomology of C Γ whenever γ is of infinite order; in this section we makethis notion precise. For any unital associative algebra A over a field containing Q – henceparticularly for the group algebra C Γ– the cyclic and Hochschild homology fit into a longexact sequence · · · HH n ( A ) HC n − ( A ) HC n +1 ( A ) HH n +1 ( A ) · · · S ∗ (3.1.1)where S is the Connes periodicity operator introduced in [9, II.1]. We will use the explicitconstruction in terms of maps of complexes that is provided in [27, Chapter 2], and soprovide the following expression for S when A = C Γ. Let b ∗ be the homomorphism inducedby the boundary map b , and define a map β : HC ∗ ( C Γ) −→ HC ∗ +1 ( C Γ) according to( βϕ )( g , g , . . . , g k +1 ) = k +1 X i =0 ( − i i ( δ i ϕ )( g , g , . . . , g k +1 ) (3.1.2)Hence ( βb ) ∗ : HC ∗ ( C Γ) −→ HC ∗ +2 ( C Γ) and similarly for the map induced by bβ . Dualto the result given in [27, Theorem 2.2.7] we have that for any cohomology class [ ϕ ] ∈ HC n ( C Γ) its image under the periodicity operator is S ∗ [ ϕ ] = [ Sϕ ] ∈ HC n +2 ( C Γ) where S = 1( n + 1)( n + 2) ( βb + bβ ) (3.1.3) bβ = X ≤ i The delocalized Connes periodicity operator S γ is obtained by the restric-tion of S to the sub-complex ( C n ( C Γ , cl( γ )) , b ). S ∗ γ : HC n ( C Γ , cl( γ )) −→ HC n +2 ( C Γ , cl( γ ))In the construction of the group cohomology H ∗ ( G, C ) = Ext ∗ Z G ( Z , C ) the minimallength of the projective Z G -resolution over Z is called the cohomological dimension of thegroup. Denoting this by cd Z ( G ), it is immediate from the definition that if cd Z ( G ) = n Note that our choice of constant differs from that of Connes [9] due to the constants involved in thedefinition (see Equation (4.4.3)) of the Chern-Connes character H k ( G, C ) = 0 for all k > n . If we consider projective Q G -resolutions instead, thenthere is notion of rational cohomology and rational cohomological dimension. Recall that Q is a flat Z -module, hence tensoring with Q preserves exactness, and cd Q ( G ) denotes theminimal length of the projective resolution defining the groups H ∗ ( G, C ) ⊗ Z Q = Ext ∗ Q G ( Q , C )When G is a group of polynomial growth then it belongs to the class C of groups satisfyingthe following conditions (see [21, Section 4])(i) G is of finite rational cohomological dimension.(ii) The rational cohomological dimension of N g = Z G ( g ) /g Z is finite whenever g is nota torsion element.It is now important to note that if γ is of infinite order then cl( γ ) is torsion free, forotherwise there exists g − γg ∈ cl( γ ) of finite order, and thus e = ( g − γg ) k = g − γ k g ,which implies γ k = e . In this torsion free case the nilpotency of S γ with respect to the longexact sequence · · · HH n − ( C Γ , cl( γ )) HC n ( C Γ , cl( γ )) HC n +2 ( C Γ , cl( γ )) HH n +2 ( C Γ , cl( γ )) · · · S ∗ (3.1.5)follows from the proof of [21, Theorem 4.2], and so we have that HC n ( C Γ , cl( γ )) = 0 for n > γ of infinite order. Lemma 3.3. If Γ is of polynomial growth, then N γ is of polynomial cohomology.Proof. Since Γ is of polynomial growth, then a theorem of Gromov [15] states that Γ isvirtually nilpotent. We take N to be a normal nilpotent subgroup of finite index, and so Z γ ∩ N ≤ N is finitely generated and nilpotent. The exact sequence Z γ ∩ N −→ Z γ −→ Γ /N shows that Z γ ∩ N is of finite index in Z γ , hence Z γ is finitely generated and admits a wordlength function l Z γ which is bounded by l Γ . It follows that Z γ is of polynomial growthwith respect to any word length function on it. Taking the quotient by a central cyclicgroup preserves polynomial growth, and thus by [20, Corollary 4.2] N γ is of polynomialcohomology.It should be made explicit that in the above proof we also obtained that Z γ was ofpolynomial cohomology– or equivalently that H ∗ ( Z γ , C ) is polynomial bounded. Extend-ing the notion of polynomial cohomology to that of delocalized cyclic cocyles, we call HC ∗ ( C Γ , cl( γ )) polynomially bounded if every cohomology class admits a representative ϕ γ ∈ ( C n ( C Γ , cl( γ )) , b ) which is of polynomial growth. Lemma 3.4. There is a rational isomorphism between the cohomology group H n ( C , C ) (seeDefinition 2.19) and H n ( Z γ , C ) for n ≥ . Proof. We begin with an alteration of the complex (( C n (Γ , Z γ , γ ) , ˆ b ) constructed in Def-inition 2.19, by removing the third condition: α ( γg , g , . . . , g n ) = α ( g , g , . . . , g n ). Thisproduces a larger cochain complex which shall be denoted by (( D n (Γ , Z γ , γ ) , ˆ b ), and thereis a natural inclusion map ı : (( C n (Γ , Z γ , γ ) , ˆ b ) ֒ → (( D n (Γ , Z γ , γ ) , ˆ b ). In the other directionwe consider an ”averaging” map R : (( D n (Γ , Z γ , γ ) , ˆ b ) −→ (( C n (Γ , Z γ , γ ) , ˆ b ) similar to thatfrom [8, Theorem 5.2], defined according to( R α )( g , g , . . . , g n ) = ord( γ ) X r ,r ,...,r n =1 α ( γ r g , γ r g , . . . , γ r n g n ) (3.1.6)where ord( γ ) is the order of γ . By the above results of this section we only need toconcern ourselves with γ being a torsion element, and thus the above sum is finite, so R is well defined. To show that R is a surjective map it is first necessary to provethat ( R α ) actually belongs to the complex (( C n (Γ , Z γ , γ ) , ˆ b ); namely if γ is torsion, then γ Z = { e, γ, . . . , γ ord( γ ) − } = γ · γ Z and( R α )( γg , g , . . . , g n ) = ord( γ ) X r ,r ,...,r n =1 α ( γ r +1 g , γ r g , . . . , γ r n g n ) ord( γ ) X r +1 ,r ,...,r n =1 α ( γ r +1 g , γ r g , . . . , γ r n g n ) = ( R α )( g , g , . . . , g n )It is similarly straightforward to prove that the following diagram commutes D n +1 (Γ , Z γ , γ ) D n (Γ , Z γ , γ ) C n +1 (Γ , Z γ , γ ) C n (Γ , Z γ , γ ) R ˆ b R ˆ b and so R is indeed a chain complex map. We now prove that the composition R ◦ ı :(( C n (Γ , Z γ , γ ) , ˆ b ) −→ (( C n (Γ , Z γ , γ ) , ˆ b ) is rationally equivalent to the identity map Id C ,and thus by extension that R is a rational surjection. Taking any α ∈ C n (Γ , Z γ , γ ) andusing the property α ( γ r g , g , . . . , g n ) = α ( g , g , . . . , g n )( R ◦ ıα )( g , g , . . . , g n ) = ord( γ ) X r ,r ,...,r n =1 α ( γ r g , γ r g , . . . , γ r n g n )= ord( γ ) X r =1 ord( γ ) X r ,...,r n =1 α ( γ r g , γ r g , . . . , γ r n g n )= ord( γ )(ord( γ ) + 1)2 ord( γ ) X r ,...,r n =1 α ( g , γ r g , . . . , γ r n g n )18HEAGAN A. K. A JOHNFor convenience denote the coefficient ord( γ )(ord( γ ) + 1) / A , then by repeated shiftingof the g element the above sum becomes= ( − A ord( γ ) X r =1 ord( γ ) X r ,...,r n =1 α ( g , g , γ r g , . . . , γ r n g n )= ( − A γ ) X r ,...,r n =1 α ( g , g , γ r g , . . . , γ r n g n ) = · · · = ( − n A n +1 α ( g , . . . , g n , g )= ( − n ( − n A n +1 α ( g , g , . . . , g n ) = A n +1 α ( g , g , . . . , g n )It thus follows that as maps from ( C n (Γ , Z γ , γ ) , ˆ b ) ⊗ Z Q to itself, we have the equality( R ◦ ı ) ⊗ Z A n +1 = Id C ⊗ Z H ∗ ( C , C ) ⊗ Z Q R ∼ = H ∗ ( D , C ) ⊗ Z Q . Thedesired result now follows from Nistor’s [32, Section 2.7] application of spectral sequencesto prove that H ∗ ( D , C ) is isomorphic to the group cohomology H ∗ ( Z γ , C ). § The isomorphism H ∗ ( D , C ) ∼ = H ∗ ( Z γ , C ) mentioned in Lemma 3.4 unfortunately does notprovide an explicit way to realize preservation of polynomial cohomology. It is, however,easy to show that as defined in Lemma 3.4 the map R behaves as desired in this respect. Proposition 3.5. The map ( R ⊗ Z /A n +1 ) ∗ preserves polynomial growth.Proof. It suffices to show that if α is of polynomial growth then so is R α , which followsdirectly from γ having finite order. | ( R α )( g , g , . . . , g n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ord( γ ) X r ,r ,...,r n =1 α ( γ r g , γ r g , . . . , γ r n g n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ord( γ ) X r ,r ,...,r n =1 | α ( γ r g , γ r g , . . . , γ r n g n ) | ≤ ord( γ ) X r ,r ,...,r n =1 R α (1+ || γ r g || ) k · · · (1+ || γ r n g n || ) k ≤ (ord( γ )) n +1 max r ,r ,...,r n ∈{ , ..., ord( γ ) } R α (1 + || γ r g || ) k · · · (1 + || γ r n g n || ) k = (ord( γ )) n +1 R α (1 + || γ m g || ) k · · · (1 + || γ m n g n || ) k Theorem 3.6. For every nontrivial conjugacy class cl( γ ) for γ of finite order, if H ∗ ( Z γ , C ) is of polynomial cohomology then H ∗ ( D , C ) is polynomially bounded. Proof. Consider the simplicial set E ∗ Γ, the nerve of Γ, with simplices E k Γ := Γ × Γ k andrelations d i ( g , . . . , g k ) = (cid:26) ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g k ) 0 ≤ i ≤ k − g , g . . . , g k − ) i = ks j ( g , . . . , g k ) = ( g , . . . , g j , e, g j +1 , . . . , g k )( g , . . . , g k − , g k ) g = ( g g, . . . , g k − , g k ) ∀ g ∈ Γ (3.2.1)The last equation defines a free Γ-action on the nerve according to right multiplication. Thecontractible space E Γ = | E ∗ Γ | – which is a locally finite CW-complex– is the geometricalrealization of the nerve under the relations | E ∗ Γ | = G k ≥ E k Γ × △ k / ∼ ( x, δ i t ) ∼ ( d i x, t ) for x ∈ E k Γ , t ∈ △ k − ( x, σ j t ) ∼ ( s j x, t ) for x ∈ E k Γ , t ∈ △ k +1 (3.2.2)where △ k = { ( t , . . . , t k ) ∈ R k +1 : t i ∈ [0 , , P ki =0 t i = 1 } is the standard k -simplexwith degeneracy maps σ j and face maps δ i . Since right multiplication is a free action, it isimmediate that Γ acts freely on E Γ, thus we define the classifying space B Γ = | B ∗ Γ | to bethe orbit space E Γ / Γ; the simplices of B ∗ Γ are thus of the form E k Γ / Γ. More generally,it is useful to view E Γ as a the universal principal bundle p : E Γ −→ B Γ, which providesthe relation of balanced products B Γ = E Γ × Γ Γ = { ( v, w ) ∈ E Γ × Γ / (( v, gw ) ∼ ( vg, w )) } Applying the above simplicial construction to Z γ we can similarly construct the universalprincipal Z γ -bundle p : EZ γ −→ BZ γ . The geometric realization is functorial, hence anygroup homomorphism induces a homomorphism on E Γ at the simplex level. Since Γ is adiscrete group then Z γ is admissible as a subgroup, and there is the principal Z γ -bundle q : Γ −→ Γ /Z γ where q is the quotient map. In particular, since p : E Γ −→ B Γ is auniversal principal Γ-bundle p ′ : E Γ × Z γ Γ −→ E Γ × Γ (Γ /Z γ ) = ( Z γ y E Γ) /Z γ is a universal principal Z -bundle, where Z γ y E Γ has the same nerve E ∗ Γ, except withthe third relation in (3.2.1) replaced by a Z γ group action( g , . . . , g k − , g k ) z = ( g z, . . . , g k − , g k ) ∀ z ∈ Z γ By universality of the principal bundle there is a homotopy equivalence between BZ γ and( Z γ y E Γ) /Z γ as CW-complexes, which implies equivalence of (co)homology groups. (See[30] for a more detailed discussion on classifying spaces and fibre bundles)Every point v in an oriented m -simplex V m of E Γ can be expressed as a formal sum,using barycentric coordinates and the vertices g i = ( e, . . . , g i , . . . , e ). m X i =0 t i = 1 and v = m X i =0 t i g i V m = g [ g | · · · | g m ]20HEAGAN A. K. A JOHNThe particular notation for the m -simplex is important in emphasizing the group action onthe first coordinate. As we shall see shortly, this is also useful when describing boundarymaps of simplicial chain complexes. First consider the CW-complex B Γ as the image of E Γ under the projection map p acting on the nerve by deletion of the first coordinate.Hence, an oriented m -simplex V m of B Γ is of the form p ( V m ) = [ g | · · · | g m ]; hence weconstruct the following chain complex, with C m ( B Γ) the free module generated by thebasis of m -simplexes. · · · C m ( B Γ) C m − ( B Γ) · · · C ( B Γ) 0 ∂ ∂ ∂ ∂ c α ∈ C , X α c α p ( V m ) α ∈ C m ( B Γ) ∂ = m X i =0 ( − i d i Thus, explicitly expressing the boundary map action shows that ∂ ( p ( V m ) α ) is equal to p ( g g [ g | · · · | g m ]) + m − X i =1 ( − i p ( g [ g | · · · | g i g i +1 | · · · | g m ]) + ( − m p ( g m g [ g | · · · | g m − ])= [ g | · · · | g m ] + m − X i =1 ( − i [ g | · · · | g i g i +1 | · · · | g m ] + ( − m [ g | · · · | g m − ]Dualizing, the associated simplicial cochain complex C ∗ i ( B Γ) := Hom ( C i ( B Γ) , C ) is ob-tained, with coboundary map b = P n +1 i =0 δ i ; recall that h δ i f, v i = h f, d i ( v ) i .0 C ∗ ( B Γ) C ∗ ( B Γ) · · · C ∗ m ( B Γ) · · · b b b b The simplicial cohomology groups H ∗ ( B Γ , C ) of this complex give precisely the same char-acteristic classes as the group cohomology H ∗ (Γ , C ). More usefully, the expression of bun-dles in the language of balanced products allows us to similarly obtain equivalences H ∗ ( EZ γ × Z γ Z γ , C ) = H ∗ ( BZ γ , C ) = H ∗ ( Z γ , C ) H ∗ ( E Γ × Γ (Γ /Z γ ) , C ) = H ∗ (( Z γ y E Γ) /Z γ , C ) = H Z γ ∗ (Γ , C ) = H ∗ ( D , C )It follows that exhibiting an explicit Z γ -equivariant map ψ : E Γ × Γ (Γ /Z γ ) −→ EZ γ × Z γ Z γ which induces a polynomial growth preserving isomorphism on cohomology ψ ∗ : H ∗ ( EZ γ × Z γ Z γ , C ) −→ H ∗ ( E Γ × Γ (Γ /Z γ ) , C ) (3.2.3)also provides an isomorphism ψ ∗ : H ∗ ( Z γ , C ) −→ H ∗ ( D , C ) preserving polynomial coho-mology. It is of course necessary to be precise by what is meant by polynomial cohomologyin the context of the classifying space construction. If for every class [ ϕ ] ∈ H m ( B Γ , C ) thereexists a representative e ϕ ∈ ( C ∗ m ( B Γ) , b ) which is of polynomial growth, then H ∗ ( B Γ , C )is polynomial cohomology. Viewing e ϕ as a function on basis elements, it is of polynomialgrowth given e ϕ = X α c α p ( V m ) α = X α c α p ( g α [ g α | · · · | g m α ]) := e ϕ ( g α , g α , . . . , g m α ) = c α | e ϕ ( g α , g α , . . . , g m α ) | = | c α | ≤ R e ϕ (1 + || g α || ) k (1 + || g α || ) k · · · (1 + || g m α || ) k (3.2.4)21IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSwhere R e ϕ and k are positive integer constants. Recall that the word length function l Z isbounded above by l Γ , hence we shall view Z γ as metrically embedded in Γ. Fix some gen-erating set S of Γ and also fix an ordering for S , then every g ∈ Γ can be lexicographicallyordered; denote by lex (Γ) the lexicographic ordering of Γ under that of S . We introduce amap f : Γ −→ Z γ defined as the following minimizing z ∈ lex (Γ) ∩ Z γ . f ( g ) := min z ∈ lex (Γ) ∩ Z γ (cid:26) min z ∈ Z γ { d Γ ( z, g ) ≤ || g ||} (cid:27) (3.2.5)Due to the lexicographic ordering this provides a unique element f ( g ) ∈ Z γ . Such anelement always exists, since there is at least the option z = e ; in particular, it is a directconsequence that f ( z ) = z for z ∈ Z γ . Moreover, using the fact that left multiplication ofany group on itself is a free and transitive action it follows that f is Z γ -equivariant, sinceif we fix z ∈ Z γ f ( z g ) := min z ∈ lex (Γ) ∩ Z γ (cid:26) min z ∈ Z γ { d Γ ( z, z g ) ≤ || z g ||} (cid:27) = min z z ∈ lex (Γ) ∩ Z γ (cid:26) min z z ∈ Z γ { d Γ ( z z, z g ) ≤ || z g ||} (cid:27) = min z z ∈ lex (Γ) ∩ Z γ (cid:26) min z z ∈ Z γ { d Γ ( z, g ) ≤ || z g ||} (cid:27) =: z f ( g )It is similarly possible to express a Z γ -invariant map ˜ f : Γ /Z γ −→ Z γ in terms of the map f constructed above. We recall that for h, g ∈ Γ d Γ ( hZ γ , g ) = min z ∈ Z γ { d Γ ( hz, g ) } and d Γ ( h Z γ , h Z γ ) = min z,z ′ ∈ Z γ { d Γ ( h z, h z ′ ) } and thus define ˜ f ( hZ γ ) to be the value of f ( hz ) present in the minimizationmin z ∈ Z γ { d Γ ( hz, f ( hz )) } By the bijection between cosets of Γ /Z γ and conjugacy classes of γ arising from the map hZ γ h − γh we ensure well-definedness of ˜ f . The map ψ acts on E Γ × Γ (Γ /Z γ ) accordingto ψ ( x, hZ γ ) = ψ X i t i g i , hZ γ ! = X i t i ( e, . . . , f ( g i ) , . . . , e ) , ˜ f ( hZ γ ) ! (3.2.6)From this we can show that ψ is Z γ -equivariant on E Γ and Z γ -invariant in the secondargument. Pick any ( z, z ′ ) ∈ Z γ × Z γ , then converting the right Γ-action on E Γ to a leftone ψ (( z, z ′ )( x, hZ γ )) = ψ ( z − x, z ′ hZ γ ) = ψ X i t i ( z − g i ) , z ′ hZ γ ! = X i t i ( z − , . . . , f ( z − g i ) , . . . , z − ) , ˜ f ( z ′ hZ γ ) ! X i t i ( z − e, . . . , z − f ( g i ) , . . . , z − e ) , ˜ f ( hZ γ ) ! = X i t i · z − ( e, . . . , f ( g i ) , . . . , e ) , ˜ f ( hZ γ ) ! = ( z, e ) · ψ ( x, hZ γ )By the definition provided by equation (3.2.4) the map ψ : E Γ × Γ (Γ /Z γ ) −→ EZ γ × Z γ Z γ preserves polynomial growth if there exists a positive integer r and constant K > ρ, d Γ ) p ( ψ ( x, h Z γ ) , ψ ( y, h Z γ )) ≤ K · [( ρ, d Γ ) p (( x, h Z γ ) , ( y, h Z γ ))] r (3.2.7)This ensures that for any change λ | φ | in the value of a cyclic cocyle representative of theclass [ ϕ ] ∈ H ∗ ( EZ γ × Z γ Z γ , C ) there exists a representative of ψ ∗ [ ϕ ] ∈ H ∗ ( E Γ × Γ (Γ /Z γ ) , C )whose absolute value | ψ ∗ φ | changes no more than K ( λ | φ | ) r . As a preliminary necessity toproving this property of ψ , we recall that a path metric can be put on E Γ such that for x, y not in the same connected component ρ ( x, y ) = 1, and otherwise if x and y are joinedby a union of paths S kl =1 α l , where each α l belongs to a single simplex ρ ( x, y ) = inf α l k X l =1 length( α l )Hence there exists points v and v ′ in this simplex such that length( α l ) = ρ ( v, v ′ ); the metric ρ is defined as min { ρ , ρ } , where ρ X i t i g i , X i t ′ i g i ! = X i | t i − t ′ i ||| g i || ρ X i t i g i , X i t ′ i g i ! = X i,j t i t ′ j d Γ ( g i , g j )The metric placed on Γ /Z γ will be the usual word metric d Γ , and we assign to E Γ × Γ (Γ /Z γ )the p -product metric ( ρ, d Γ ) p , for p ∈ [1 , ∞ ). By the left Γ-invariance of d Γ , and the factthat ( xg, w ) ∼ ( x, gw ) it follows that ( ρ, d Γ ) p is also left Γ-invariant. Without loss ofgenerality we may take x and y to belong to the same connected component, and sincethe collection of left cosets partition Γ, we assume that h Z γ is distinct from h Z γ . Theproof of the inequality (3.2.7) thus follows immediately from the definition of ψ : explicitly,( ρ , d Γ ) p ( ψ ( x, h Z γ ) , ψ ( y, h Z γ )) has the expression (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ k X l =1 X i l t i l f ( g i l ) , k X l =1 X i l t ′ i l f ( g i l ) , d Γ ( ˜ f ( h Z γ ) , ˜ f ( h Z γ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X l =1 X i l ,j l t i l t ′ j l d Γ ( f ( g i l ) , f ( g j l )) , d Γ ( ˜ f ( h Z γ ) , ˜ f ( h Z γ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X l =1 X i l ,j l t i l t ′ j l d Γ ( g i l , g j l ) , d Γ ( h Z γ , h Z γ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p d Γ ( f ( g i ) , f ( g j ) ≤ d Γ ( g i , g j ) provenbelow in Proposition 3.8 and the analogous one for ˜ f . Similarly, with respect to the metric ρ we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ k X l =1 X i l t i l f ( g i l ) , k X l =1 X i l t ′ i l f ( g i l ) , d Γ ( ˜ f ( h Z γ ) , ˜ f ( h Z γ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X l =1 X i l | t i l − t ′ i l ||| f ( g i l ) || , d Γ ( ˜ f ( h Z γ ) , ˜ f ( h Z γ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X l =1 X i l | t i l − t ′ i l ||| g i l || , d Γ ( h Z γ , h Z γ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p The factor of 2 present in the last line stems from the fact that || f ( g i ) || = d Γ ( f ( g i ) , e ) ≤ d Γ ( f ( g i ) , g i ) + d Γ ( g i , e ) ≤ d Γ ( e, g i ) + d Γ ( g i , e ) = 2 || g i || In combination with the result for ρ this proves the inequality (3.2.7) for K = 4 and r = 1. Corollary 3.7. Every delocalized cyclic cocyle class [ ϕ γ ] ∈ HC ∗ ( C Γ , cl( γ )) has a repre-sentative ϕ α,γ of polynomial growth, hence H ∗ ( C Γ , cl( γ )) is polynomially bounded.Proof. Since Γ is of polynomial growth, then the proof of Lemma 3.3 asserts that H ∗ ( Z γ , C )is of polynomial cohomology. Furthermore, if we denote by R − the inverse isomorphismto that constructed in Lemma 3.4, then by Proposition 3.5 and Theorem 3.6 there exists achain of isomorphisms which preserve polynomial cohomology for all n ≥ H n ( Z γ , C ) ⊗ Z Q ( ψ ⊗ Z ∗ −−−−−→ H n ( D , C ) ⊗ Z Q ( R − ⊗ Z /A n +1 ) ∗ −−−−−−−−−−−→ H n ( C , C ) ⊗ Z Q The desired result now follows from recalling that by Definition 2.19 there exists an explicitrepresentation ϕ α,γ ∈ ( C n (Γ , Z γ , γ ) , ˆ b ) for each ϕ γ ∈ ( C n ( C Γ , cl( γ )) , b ). Proposition 3.8. Let f : Γ −→ Z γ and ˜ f : Γ /Z γ −→ Z γ be as described in Theorem 3.6,then both have a Lipschitz constant of 4.Proof. Given distinct g i , g j ∈ Γ with word representations g i = s i s i · · · s i k and g j = s j s j · · · s j k d Γ ( g i , g j ) = || g i g − j || = l Γ ( s i s i · · · s i k − m s − j k − m · · · s − j s − j ) = j k + i k − m where m is the cancellation length. By definition of f provided above in (3.2.5) there existlexicographically minimal z i = f ( g i ) and z j = f ( g j ) such that d Γ ( z i , g i ) and d Γ ( z j , g j ) areminimized. By the properties of the metric, it is immediate that d Γ ( z i , z j ) ≤ d Γ ( z i , g i ) + d Γ ( g i , g j ) + d Γ ( g j , z j ) ≤ i k + ( j k + i k − m ) + j k ≤ i k + j k )24HEAGAN A. K. A JOHNOn the other hand, expressing z i and z j as the words z i = s ′ i s ′ i · · · s ′ i k ′ and z j = s ′ j s ′ j · · · s ′ j k ′ d Γ ( z i , g i ) = || z i g − i || = l Γ ( s ′ i s ′ i · · · s ′ i k ′ − b s − i k − b · · · s − i s − i ) = i k ′ + i k − bd Γ ( g j , z j ) = || g j z − j || = l Γ ( s j s j · · · s j k − a s ′− j k ′ − a · · · s ′− j s ′− j ) = j k ′ + j k − a We can thus make use of the decomposition z i z − j = z i g − i g i g − j g j z − j and obtain thereduced word expression for z i z − j as s ′ i · · · s ′ i k ′ − b s − i k − b · · · s − i s i · · · s i k − m s − j k − m · · · s − j s j · · · s j k − a s ′− j k ′ − a · · · s ′− j = s ′ i s ′ i · · · s ′ i k ′ − b s − i k − b · · · s − i k − m − s j k − m − · · · s j k − a s ′− j k ′ − a · · · s ′− j s ′− j where– without loss of generality– we have assumed that m ≥ a, b . It follows that l Γ ( z i z − j ) = i k ′ − b + ( i k − b − i k + m + 1) + ( j k − a − j k + m + 1) + j k ′ − a = i k ′ + j k ′ − a − b + 2 m + 2However, since the identity element is always a possible choice for z i we know that i k ′ + i k − b ≤ i k , from which it follows that i k ′ − b ≤ 0, and analogously j k ′ + j k − a ≤ j k implies that j k ′ − a ≤ 0; this provides the bound d Γ ( z i , z j ) ≤ m + 2. We thus have thetwo following cases:(i) If m ≥ i k + j k then d Γ ( g i , g j ) = j k + i k − m ≥ ( j k + i k ), and the bound d Γ ( z i , z j ) ≤ i k + j k ) provides: d Γ ( z i , z j ) ≤ d Γ ( g i , g j )(ii) If m ≤ i k + j k then d Γ ( g i , g j ) = j k + i k − m ≥ m , and the bound d Γ ( z i , z j ) ≤ m + 2provides: d Γ ( z i , z j ) ≤ d Γ ( g i , g j )An analogous result is obtained for ˜ f using its definition with respect to f and the in-equalities already proven for f . The proof follows along the exact same lines, explicitlyproviding min z,z ′ ∈ Z γ { d Γ ( f ( h z ) , f ( h z ′ )) } ≤ z,z ′ ∈ Z γ { d Γ ( h z, h z ′ ) } d Γ ( ˜ f ( h Z γ ) , ˜ f ( h Z γ )) ≤ d Γ ( h Z γ , h Z γ )When choosing a representative of polynomial growth for the purposes of the followingsections, it is paramount that this choice is independent of representative in a way thatrespects polynomial growth. Explicitly, let [ ϕ γ ] be a delocalized cyclic cocyle class withpolynomial growth representatives ψ γ, , ψ γ, ∈ ( C n ( C Γ , cl( γ )) , b ). Since by definition thegroup H n ( C Γ , cl( γ )) is the quotient ZC n ( C Γ , cl( γ )) /BC n ( C Γ , cl( γ ))then ψ γ, and ψ γ, being cohomologous implies existence of a delocalized cyclic cocyle φ belonging to ( C n − ( C Γ , cl( γ )) , b ) such that ψ γ, − ψ γ, = bφ .25IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPS Remark 3.9. If [ ϕ γ ] ∈ H n ( C Γ , cl( γ )) has polynomial growth representatives ψ γ, and ψ γ, ,then there exists a cyclic cocyle φ of polynomial growth such that ψ γ, − ψ γ, = bφ .Proof. For any length function l on Γ it is proven by Ji [20, Theorem 2.23] that the inclusion i : C Γ ֒ → S l (Γ) induces an isomorphism between the Schwartz cohomology s H n (Γ , C ) = H n ( S l (Γ) , C ) and the group cohomology H n (Γ , C ) for Γ a discrete countable group ofpolynomial growth. In particular, we have s H nl ( Z γ , C ) ∼ = H n ( Z γ , C ) and so Proposition 3.5along with the strategy of Theorem 3.6 provides for a polynomial growth preserving mapsuch that φ is the image of a representative of an element in s H nl ( Z γ , C ). § § Let e D be the Dirac operator lifted to f M , s a section of the spinor bundle S , and ∇ : C ∞ ( f M , S ) −→ C ∞ ( f M , T ∗ f M ⊗ S ) the connection on S . Since M has positive scalar curva-ture κ > e g , then Lichnerowicz’s formula [26] e D s = ∇∇ ∗ ψ + κ ( s )4 (4.1.1)implies that e D is invertible. Moreover, e D is a self-adjoint elliptic operator, and so possessesa real spectrum: σ ( e D ) ⊂ R . The invertibility condition particularly provides existence of aspectral gap at 0, which will be necessary in ensuring convergence of the integral introducedin Definition 4.2. This will be a higher analogue of the delocalized higher eta invariant Lott[29] introduced in the case of 0-dimensional cyclic cocyles– that is for traces. Given a non-trivial conjugacy class cl( γ ) of the fundamental group Γ = π ( M ) Lott’s delocalized highereta invariant can be formally defined as the pairing between Lott’s higher eta invariantand traces. η tr γ ( e D ) := 2 √ π Z ∞ tr γ (cid:16) e De − t e D (cid:17) (4.1.2)Here the trace map tr : C Γ −→ C continuously extends to a suitable smooth dense subal-gebra of C ∗ r (Γ) to which e De − t e D belongs. Generally, if F is a fundamental domain of f M under the action of Γ, then for Γ-equivariant kernels A ∈ C ∞ ( f M × f M )tr γ ( A ) = X g ∈ cl( γ ) Z F A ( x, gx ) dx (4.1.3)Under the assumption of hyperbolicity or polynomial growth of the conjugacy class of γ , Lott [29] showed convergence of the above integral. Invertibility of e D is in general anecessary condition for this convergence, as was shown by the construction of a divergentcounterexample by Piazza and Schick [33, Section 3]. However, it was proven by Cheng,Wang, Xie and Yu [8, Theorem 1.1] that as long the spectral gap of e D is sufficiently large,then η tr γ ( e D ) converges absolutely, and does not require any restriction on the fundamentalgroup of the manifold. 26HEAGAN A. K. A JOHNSince we shall have occasion to use their properties often, we shall briefly recall themost important aspects of the space S ( R ) of Schwartz functions. By definition, f belongsto S ( R ) if f : R −→ C is a smooth function such that for every k, m ∈ N lim | x |−→∞ x k d m dx m ( f ( x )) = 0This implies that f is bounded with respect to the family of semi-norms || f || k,m = sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) x k d m dx m ( f ( x )) (cid:12)(cid:12)(cid:12)(cid:12) (4.1.4)Moreover, the Fourier transform f ˆ f is an automorphism of the Schwartz space, thusˆ f ∈ S ( R ) for every Schwartz function f , whereˆ f ( ξ ) = 12 π Z ∞−∞ f ( x ) e − iξx dx (4.1.5) Lemma 4.1. If Φ is a Schwartz function and e D is the lifted Dirac operator associated to f M , then Φ( e D ) ∈ A ( f M , S ) Γ .Proof. This is proven as Proposition 4.6 in [44] Definition 4.2. For any delocalized cyclic cocyle class [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) the delo-calized higher eta invariant of e D with respect to [ ϕ γ ] is defined as η ϕ γ ( e D ) := m ! πi Z ∞ η ϕ γ ( e D, t ) dt (4.1.6)where η ϕ γ ( e D, t ) = ϕ γ (( ˙ u t ( e D ) u − t ( e D )) ˆ ⊗ (( u t ( e D ) − ) ˆ ⊗ ( u − t ( e D ) − )) ˆ ⊗ m ) and F t ( x ) = 1 √ π Z tx −∞ e − s ds u t ( x ) = e πiF t ( x ) ˙ u t ( x ) = ddt u t ( x )Note that the arguments of η ϕ γ ( e D, t ) all belong to A ( f M , S ) Γ , since u t ( x ) − , u − t ( x ) − u t ( x ) u − t ( x ) are all Schwartz functions. In particular, we have the simplification˙ u t ( x ) u − t ( x ) = 2 πi (cid:18) ddt F t ( x ) (cid:19) e πiF t ( x ) e − πiF t ( x ) = 2 πi (cid:18) ddt F t ( x ) (cid:19) = 2 i √ πxe − t x It is also useful to consider the representation of the delocalized higher eta invariant interms of smooth Schwartz kernels, namely if L i is an element of the convolution algebra L ( f M , S ) Γ , the action of ϕ γ on C Γ can be extended to L ( f M , S ) Γ by– abusing notation alittle we will denote the Schwartz kernel of L i by L i also– defining ϕ γ ( L ˆ ⊗ L ˆ ⊗ . . . ˆ ⊗ L n ) tobe X g g ··· g n ∈ cl( γ ) ϕ γ ( g , . . . , g n ) Z F n +1 tr n Y i =0 L i ( x i , g i x i +1 ) ! dx · · · dx n : x n +1 = x (4.1.7)27IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSwhere F is the fundamental domain of f M under the action of Γ = π ( M ), and tr denotesthe pointwise matrix trace, not to be confused with the trace norm || · || tr for trace classoperators. Denoting by a t ( x, y ) , b t ( x, y ) and k t ( x, y ) the Schwartz kernels of the operators u t ( e D ) − , u − t ( e D ) − and ˙ u t ( e D ) u − t ( e D ) respectively, then η ϕ γ ( e D, t ) is given by X g g ··· g m ∈ cl( γ ) ϕ γ ( g m ) Z F m +1 tr k t ( x , g x ) m − Y i =1 a t ( x i , g i x i +1 ) b t ( x i +1 , g i +1 x i +2 ) ! d x m g m := ( g , . . . , g m ) d x m := dx · · · dx m (4.1.8)In this form we can better exploit the properties of A ( f M , S ) Γ , in order to prove that η ϕ γ ( e D ) converges for e D invertible and Γ of polynomial growth. The first step is provingextension of delocalized cyclic cocycles on the smooth dense subalgebra A ( f M , S ) Γ , interms of kernel operators. Since the fundamental group of a manifold has a cocompact,isometric, and properly discontinuous action on the universal cover, by the ˇSvarc-Milnorlemma [37, 31] there is a quasi-isometry f : π ( M ) −→ f M ; for every g, h ∈ Γ there exists K ≥ , ℓ ≥ d Γ ( g, h ) − Kℓ ≤ Kd f M ( f ( g ) , f ( h )) ≤ K d Γ ( g, h ) + Kℓ and for every y ∈ f M there exists g y ∈ Γ such that d f M ( f ( g y ) , y ) ≤ ℓ . In particular, wemay fix some p ∈ f M and define f ( g ) = gp ; moreover, restricting our attention to pointsbelonging to F the value of ( K, ℓ ) can be taken to be (1 , diam( F )) since each orbit iscobounded.We note that in [8, Section 8], under the assumption that π ( M ) is of polynomialgrowth, the authors used the techniques of [10, 11] to establish that the above definitionof the de-localized higher eta invariant agrees with Lotts higher eta invariant [28, Section4.4 & 4.6] up to a constant. Theorem 4.3. Let Γ = π ( M ) and ϕ γ ∈ ( C n ( C Γ , cl( γ )) , b ) be a delocalized cyclic cocyleof polynomial growth, then ϕ γ extends continuously on the algebra ( A ( f M , S ) Γ ) ˆ ⊗ n +1 π .Proof. Denote by ˆ ρ : f M −→ [0 , ∞ ) the distance function ˆ ρ ( x ) = ˆ ρ ( x, y ) for some fixedpoint y ∈ f M , with ρ being the modification of ˆ ρ near y to ensure smoothness. Let B ∈ A ( f M , S ) Γ and recall that we have the norm || B || A ,k = || e ∂ k ( B ) ◦ ( e D n + 1) || op ; forany f ∈ L ( f M , S ) the Sobolev embedding theorem provides existence of some constant C such that | B ( f )( x ) | ≤ C || (1 + e D n ) B ( f ) || L ( f M, S ) ≤ C || ( e D n + 1) B || op || f || L ( f M, S ) In particular, since f is arbitrary, the bound || B ( x, · ) || L ( f M, S ) ≤ C || ( e D n + 1) B || op showsthat taking the supremum over all ( x, y ) ∈ f M × f M the Schwartz kernel e ∂ k ( B )( x, y ) = ( ρ ( x ) − ρ ( y )) k B ( x, y )has operator norm bounded by || B || A ,k , hence it is a uniformly bounded continuous func-tion for all k ∈ N . Now view B ( x, y ) as a matrix acting on the spinors f ( y ), where each28HEAGAN A. K. A JOHNsection f has a representation as a matrix in the complex Clifford algebra C ℓ dim( M ) . If I is the identity matrix, then by the Holder inequality for Schatten p -norms | tr ( B ( x, y )) | ≤ || B ( x, y ) || tr = || B ( x, y ) I || tr ≤ || B ( x, y ) || op || I || tr < n || B ( x, y ) || op Since all points of f M belong to some orbit of the fundamental domain we have the bound | ρ ( x i ) − ρ ( x i +1 ) | ≤ diam( F ), and by quasi-isometry of Γ and the universal cover, we have(taking a family of quasi-isometries f i ( g ) = gx i ) | ρ ( x i +1 ) − ρ ( gx i +1 ) | ≥ d Γ ( e, g ) − diam( F ) = || g || − diam( F )From this, an application of the reverse triangle inequality provides the bound | ρ ( x i ) − ρ ( gx i +1 ) | = | ρ ( x i ) − ρ ( x i +1 ) + ρ ( x i +1 ) − ρ ( gx i +1 ) |≥ | ρ ( x i +1 ) − ρ ( gx i +1 ) | − | ρ ( x i ) − ρ ( x i +1 ) | ≥ || g || − diam( F ) − diam( F )Denoting the matrix norm || · || op by | · | , the boundedness properties of the Schwartz kernelimplies existence of a constant C k > k ∈ N C k | e ∂ k ( B i )( x i , gx i +1 ) | = ( ρ ( x i ) − ρ ( gx i +1 )) k | B i ( x i , gx i +1 ) | ≥ (1 + || g || ) k | B i ( x i , gx i +1 ) | We will use the explicit representation ϕ α,γ of ϕ γ , and for ease of notation, we shorten theargument of α by writing α ( g n ); we wish to prove convergence of the following sum. ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) = X g g ··· g n ∈ cl( γ ) α ( g ) Z F n +1 tr n Y i =0 B i ( x i , g i x i +1 ) ! dx · · · dx n (4.1.9)From the fact that α is of polynomial growth, and using the above inequalities coupledwith Cauchy-Schwartz, | ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) | is bounded above by X g g ··· g n ∈ cl( γ ) | α ( g n ) | Z F n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tr n Y i =0 B i ( x i , g i x i +1 ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx · · · dx n ≤ X g g ··· g n ∈ cl( γ ) n | α ( g n ) | Z F n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y i =0 B i ( x i , g i x i +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx · · · dx n ≤ X g g ··· g n ∈ cl( γ ) R α n Y i =0 (1 + || g i || ) k n Y i =0 (cid:18)Z F | B i ( x i , g i x i +1 ) | dx i dx i +1 (cid:19) / = X g g ··· g n ∈ cl( γ ) R α n Y i =0 (cid:18)Z F (1 + || g i || ) k | B i ( x i , g i x i +1 ) | dx i dx i +1 (cid:19) / ≤ R α C / k n Y i =0 X g i ∈ Γ (1 + || g i || ) − k Z F sup ( x i ,g i x i +1 ) ∈F×F | e ∂ k ( B i )( x i , g i x i +1 ) | dx i dx i +1 / g i the integral over the fundamental domain is finite, since e ∂ k ( B i )( x i , g i x i +1 ) isuniformly bounded; explicitly there exists a constant Λ k such that for g g · · · g n ∈ cl( γ )the value | ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) | is bounded above by R α C / k n Y i =0 X g i ∈ Γ (1 + || g i || ) − k Z F Λ k || B i || A ,k dx i dx i +1 / ≤ R α C / k diam( F )Λ k n Y i =0 X g i ∈ Γ (1 + || g i || ) − k || B i || A ,k / (4.1.10)where we denote R α,k = R α C / k diam( F )Λ k . Moreover, due to Γ being of polynomialgrowth, there exists k i such that(1 + || g i || ) − k i |{ g i ∈ Γ : || g i || ≤ c }| < c It follows that each of the sums in the final expression (4.1.10) are finite for sufficiently large k , and thus so is any finite product of them. Now, by construction L ( f M ) Γ is a smooth densesub-algebra of A ( f M ) Γ , and this relationship also extends when considering their projectivetensor products. We have just proven that ϕ α,γ is continuous on ( A ( f M ) Γ ) ˆ ⊗ n +1 π , hence toobtain the desired result it suffices to prove that for operators B , . . . , B n ∈ L ( f M ) Γ sgn( σ ) ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) = ϕ α,γ ( B σ (0) ˆ ⊗ B σ (1) ˆ ⊗ · · · ˆ ⊗ B σ ( n ) ) (4.1.11)whenever σ ∈ S n +1 is a cyclic shift. By application of the Fubini-Tonelli theorem, and since ϕ α,γ is a cyclic cocyle on C Γ we obtain ϕ α,γ ( B σ (0) ˆ ⊗ B σ (1) ˆ ⊗ · · · ˆ ⊗ B σ ( n ) ) is equal to X g g ··· g n ∈ cl( γ ) sgn( σ ) ϕ α,γ ( g n ) Z F n +1 tr n Y i =0 B σ ( i ) ( x σ ( i ) , g σ ( i ) x σ ( i +1) ) ! dx σ (0) · · · dx σ ( n ) = X g g ··· g n ∈ cl( γ ) sgn( σ ) ϕ α,γ ( g n ) Z F n +1 tr n Y i =0 B i ( x i , g i x i +1 ) ! dx · · · dx n The following technical result is one which we will have occasion to use often, both inthe remainder of this section and elsewhere. Proposition 4.4. For any collection of Schwartz functions f , f , . . . , f n ∈ S ( R ) and anydelocalized cyclic cocyle ϕ γ ∈ ( C n ( C Γ , cl( γ )) , b ) of polynomial growth lim t −→ ϕ γ ( f ( t e D ) ˆ ⊗ f ( t e D ) ˆ ⊗ · · · ˆ ⊗ f n ( t e D )) = 030HEAGAN A. K. A JOHN Proof. Fix some t = 0 and consider the Schwartz functions f i,t ( x ) = f i ( tx ) for 1 ≤ i ≤ n ;since the Fourier transform is an automorphism of S ( R ) there exists g i,t ∈ S ( R ) such thatˆ g i,t ≡ f i,t . Using the change of variables x = y/t , and the definition from (4.1.5), we obtain f i,t ( ξ ) = ˆ g i,t ( ξ ) = 12 π Z ∞−∞ g i,t ( x ) e − iξx dx = 12 π Z ∞−∞ g i ( tx ) e − iξy/t dyt = 12 πt Z ∞−∞ g i ( y ) e − iy ( ξ/t ) dy = ˆ g i ( ξ/t ) t = f i ( tξ )Now since each of these functions is Schwartz, (4.1.4) asserts that the following limit existsand is finite; in particular, ˆ g i ( tξ ) −→ t as t −→ ∞ , fromwhich it follows thatlim t −→ f i,t ( ξ ) = lim t −→ f i ( tξ ) = lim t −→ ˆ g i ( ξ/t ) t = lim t −→∞ t · ˆ g i ( tξ ) = 0Turning to functional calculus, by Lemma 4.1 each f i ( t e D ) belongs to A ( f M , S ) Γ , and thespectral gap at 0 of e D ensures that || f i ( t e D ) || A ,k converges to 0 as t −→ 0. From (4.1.10)of Theorem 4.3 it follows that there exists a positive constant R α,k such thatlim t −→ | ϕ γ ( f ( t e D ) ˆ ⊗ f ( t e D ) ˆ ⊗ · · · ˆ ⊗ f n ( t e D )) |≤ lim t −→ R α,k n Y i =0 X g i ∈ Γ (1 + || g i || ) − k || f i ( t e D ) || A ,k / = 0 Lemma 4.5. Let [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) , then if e D is invertible and ϕ γ is of polynomialgrowth, then η ϕ γ ( e D ) converges absolutely.Proof. The higher delocalized eta invariant can be split into two integrals, as follows. η ϕ γ ( e D ) := m ! πi Z ∞ η ϕ γ ( e D, t ) dt = m ! πi (cid:18)Z η ϕ γ ( e D, t ) dt + Z ∞ η ϕ γ ( e D, t ) dt (cid:19) For the first integral, absolute convergence follows from Theorem 4.3 and Proposition 4.4,using the Schwartz kernel expression Z | η ϕ γ ( e D, t ) | dt ≤ sup t ∈ [0 , | ϕ γ (( ˙ u t ( e D ) u − t ( e D )) ˆ ⊗ (( u t ( e D ) − ) ˆ ⊗ ( u − t ( e D ) − )) ˆ ⊗ m ) | < ∞ For the second integral it is useful to work in the unitization ( A ( f M , S ) Γ ) + , for which ϕ γ is well defined and continuous on the projective tensor product (( A ( f M , S ) Γ ) + ) ˆ ⊗ m +1 π byTheorem 4.3. Then following an argument similar to that of [8, Proposition 3.30] we havethat R ∞ η ϕ γ ( e D, t ) dt is bounded above by (cid:20) sup t ∈ [1 , ∞ ) n Y i =0 X g i ∈ Γ (1 + || g i || ) − k || e De − e D ˆ ⊗ (¯ u t ( e D ) ˆ ⊗ ¯ u − t ( e D )) ˆ ⊗ m || A + ) ˆ ⊗ m +1 ,k / × Z ∞ R α,k Ce − ( t − r / dt (cid:21) < ∞ where C, R α,k and r are positive constants. Theorem 4.6. The higher delocalized eta invariant is independent of the choice of cocyclerepresentative. Explicitly, if [ ϕ γ ] = [ φ γ ] ∈ HC m ( C Γ , cl( γ )) , then η ϕ γ ( e D ) = η φ γ ( e D ) Proof. By hypothesis, ϕ γ and φ γ are cohomologous via a coboundary bϕ belonging to BC m ( C Γ , cl( γ )). By the results of Section 3.2 we can assume ϕ ∈ ( C m − ( C Γ , cl( γ )) , b ) tobe a skew cochain of polynomial growth, and it suffices to prove that η bϕ ( e D ) = 0. Workingin ( A ( f M , S ) Γ ) + we obtain the transgression formula of [8, eq(3.23)] mη bϕ ( e D, t ) = ddt ϕ ((¯ u t ( e D ) ˆ ⊗ ¯ u − t ( e D )) ˆ ⊗ m ) (4.1.12)Ignoring the constant term m ! πi in the definition of the delocalized higher eta invariant andintegrating both sides with respect to tmη bϕ ( e D ) = m lim T −→∞ Z T /T η bϕ ( e D, t ) dt = m lim T −→∞ Z T /T ddt ϕ ((¯ u t ( e D ) ˆ ⊗ ¯ u − t ( e D )) ˆ ⊗ m ) dt = m lim T −→∞ ϕ (( u T ( e D ) − ˆ ⊗ u − T ( e D ) − ) ˆ ⊗ m − m lim T −→ ϕ (( u T ( e D ) − ˆ ⊗ u − T ( e D ) − ) ˆ ⊗ m )That lim T −→ ϕ (( u T ( e D ) − ˆ ⊗ u − T ( e D ) − ) ˆ ⊗ m ) = 0 follows from Proposition 4.4. For T −→∞ , since e D has a spectral gap at 0, we have by the properties of holomorphic functionalcalculus that for x ∈ (0 , ∞ ) both u T ( e D ) − and u − T ( e D ) − converge in the || · || A ,k normto 0. It thus follows from the bounds of Theorem 4.3 thatlim T −→∞ | ϕ (( u T ( e D ) − ˆ ⊗ u − T ( e D ) − ) ˆ ⊗ m | = 0 (4.1.13) Proposition 4.7. Let S ∗ γ : HC m ( C Γ , cl( γ )) −→ HC m +2 ( C Γ , cl( γ )) be the delocalizedConnes periodicity operator, then η [ ϕ γ ] ( e D ) = η [ S γ ϕ γ ] ( e D ) for every [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) .Proof. We may assume by Corollary 3.7 that ϕ γ is of polynomial growth; by the definitionof S γ the cocyle S γ ϕ γ is also of polynomial growth. Since our expression for S γ coincideswith that of [8, Definition 3.32] the result follows from [8, Proposition 3.33] § To begin with, we recall the assumptions put on the spin manifold M , namely that itis closed and odd dimensional with a positive scalar curvature metric g . Let e D be theDirac operator lifted to the universal cover f M , s a section of the spinor bundle S , and ∇ : C ∞ ( f M , S ) −→ C ∞ ( f M , T ∗ f M ⊗ S ) the connection on S . Since f M has positive scalarcurvature κ > e g , then Lichnerowicz’s formula shows that e D is invertible,hence there exists a spectral gap at 0. Since e D is an elliptic essentially self-adjoint operator,32HEAGAN A. K. A JOHNusing a suitable normalizing function ψ , the operator ψ ( e D ) is bounded pseudo-local andself-adjoint. In particular, to emphasize the relationship with the delocalized higher etainvariant it is particularly useful that for t ∈ (0 , ∞ ) we consider ψ ( tx ) = 2 √ π Z tx e − s ds (4.2.1)Since there exist a spectral gap at 0 with respect to e D , the limit || lim t −→ ψ ( e D/t ) || op exists and converges to || ( e D | e D | − ) || op , where e D | e D | − = signum( e D ). Define an operator H = ( + e D | D | − ) and let { φ s,j } be a partition of unity subordinate to the Γ-invariantlocally finite open cover { U s,j } s,j ∈ N of f M . For each s we take diam( U s,j ) < s , and so for t ≥ H ( t ) = X j ( s + 1 − t ) φ / s,j H φ / s,j + ( t − s ) φ / s +1 ,j H φ / s +1 ,j : t ∈ [ s, s + 1] (4.2.2)Since the support supp( φ s,j ) of each member of the partition of unity is a subset of U s,j the propagation of H ( t ) tends to 0 as t −→ ∞ . Together with H ( t ) being a pseudo-local,self-adjoint bounded operator, this gives us that H ( t ) ∈ D ∗ ( f M , S ) Γ ; moreover, due to thechoice of χ we have H ′ ( t ) , H ( t ) − ∈ C ∗ ( f M , S ) Γ . Moreover, as H ( t ) is a projection, thepath of invertibles S = { u ( t ) = exp(2 πiH ( t )) | t ∈ [0 , ∞ ) } belong to ( C ∗ ( f M , S ) Γ ) + , and since exp(2 πi · signum( x )) = 1 for any x = 0 we have byconstruction that u (0) = . It follows that u belongs to the kernel of the evaluation map ev : ( C ∗ L ( f M , S ) Γ ) + −→ ( C ∗ ( f M , S ) Γ ) + and so the path S gives rise to a K-theory class [ u ] ∈ K ( C ∗ L, ( f M , S ) Γ ), which is bydefinition the higher rho invariant ρ ( e D, e g ) of Higson and Roe [17, 18, 19]. Before definingthe pairing between cyclic cocyles and the higher rho invariant it is useful to introduce a fewtechnical notions which will be needed later on. By Proposition 2.10 any class of invertible[ u ] ∈ K ( C ∗ L, ( f M , S ) Γ ) is directly equivalent to a class of invertible [ u ] ∈ K ( B L, ( f M , S ) Γ ),and we also recall that the (localized)-equivariant Roe algebra is independent of the choiceof admissible module, hence we will work within the framework of B ( f M ) Γ . Definition 4.8. Consider the unitization ( B ( f M ) Γ ) + of the algebra B ( f M ) Γ , and its sus-pension S B ( f M ) Γ . If A is a C ∗ -algebra recall that the suspension S A is defined as { f ∈ C ([0 , , A ) | f (0) = f (1) = 0 } Identify S with the quotient space [0 , / (0 ∼ f ∈ S B ( f M ) Γ invertible if it is a piecewise smooth loop f : S −→ ( B ( f M ) Γ ) + of invertible elementssatisfying f (0) = f (1) = . The map f is local if there exists f L ∈ S B L ( f M ) Γ such thatthe following hold(i) f L : S −→ ( B L ( f M ) Γ ) + is a loop of invertible elements satisfying f L (0) = f L (1) = .33IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPS(ii) f is the image of f L under the evaluation map ev : S B L ( f M ) Γ −→ S B ( f M ) Γ Recall that identifying the Bott generator b as the class [ e πiθ ] ∈ K ( C ( R )) the Bottperiodicity map β provides the following relationship between idempotents of a C ∗ -algebra A and invertibles of the suspension β : K ( A ) −→ K ( S A ) β [ p ] = [ bp + (1 − p )] (4.2.3)Combining this with the Baum-Douglas geometric description of K-homology we obtainthe following result concerning the propagation properties of local loops which is essentiallythe same as [44, Lemma 3.4], and we refer the reader to the proof given in that paper. Lemma 4.9. If f ∈ S B ( f M ) Γ is a local invertible then for any ε > there exists anidempotent p ∈ B ( f M ) Γ such that prop ( p ) ≤ ε and f ( θ ) is homotopic to the element ψ ( θ ) = e πiθ p + (1 − p ) through a piecewise smooth family of invertible elements. We now go through the process of assigning to any class [ u ] ∈ K ( C ∗ L, ( f M , S ) Γ ) aspecial representative which will enable the calculations later on in this section. Makinguse of the results proved in [44, Proposition 3.5], there exist a piecewise smooth pathof invertible elements h ( t ) ∈ B ( f M ) Γ connecting u (1) and e πi E (1)+12 where the operator E : [1 , ∞ ) −→ D ∗ ( f M ) Γ has uniformly bounded operator norm and satisfieslim t −→∞ prop ( E ( t )) = 0 E ′ ( t ) ∈ B ( f M ) Γ E ( t ) − ∈ B ( f M ) Γ E ∗ ( t ) = E ( t )where there exists a twisted Dirac operator e D over a spin c manifold, along with smoothnormalizing function χ : R −→ [ − , 1] such that E ( t ) = χ ( e D/t ). We can thus define the regularized representative of u to be w ( t ) = u ( t ) 0 ≤ t ≤ h ( t ) 1 ≤ t ≤ e πi E ( t − t ≥ v is another such representativethen there exist a family of piecewise smooth maps { E s } s ∈ [0 , belonging to D ∗ L, ( f M ) Γ and having the same properties as E . In particular, the propagation of E s ( t ) goes to zerouniformly in s as t −→ ∞ , and ∂∂t E s ( t ) ∈ B ( f M ) Γ E s ( t ) − ∈ B ( f M ) Γ E s ( t ) ∗ = E s ( t )Furthermore there exists piecewise smooth family of invertibles { v s } s ∈ [0 , belonging to( B L, ( f M ) Γ ) + and which satisfy(i) v ( t ) = w ( t ) for t ∈ [0 , ∞ ), and v ( t ) = v ( t ) for all t / ∈ (1 , v s ( t ) = exp(2 πi E s ( t − ) for all t ≥ v v − : [1 , −→ ( B ( f M ) Γ ) + is a local loop of invertible elements.34HEAGAN A. K. A JOHN Definition 4.10. Let a i = P g i ∈ Γ c g i · g i be an element of the group algebra C Γ, and ω i belong to the algebra R of smooth operators on a closed oriented Riemannian manifold.Denoting W i = a i ⊗ ω i , the action of ϕ γ on C Γ can be extended to C Γ ⊗ R by ϕ γ ( W ˆ ⊗ W ˆ ⊗ · · · ˆ ⊗ W n ) = tr ( ω ω · · · ω n ) · ϕ γ ( a , a , . . . , a n ) Definition 4.11. Given [ ρ ( e D, e g )] ∈ K ( B L, ( f M ) Γ ) with w being its regularized represen-tative, associated to each delocalized cyclic cocyle [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) the determinantmap τ ϕ γ is defined by τ ϕ γ ( ρ ( e D, e g )) := 1 πi Z ∞ ϕ γ ( e ch ( w ( t ) , ˙ w ( t )) dt e ch ( w, ˙ w ) = ( − m ( m − m X j =1 (cid:16) ( w − ˆ ⊗ w ) ˆ ⊗ j ˆ ⊗ ( w − ˙ w ) ˆ ⊗ ( w − ˆ ⊗ w ) ˆ ⊗ ( m − j ) (cid:17) We remark that the definition of e ch is directly modeled upon the secondary odd Cherncharacter pairing invertibles of GL( N, C ) and traces (see, for example, [25, Section 1.2]).Moreover, by the property of cyclic cocyles this expression can be simplified so that ourcoefficients exactly resemble that of the delocalized higher eta invariant. By the action ofthe cyclic operator t we obtain from applying the n = 2( m − j ) + 1 fold composition t n foreach j , that the integrand can be written as( − m ( m − πi m X j =1 ( − m (2 m − j +1) ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) from which it follows that there is the simplified expression for the determinant map τ ϕ γ ( ρ ( e D, e g )) := ( − m m ! πi Z ∞ ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) dt (4.2.5)It is not at all obvious why the above pairing is well defined, the resolving of this doubtoccupying the remainder of this section. Theorem 4.12. Let Γ = π ( M ) and ϕ γ ∈ ( C n ( C Γ , cl( γ )) , b ) be a delocalized cyclic cocyleof polynomial growth, then ϕ γ extends continuously on the algebra ( B ( f M ) Γ ) ˆ ⊗ n +1 π .Proof. Again using the explicit representation for delocalized cyclic cocyles, we show that ϕ α,γ extends to a continuous multi-linear map on ( B ( f M ) Γ ) ˆ ⊗ n +1 π . By fixing a basis, let B k ∈ B ( f M ) Γ be represented by the matrix ( β kij ) i,j ∈ N with β kij ∈ C ∗ r (Γ). We wish to proveconvergence of ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) = ϕ α,γ X g ∈ Γ B ( g ) · g ˆ ⊗ · · · ˆ ⊗ X g n ∈ Γ B n ( g n ) · g n = X g ∈ Γ X g ∈ Γ · · · X g n ∈ Γ tr ( B ( g ) B ( g ) · · · B n ( g n )) · ϕ α,γ ( g , g , · · · , g n )= X g g ··· g n ∈ cl( γ ) tr ( C ( g , . . . , g n )) · α ( h, hg , . . . , hg g · · · g n − ) (4.2.6)35IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSStraight forward matrix multiplication gives the product C = ( c ij ) i,j ∈ N as having entries c ij ( g , . . . , g n ) = X k n − ∈ N · · · X k ∈ N X k ∈ N (cid:16) β ik ( g ) β k k ( g ) · · · β nk n − j ( g n ) (cid:17) For ease of notation, we shorten the argument of α by writing α ( g ); in addition we suppressthe argument of the functions c ij . Taking the desired trace in the above expression (4.2.6),and using the fact that α is of polynomial growth, we obtain the inequality | ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) | ≤ X j ∈ N X g g ··· g n ∈ cl( γ ) | c jj || α ( g ) |≤ X j ∈ N X g g ··· g n ∈ cl( γ ) R α (1 + || g || ) k (1 + || g || ) k · · · (1 + || g n || ) k | c jj | (4.2.7)for some positive constant R α . Next, considering the following inequality for | c jj | X j ∈ N | c jj | ≤ X k ,...,k n − ,j ∈ N (cid:12)(cid:12)(cid:12)(cid:16) β jk ( g ) · · · β nk n − j ( g n ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ n Y i =0 X k i − ,k i ∈ N | β ik i − k i ( g i ) | / where j = k − = k n . Substituting the above final product into the second line of (4.2.7), | ϕ α,γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) | is bounded above by X g g ··· g n ∈ cl( γ ) R α n Y i =0 (1 + || g i || ) k n Y i =0 X k i − ,k i ∈ N | β ik i − k i ( g i ) | / = R α n Y i =0 X g g ··· g n ∈ cl( γ ) X k i − ,k i ∈ N (1 + || g i || ) k | β ik i − k i ( g i ) | / ≤ R α n Y i =0 X g i ∈ Γ X k i − ,k i ∈ N (1 + || g i || ) k | β ik i − k i ( g i ) | / : g g · · · g n ∈ cl( γ )In particular, the proof of [13, Lemma 6.4] shows that each of the double sums in the finalexpression are in fact bounded by the norm || B i || B ,k hence finite for all k , and thus so isany finite product of them. Since C Γ ⊗ R is a smooth dense sub-algebra of B ( f M ) Γ and ϕ α,γ has been proven to be continuous on B ( f M ) Γ , it suffices to prove that for operators W , . . . , W n ∈ C Γ ⊗ R sgn( σ ) ϕ α,γ ( W ˆ ⊗ W ˆ ⊗ · · · ˆ ⊗ W n ) = ϕ α,γ ( W σ (0) ˆ ⊗ W σ (1) ˆ ⊗ · · · ˆ ⊗ W σ ( n ) ) (4.2.8)whenever σ ∈ S n +1 is a cyclic shift. Write W i = a i ⊗ ω i , then using the fact that the traceis invariant under cyclic shifts and that ϕ α,γ is a cyclic cocyle on C Γ ϕ α,γ ( W σ (0) ˆ ⊗ W σ (1) ˆ ⊗ · · · ˆ ⊗ W σ ( n ) ) = trace ( ω σ (0) ω σ (1) · · · ω σ ( n ) ) · ϕ α,γ ( a σ (0) , a σ (1) , . . . , a σ ( n ) )= trace ( ω ω · · · ω n ) · sgn( σ ) ϕ α,γ ( a , a , . . . , a n )= sgn( σ ) ϕ α,γ ( a ⊗ ω , a ⊗ ω , . . . , a n ⊗ ω n ) = sgn( σ ) ϕ α,γ ( W ˆ ⊗ W ˆ ⊗ · · · ˆ ⊗ W n )36HEAGAN A. K. A JOHN Proposition 4.13. Let w be a regularized representative of some class [ u ] ∈ K ( B L, ( f M ) Γ ) .For all t ≥ there exists a finite propagation operator ̟ ∈ B ( f M ) Γ such that for every k > and any ε > || w ( t ) − ̟ ( t ) || B ,k < C k /t prop ( ̟ ( t )) < ε whenever t ≥ max { e , r } , where C k and r are positive constants .Proof. For real valued x , finite r > z = πi ( x + 1) ∈ B r (0) define the boundedholomorphic function f ( z ) = e πi x +12 . Consider the Taylor series expansion of f ( z ) centeredat the origin, and for each m ∈ N define P m ( z ) = m X k =0 (cid:0) πi x +12 (cid:1) k k ! R m ( z ) = ∞ X k = m +1 (cid:0) πi x +12 (cid:1) k k ! A m ( t ) = P m ( G ( t )) = P m (cid:18) πi E ( t − 1) + 12 (cid:19) (4.2.9)We know that E ( t ) has compact real spectrum which is symmetric around λ = 0; inparticular the spectral radius rad( σ ( E ( t ))) is bounded above by || E ( t ) || op . It is clear thatthe same holds true for G ( t ), so choose r > sup t ≥ {|| G ( t ) || op } so that σ ( G ( t )) ⊂ B r (0),then f ∈ H ∞ ( B r (0)) and since B ( f M ) Γ is closed under holomorphic functional calculus,the usual remainder bound | R m ( z ) | ≤ c | z | m +1 ( m + 1)! if | f ( m +1) ( z ) | ≤ c, ∀ z ∈ B r (0)has a functional equivalent with respect to every || · || B ,k . In particular for every k > C k,t > || w ( t ) − A m ( t ) || B ,k = || R m ( G ( t )) || B ,k ≤ C k,t || R m || ∞ = C k,t sup z ∈ B r (0) | R m ( z ) |≤ C k,t · c sup z ∈ B r (0) | z | m +1 ( m + 1)! ≤ C k,t · c r m +1 ( m + 1)! = C k,t r m +1 ( m + 1)!Note that we can take c = 1 since | f ( m +1) ( z ) | = | e z | = | e i ( πx + π ) | = 1. Suppose that m ≥ max { e , r } , then by applying Stirling’s approximation C k,t r m +1 ( m + 1)! ≤ C k,t r m +1 √ π ( m + 1) m +3 / e − ( m +1) < C k,t m Since the exponential is an entire function its power series converges uniformly on thecompact set B r (0), and thus we can choose a finite C k ≥ sup t ∈ [2 , ∞ ) { C k,t } . Finally, for all t ≥ ̟ ( t ) according to ̟ ( t ) = ∞ X m =2 [ m,m +1) ( t ) · A m ( t ) (4.2.10)37IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSFrom the definition of E ( t ) the propagation tends to 0 as t −→ ∞ ; hence for all ε > N ε ∈ N such that prop ( E ( t )) < ε/N ε whenever t ≥ N ε . Recall that if S and T are bounded operators on some module H X then prop ( ST ) ≤ prop ( S ) + prop ( T ) prop ( S + T ) ≤ max { prop ( S ) , prop ( T ) } thus by the definition of ̟ ( t ), if t ∈ [ m, m + 1] for m ≤ N ε ≤ t prop( ̟ ( t + 1)) = prop( A m ( t + 1)) = prop m X k =0 (cid:16) πi E ( t )+12 (cid:17) k k ! ≤ prop (cid:16) πi E ( t )+12 (cid:17) m m ! = prop (cid:18) E ( t ) + 12 (cid:19) m ≤ m · prop (cid:18) E ( t ) + 12 (cid:19) ≤ m · prop( E ( t )) < mεN ε ≤ ε Corollary 4.14. Let w be a regularized representative of some class [ u ] ∈ K ( B L, ( f M ) Γ ) .For all t ≥ there exists a finite propagation operator ̟ ∈ B ( f M ) Γ such that for every k > and any ε > || w − ( t ) − ̟ − ( t ) || B ,k < C k /t prop ( ̟ − ( t )) < ε whenever t ≥ max { e , r } , where C k and r are positive constants .Proof. Take f ( z ) = e − πi x +12 and apply the same argument as above.For each member of the family of invertibles { v s } s ∈ [0 , the above also leads to finitepropagation operators ̟ s ( t ) and ̟ − s ( t ) having analogous properties. The following tech-nical result will be of similar importance in establishing well-definedness of the determinantmap. Remark 4.15. For ϕ γ ∈ C n (( C Γ , cl( γ )) , b ) and B , . . . , B n ∈ B ( f M , S ) Γ – or equivalentlyfor A , . . . , A n ∈ A ( f M , S ) Γ – there exists ε f M which depends only on M such that ϕ γ ( B ˆ ⊗ B ˆ ⊗ · · · ˆ ⊗ B n ) = 0 whenever prop ( B i ) < ε f M for each ≤ i ≤ n . Theorem 4.16. Let [ u ] ∈ K ( B L, ( f M ) Γ ) and w be a regularized representative of u , thenthe determinant map τ ϕ converges absolutely for any ϕ γ ∈ ( C m ( C Γ , cl( γ )) , b ) of polynomialgrowth.Proof. Using the simplified expression of Equation (4.2.5) we can write τ ϕ ( u ) as( − m m ! πi Z ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) dt + ( − m m ! πi Z ∞ ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) dt || B || B ,k and the results ofTheorem 4.12, being bounded above by2 R α m ! πi sup t ∈ [0 , || w ( t ) || B ,k || w − ( t ) || B ,k || w − ( t ) ˙ w ( t ) || B ,k < ∞ By Proposition 4.13 and its corollary there exist operators ̟ ( t ) and ̟ − ( t ) belonging to( B ( f M ) Γ ) + such that for any ε > t large enough such that prop ( ̟ ( t )) , prop ( ̟ − ( t )) < ε By basic distribution of tensors over addition and mutlilinearity of cyclic cocyles we obtainthe following expansion– ignoring the constant for the integral over t ∈ [2 , ∞ ) Z ∞ ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) − ̟ − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) dt + Z ∞ ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t ) − ̟ ( t )) ˆ ⊗ m (cid:17) dt + Z ∞ ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( ̟ − ( t ) ˆ ⊗ ̟ ( t )) ˆ ⊗ m (cid:17) dt (4.2.11)Since t ≥ w − ( t ) ˙ w ( t ) = πiE ′ ( t − t −→ ∞ . By Remark 4.15 it followsthat the integrand ϕ γ (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( ̟ − ( t ) ˆ ⊗ ̟ ( t )) ˆ ⊗ m (cid:17) = 0 once the propagation of allthese operators is less than some ε f M , which occurs for large enough t . Hence there existssome t ε such that the last integral of (4.2.11) is bounded above by Z t ε πi (cid:12)(cid:12)(cid:12) ϕ γ (cid:16) E ′ ( t − 1) ˆ ⊗ ( ̟ − ( t ) ˆ ⊗ ̟ ( t )) ˆ ⊗ m (cid:17)(cid:12)(cid:12)(cid:12) dt which is finite from the bounds of Theorem 4.12. Similarly, by Proposition 4.13 there existssome constant r > t ≥ r || ̟ ( t ) − w ( t ) || B ,k and || ̟ ( t ) − − w − ( t ) || B ,k < C k t The norm boundedness of all B ∈ ( B ( f M ) Γ ) + and the results of Theorem 4.12 provideexistence of M k , N k > r − R α N k + R α Z ∞ r || w − ( t ) ˙ w ( t ) || B ,k || w ( t ) || m B ,k || ̟ ( t ) − w ( t ) || m B ,k dt ≤ (6 r − R α N k + R α Z ∞ r M k || ̟ ( t ) − w ( t ) || m B ,k dt The finiteness of the integral follows directly from Z ∞ r M k || ̟ ( t ) − w ( t ) || B ,k dt < M k C k Z ∞ r t m dt An exact replica of this argument applied to w − ( t ) − ̟ − ( t ) finishes our proof.39IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSThe following three results show that τ ϕ ([ u ]) is independent of the choice of regularizedrepresentatives, with Theorem 4.19 providing the proof that the replacement of u by w through the path of local loops behaves as intended. Lemma 4.17. Let [ u ] ∈ K ( B ∗ L, ( f M ) Γ ) with v and w both being regularized representatives,and { v s } s ∈ [0 , the associated family of piecewise smooth invertibles. For any delocalizedcyclic cocyle ϕ γ ∈ ( C m ( C Γ , cl( γ )) , b ) ∂∂s ϕ γ (cid:16) ( v − s · ∂ t v s ) ˆ ⊗ ( v − s ˆ ⊗ v s ) ˆ ⊗ m (cid:17) = ∂∂t ϕ γ (cid:16) ( v − s · ∂ s v s ) ˆ ⊗ ( v − s ˆ ⊗ v s ) ˆ ⊗ m (cid:17) Proof. Working in the unitization ( B ∗ ( f M ) Γ ) + and noting that every invertible element in( B ∗ ( f M ) Γ ) + can be viewed as one in ( B ∗ L, ( f M ) Γ ), we wish to show vanishing of ∂∂s ϕ γ (cid:16) ( v − s ∂ t v s ) ˆ ⊗ ( v − s ˆ ⊗ v s ) ˆ ⊗ m (cid:17) − ∂∂t ϕ γ (cid:16) ( v − s ∂ s v s ) ˆ ⊗ ( v − s ˆ ⊗ v s ) ˆ ⊗ m (cid:17) (4.2.12)By definition every cyclic cocyle ϕ γ belongs to the kernel of the boundary map b ; in addition ϕ γ ( A ˆ ⊗ · · · ˆ ⊗ A n ) vanishes if A i = for any A i ∈ A + .0 = m X k =0 bϕ γ ) (cid:16) ( v − s · ∂ s v s ) ˆ ⊗ ( v − s ˆ ⊗ v s ) ˆ ⊗ k ˆ ⊗ ( v − s · ∂ t v s ) ˆ ⊗ ( v − s ˆ ⊗ v s ) ˆ ⊗ ( m − k ) (cid:17) Through a tedious but straightforward computation we obtain that the above sum givesprecisely the expression for (4.2.12) which thus proves the result. Corollary 4.18. Let [ u ] ∈ K ( B ∗ L, ( f M ) Γ ) with v and w both being regularized represen-tatives and { v s } s ∈ [0 , the associated family of piecewise smooth invertibles, then τ ϕ ( v ) = τ ϕ ( w ) for any polynomial growth ϕ γ ∈ ( C m ( C Γ , cl( γ )) , b ) .Proof. Taking a double integral R T R ds dt over the derivative equality in (4.2.12) gives Z T ϕ γ (cid:16) ( v ( t ) − ∂ t v ( t )) ˆ ⊗ ( v − ( t ) ˆ ⊗ v ( t )) ˆ ⊗ m (cid:17) dt − Z T ϕ γ (cid:16) ( v ( t ) − ∂ t v ( t )) ˆ ⊗ ( v − ( t ) ˆ ⊗ v ( t )) ˆ ⊗ m (cid:17) dt = Z ϕ γ (cid:16) ( v s ( t ) − ∂ s v s ( t )) ˆ ⊗ ( v − s ( t ) ˆ ⊗ v s ( t )) ˆ ⊗ m (cid:17)(cid:12)(cid:12)(cid:12) T ds Now ϕ γ (cid:16) ( v s (0) − ∂ s v s (0)) ˆ ⊗ ( v − s (0) ˆ ⊗ v s (0)) ˆ ⊗ m (cid:17) = 0 since v s (0) ≡ v − s (0) ≡ for all s ∈ [0 , v ( t ) = w ( t ) and letting T −→ ∞ we obtain τ ϕ ( v ) − τ ϕ ( w ) = ( − m m ! πi Z lim T −→∞ ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( v − s ( T ) ˆ ⊗ v s ( T )) ˆ ⊗ m (cid:17) ds It thus only remains to prove that for all s , uniformly with respect to the norm || · || B ,k lim T −→∞ ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( v − s ( T ) ˆ ⊗ v s ( T )) ˆ ⊗ m (cid:17) = 0 (4.2.13)40HEAGAN A. K. A JOHNBy basic distribution of tensors over addition and mutlilinearity of cyclic cocyles the abovedecomposes as lim T −→∞ ϕ γ ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( v − s ( T ) − ̟ − ( T ) ˆ ⊗ v s ( T )) ˆ ⊗ m (cid:17) + lim T −→∞ ϕ γ ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( ̟ − ( T ) ˆ ⊗ v s ( T ) − ̟ ( T )) ˆ ⊗ m (cid:17) + lim T −→∞ ϕ γ ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( ̟ − ( T ) ˆ ⊗ ̟ ( T )) ˆ ⊗ m (cid:17) By Proposition 4.13 and its corollary there exist operators ̟ ( t ) , ̟ − ( t ) ∈ ( B ( f M ) Γ ) + suchthat for any ε > t large enough such that the following hold. || ̟ ( t ) − v s ( t ) || B ,k , || ̟ ( t ) − − v − s ( t ) || B ,k < C k t , prop ( ̟ ( t )) , prop ( ̟ − ( t )) < ε We may assume that T ≥ v − s ( T ) · ∂ s v s ( T ) = πi∂ s E s ( T ), the propagation of which tends to 0 as T −→ ∞ . Thusfor large enough T the propagation of all the operators is less than some ε f M and byRemark 4.15 lim T −→∞ ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( ̟ − ( T ) ˆ ⊗ ̟ ( T )) ˆ ⊗ m (cid:17) = 0By the results of Theorem 4.12 and the norm boundedness of all B ∈ ( B ( f M ) Γ ) + lim T −→∞ (cid:12)(cid:12)(cid:12) ϕ γ (cid:16) ( v − s ( T ) ∂ s v s ( T )) ˆ ⊗ ( v − s ( T ) − ̟ − ( T ) ˆ ⊗ v s ( T )) ˆ ⊗ m (cid:17)(cid:12)(cid:12)(cid:12) ≤ R α lim T −→∞ || v − s ( T ) ∂ s v s ( T ) || B ,k || ̟ ( t ) − − v − s ( T ) || m B ,k || v s ( T ) || m B ,k ≤ R α C mk lim T −→∞ || v − s ( T ) ∂ s v s ( T ) || B ,k || v s ( T ) || m B ,k T m = 0An exact replica of this argument holds for the term involving v − s ( T ) − ̟ − ( T ). Theorem 4.19. Let [ u ] ∈ K ( B L, ( f M ) Γ ) with v and w both being regularized representa-tives, then τ ϕ ( v ) = τ ϕ ( w ) for any polynomial growth ϕ γ ∈ ( C m ( C Γ , cl( γ )) , b ) .Proof. We have proven in the above corollary that τ ϕ ( v ) = τ ϕ ( w ); by construction v ( s ) = v ( s ) for all s ∈ R ≥ \ (1 , v v − : [1 , −→ ( B ( f M ) Γ ) + being a local loop of invertibleelements. Thus there exists a local invertible f : S −→ ( B ( f M ) Γ ) + – which by Lemma 4.9is homotopic to e πiθ P ( t ) + ( − P ( t )) for some idempotent P ( t )– such that v ( s ) and v ( s )differ by f ( θ ) as elements in K ( B ( f M ) Γ ). τ ϕ ( v ) = τ ϕ ( v ) + τ ϕ ( v − v ) = τ ϕ ( w ) + τ ϕ ( v v − ) = τ ϕ ( w ) + τ ϕ ( f L )= τ ϕ ( w ) + ( − m m ! πi Z ∞ Z ϕ γ (cid:16) ( f − ( θ ) ˙ f ( θ )) ˆ ⊗ ( f − ( θ ) ˆ ⊗ f ( θ )) ˆ ⊗ m (cid:17) dθ dt f ( θ ) = 2 πie πiθ P ( t ) refers to the derivative with respect to θ ; it is easy to verify that f − ( θ ) = ( e − πiθ P ( t ) + ( − P ( t ))). It follows that the above integral is equal to Z ∞ Z ϕ γ (cid:16) πiP ( t ) ˆ ⊗ (( e − πiθ − P ( t ) + ˆ ⊗ ( e πiθ − P ( t ) + ) ˆ ⊗ m (cid:17) dθ Using multilinearity of cyclic cocyles and the fact that ϕ γ vanishes on the unit the integrandsimplifies to 2 πi ( e − πiθ − m ( e πiθ − m ϕ γ (cid:16) P ( t ) ˆ ⊗ m +1 (cid:17) Vanishing of the double integral now follows from Remark 4.15 and the fact that for all ε > t ε large enough such that prop ( P ( t ε )) ≤ ε . Theorem 4.20. The determinant map pairing the higher rho invariant is independent ofthe choice of delocalized cyclic cocycle representative. Explicitly, if [ ϕ γ ] = [ φ γ ] ∈ HC m ( C Γ , cl( γ )) ,then τ ϕ γ ( ρ ( e D, e g )) = τ φ γ ( ρ ( e D, e g )) Proof. By the previous results we are able to fix a regularized representative w , and byhypothesis, ϕ γ and φ γ are cohomologous via a coboundary bϕ ∈ BC m ( C Γ , cl( γ )). Weobtain an identical transgression formula as was calculated in Theorem 4.6, (4.1.12) m ( bϕ ) (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) = ddt ϕ (( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m ) (4.2.14)Using the simplified definition of the determinant map, provided by (4.2.5) mτ bϕ ( ρ ( e D, e g )) := ( − m m ! πi Z ∞ m ( bϕ γ ) (cid:16) ( w − ( t ) ˙ w ( t )) ˆ ⊗ ( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m (cid:17) dt (4.2.15)Thus by the transgression formula τ bϕ ( ρ ( e D, e g )) is equal (up to a constant) tolim t −→∞ ϕ (( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m ) − lim t −→ ϕ (( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m ) (4.2.16)Now lim t −→ ϕ (( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m ) = 0 since by construction w (0) = and ϕ vanishes onthe unit . Moreover, for all t ≥ χ can be chosen suchthat w ( t ) = exp (cid:18) πi E ( t ) + 12 (cid:19) = exp πi χ ( e D/t ) + 12 ! Denote ψ ( x ) = exp (cid:16) πi χ ( x )+12 (cid:17) − e i ( πχ ( x )+ π ) − K -theory represen-tative is independent of the class of smooth normalizing function, without loss of generalityassume that lim x −→±∞ χ ( x ) converges to ± ψ is a Schwartz function and thus by Proposition 4.4lim t −→∞ ϕ (( w − ( t ) ˆ ⊗ w ( t )) ˆ ⊗ m ) = 0 Proposition 4.21. Let S ∗ γ : HC m ( C Γ , cl( γ )) −→ HC m +2 ( C Γ , cl( γ )) be the delocal-ized Connes periodicity operator, then τ [ S γ ϕ γ ] ( ρ ( e D, e g )) = τ [ ϕ γ ] ( ρ ( e D, e g )) for every [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) .Proof. The proof exactly mirrors that of Proposition 4.7.42HEAGAN A. K. A JOHN § The following discussion and the proof of Proposition 4.22 closely align with the proofof Theorem 4.3 in [44]. We first recall the construction at the beginning of the previoussection of a representative of ρ ( e D, e g ) using the path of invertibles S = { U ( t ) = exp(2 πiH ( t )) | t ∈ [0 , ∞ ) } This construction can be altered by using the following smooth normalizing function ψ ,where F t is as defined in the construction of Lott’s higher eta invariant. ψ ( t − x ) = F /t ( x ) = 1 √ π Z x/t −∞ e − s ds t > e D implies that ψ ( t − D ) converges in operator normto ( + e D | e D | − ) as t → 0. Since ψ is a smooth normalizing function the operator ψ ( e D ) − is locally compact, hence e πiF t ( e D ) ≡ modulo locally compact operators. Moreover, ψ canbe approximated by smooth normalizing functions with compactly supported distributionalFourier transforms, hence the inverse Fourier transform relation ψ ( e D ) = Z ∞−∞ ˆ ψ ( s ) e is e D ds (4.3.2)is well defined, and by finite propagation property of the wave operator e is e D it follows thatthe path U ∈ ( C ∗ L, ( f M , S ) Γ ) + defined by U ( t ) = U t ( e D ) = e πiψ ( e D/t ) ; t ∈ (0 , ∞ ) U ≡ (4.3.3)can be uniformly approximated by paths of invertible elements with finite propagation. Intotality, we have that U is an invertible element of ( C ∗ L, ( f M , S ) Γ ) + and gives rise to a classin K ( C ∗ L, ( f M , S ) Γ ). Proposition 4.22. The element U is invertible in ( A L, ( f M , S ) Γ ) + .Proof. Firstly, we note that by the proof of Theorem 4.20 the function U t ( x ) − U t ( e D ) − belongs to A ( f M , S ) Γ ) for all t ∈ [0 , ∞ ).Being of the form e f ( e D ) this further proves that U t ( e D ) ∈ ( A ( f M , S ) Γ ) + is invertible for all t . By definition of the localization algebra, to prove that U is an invertible element of( A L, ( f M , S ) Γ ) + it suffices to show that U − is a piecewise smooth function on the half-line. By the description of F /t this is immediate for all t ∈ (0 , ∞ ), hence we only need toshow smoothness at t = 0 with respect to the Frechet topology generated by the seminorms || · || A ,k . Denoting by ˙ ψ ( s ) the derivative with respect to s , for t ∈ (0 , ∞ ) || U t ( e D ) − || A ,k = exp (cid:18)Z t πi ˙ ψ ( e D/s ) ds (cid:19) − = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 n ! (cid:18)Z t πi ˙ ψ ( e D/s ) ds (cid:19) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ,k ≤ ∞ X n =1 (2 π ) n n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)Z t ˙ ψ ( e D/s ) ds (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n A ,k ≤ ∞ X n =1 (2 π ) n n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) √ π − s e De − e D /s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n A ,k Following the arguments on [44, Page 21] for 1 /s ∈ [ n, n + 1) there exists n large enoughsuch that for some constants C , C > || C e − e D /s || A ,k ≤ || e − n e D || A ,k ≤ e − nr / ≤ C e − r / s Thus for s > C we obtain– taking t > t −→ || U t ( e D ) − || A ,k ≤ lim t −→ ∞ X n =1 (2 π ) n n ! (cid:18) − t √ π C C e − r / t − lim s −→ − s √ π C C e − r / s (cid:19) n = lim t −→ ∞ X n =1 (2 π ) n n ! (cid:18) − t √ π C C e − r / t (cid:19) n = lim t −→ exp (cid:18) t Ce − r / t (cid:19) − U ( e D ) = it follows that U − is continuous with respect to the family of seminorms;the same holds true for all orders of its derivatives according to the expansion ddt ( U t ( x ) − 1) = X m +2 m + ··· + km k = k C m ,...,m k k Y l =1 C m l k (cid:18) d l dt l ψ ( x/t ) (cid:19) m l ( U t ( x ) − ≤ m l ≤ k . Thus U − is smooth with respect to || · || A ,k which proves that U ∈ ( A L, ( f M , S ) Γ ) + as desired. Corollary 4.23. Let ϕ γ ∈ ( C m C Γ , cl( γ ) , b ) be of polynomial growth, with e D being invert-ible, then the integral Z ∞ ϕ γ (cid:16) ˙ U t ( e D ) U − t ( e D ) ˆ ⊗ ( U t ( e D ) ˆ ⊗ U − t ( e D )) ˆ ⊗ m (cid:17) dt converges absolutely.Proof. This is a direct consequence of Lemma 4.5.The construction of a regularized representative of ρ ( e D, e g ) involves the choice of somesmooth normalizing function χ with compactly supported distributional Fourier transformˆ χ , such that E ( t ) = ( χ ( e D/t )+ ) / E has the properties outlined on page 34. Adaptingthe argument preceding [8, Proposition 6.11] we can thus construct a path ww ( t ) = U ( t ) 0 ≤ t ≤ e πi ((2 − t ) ψ ( e D )+( t − E (1)) ≤ t ≤ e πiE ( t − t ≥ U is itsown regularized representative and the equality of [ U ] and [ w ] as K-theory classes in44HEAGAN A. K. A JOHN K ( C ∗ L, ( f M , S ) Γ ) follows from their being homotopic in ( B L, ( f M , S ) Γ ) + . Explicitly, wehave the homotopy induced by the family of invertibles h s : s ∈ [0 , 1] defined by h s ( t ) = U ( t ) 0 ≤ t ≤ e πi ((2 − t ) ψ ( e D )+( t − sE (1)+(1 − s ) ψ ( e D ))) ≤ t ≤ se πi ( sE ( t − − s ) ψ ( e Dt − )) t ≥ s (4.3.5)Thus by Corollary 4.23 and the proofs of Lemma 4.17 and Corollary 4.18 we obtain that τ [ ϕ γ ] ( ρ ( e D, e g )) = τ [ ϕ γ ] ( w ) = τ ϕ γ ( U )= ( − m m ! πi Z ∞ ϕ γ (cid:16) U t ( e D ) − ˙ U t ( e D ) ˆ ⊗ ( U − t ( e D ) ˆ ⊗ U t ( e D )) ˆ ⊗ m (cid:17) dt = ( − m m ! πi Z ∞ ϕ γ (cid:16) ˙¯ u t ( e D )¯ u − t ( e D ) ˆ ⊗ (¯ u t ( e D ) ˆ ⊗ ¯ u − t ( e D )) ˆ ⊗ m (cid:17) dt = ( − m η [ ϕ γ ] ( e D )where ¯ u t = U /t and we have used the substitution u t ←→ u − t . § Consider smooth vector bundles V and V over a compact smooth manifold M – withoutboundary– and an elliptic differential operator D : V −→ V which acts on the smoothsections of these vector bundles. Since every such D has a pseudo inverse it is a Fredholmoperator, with analytical index defined byind( D ) = dim ker( D ) − dim ker( D ∗ ) (4.4.1)Let us also recall the topological index of D with respect to a non-commutative K -theoreticframework Z M ch ( D ) Td ( T ∗ M ⊗ C ) (4.4.2)where Td ( T ∗ M ⊗ C ) is the Todd class of the complexified tangent bundle of M , and ch ( D )is the pullback of the Chern character on a particular K -theory class associated to D . Theoriginal Atiyah-Singer index theorem [3] was proven through cohomological means, andasserts that the topological index of D is equal to its analytical index; a K -theoretic proof[4, 5] was later provided and shown to be equivalent to the one employing cohomology. TheAtiyah-Patodi-Singer index theorem [1, 2] generalizes this statement to include manifoldswith boundary under satisfaction of certain global boundary conditions. By consideringInd G ( D ) rather than ind( D ), this further admits a kind of (delocalized) higher analogue,in our case modeled on that of Lott [28] (see the relationship between equations (1) and(66)).Prior to stating and proving the delocalized version of a higher Atiyah-Patodi-Singerindex theorem we will first exhibit a necessary relationship between the determinant map τ ϕ γ of the previous section and the Chern-Connes character map. In the remainder of thissection we will work within the restriction of even dimensional cyclic cocyles and underthe condition of a compact spin manifold M having fundamental group Γ of polynomial45IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPSgrowth. In particular, given a delocalized cyclic cocyle ϕ γ ∈ ( C m C Γ , cl( γ ) , b ) we will definethe ϕ γ -component of the Chern-Connes character of an idempotent p ∈ B ( f M ) Γ accordingto that of [27, Chapter 8] ch ϕ γ ( p ) := ( − m (2 m )! m ! ϕ γ (cid:16) p ˆ ⊗ m +1 (cid:17) (4.4.3)Firstly let us prove the usual well-definedness properties. Proposition 4.24. Let [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) , then the [ ϕ γ ] -component of the Chern-Connes character ch [ ϕ γ ] : K ( B ( f M ) Γ ) −→ C is well defined, particularly being independent of the choices of cocycle representative and K -theory class representative.Proof. Since ϕ γ can be chosen to be of polynomial growth and p ∈ ( B ( f M ) Γ ) + thenTheorem 4.12 asserts that the formula for the ϕ γ -component of the Chern-Connes char-acter makes sense. Suppose that ϕ γ and φ γ belong to the same cohomology class in HC m ( C Γ , cl( γ )); by hypothesis, ϕ γ and φ γ are cohomologous via a coboundary bϕ ∈ BC m ( C Γ , cl( γ )). Independence with respect to cyclic cocyle representatives thus followsfrom showing that ch bϕ ( p ) = 0 for any idempotent p . A direct computation gives( bϕ ) (cid:16) p ˆ ⊗ m +1 (cid:17) = ϕ (cid:16) p ˆ ⊗ m (cid:17) = 0 (4.4.4)since by the definition of the cyclic operator t ϕ (cid:16) p ˆ ⊗ m (cid:17) = ( − m − ϕ (cid:16) p ˆ ⊗ m (cid:17) . Let usnow turn our attention to proving that if p , p ∈ ( B ( f M ) Γ ) + belong to the same classin K ( B ( f M ) Γ ) then ch [ ϕ γ ] ( p ) = ch [ ϕ γ ] ( p ) By hypothesis there exist a piecewise smoothfamily of idempotents p t : t ∈ (0 , 1) connecting p and p , which allows for the usual trickof taking the derivative. Using the fact that ϕ γ belongs to the kernel of the boundary map b a direct calculation gives ddt ϕ γ (cid:16) p ˆ ⊗ m +1 t (cid:17) = (2 m + 1) ϕ γ (cid:16) ˙ p t ˆ ⊗ p ˆ ⊗ mt (cid:17) = ( bϕ γ ) (cid:16) ( ˙ p t p t − p t ˙ p t ) ˆ ⊗ p ˆ ⊗ m +1 t (cid:17) = 0The desired result now follows immediately from integration0 = Z ( − m (2 m )! m ! ddt ϕ γ (cid:16) p ˆ ⊗ m +1 t (cid:17) dt = ch [ ϕ γ ] ( p ) − ch [ ϕ γ ] ( p ) Proposition 4.25. Let S ∗ γ : HC m ( C Γ , cl( γ )) −→ HC m +2 ( C Γ , cl( γ )) be the delocalizedConnes periodicity operator, then ch [ ϕ γ ] = ch [ S γ ϕ γ ] for every [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) .Proof. Recalling the definition of the map β as given in (3.1.3) of Section 3.1 it is straight-forward to compute the action of βb and bβ as refers to the Chern-Connes character.( β ◦ bϕ γ ) (cid:16) p ˆ ⊗ m +3 (cid:17) = 0 and ( b ◦ βϕ γ ) (cid:16) p ˆ ⊗ m +3 (cid:17) = − ( m + 1) ϕ γ (cid:16) p ˆ ⊗ m +1 (cid:17) Using the relation S γ = m +1)(2 m +2) ( βb + bβ ) as in Definition 3.2 we obtain the desiredresult. 46HEAGAN A. K. A JOHN Lemma 4.26. Let [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) , then the following diagram commutes K ( C L, ( f M ) Γ ) C K ( C ∗ ( f M ) Γ ) C τ [ ϕγ ] ∂ ch [ ϕγ ] × ( − Proof. By Proposition 2.10 we know that the K -theory of C ∗ ( f M ) Γ coincides with thatof B ( f M ) Γ , and likewise with respect to the localization algebras. Thus we can view everyelement of K ( C ∗ ( f M ) Γ ) as a formal difference of two idempotents belonging to ( B ( f M ) Γ ) + .Each idempotent p ∈ B ( f M ) Γ defines an element F ∈ B L ( f M ) Γ F ( t ) = (cid:26) (1 − t ) p t ∈ [0 , t ∈ (1 , ∞ ) (4.4.5)and ∂ [ p ] = [ u ] defines a K -theory class of invertibles in K ( B ∗ L, ( f M ) Γ ), where ∂ : K ( C ∗ ( f M ) Γ ) −→ K ( C L, ( f M ) Γ ) is the K -theoretical connecting map. The proof now follows by mirroringthe calculations in [8, Proposition 7.2]Now we shall set up the necessary preliminaries for a delocalized version of the Atiyah-Patodi-Singer index theorem. To begin with, let W be a compact n -dimensional spin man-ifold with boundary ∂W = M which is closed, and naturally is an n − W is endowed with a Riemannian metric g which has product structurenear M and is of positive scalar curvature metric when restricted to M . Let e D W be theDirac operator lifted to the universal cover f W , e g be the metric lifted to f W , and by ∂ f W = f M denote the lifting of M with respect to the covering map p : f W −→ W . As shown in [42,Section 3] the operator e D W defines a higher index Ind π ( W ) ( e D W ) ∈ K n ( C ∗ ( f W ) π ( W ) ), andas we have already detailed in Section 4.2 in the case of n − e D M defines a higher rho invariant ρ ( e D M , e g ) in K n − ( C ∗ L, ( f M ) π ( W ) ). Recall that everyequivariant coarse map f : X −→ Y induces a homomorphism C ( f ) : C ∗ ( X ) G −→ C ∗ ( Y ) G ,which itself induces a functorial map K ( f ) on the K -theory. Clearly the lifted inclusionmap e ı : f M ֒ → f W is equivariantly coarse and so gives rise to a natural homomorphism K ( e ı ) : K n − ( C ∗ L, ( f M ) π ( W ) ) −→ K n − ( C ∗ L, ( f W ) π ( W ) ) (4.4.6)We will denote the image of ρ ( e D M , e g ) under this map to also be ρ ( e D M , e g ). Theorem 4.27 (Delocalized APS Index Theorem) . Let W be a compact even dimensionalspin manifold with closed boundary ∂W = M , and endowed with a Riemannian metric g which has product structure near M and is of positive scalar curvature metric whenrestricted to M . If π ( W ) is countable discrete, finitely generated, and of polynomial growth ch [ ϕ γ ] (cid:16) Ind π ( W ) ( e D W ) (cid:17) = ( − m +1 η [ ϕ γ ] ( e D M ) for any [ ϕ γ ] ∈ HC m ( C Γ , cl( γ )) 47IGHER INVARIANTS AND POLYNOMIAL GROWTH GROUPS Proof. The proof of Lemma 4.26 did not depend on the dimension or boundary structureof M , the only necessity being that f M admit a proper and co-compact isometric action ofΓ (see Definition 2.6); thus the following diagram also commutes. K ( C L, ( f W ) π ( W ) ) C K ( C ∗ ( f W ) π ( W ) ) C τ [ ϕγ ] ∂ ch [ ϕγ ] × ( − (4.4.7)Moreover since dim( W ) = n is even, by [34, Theorem 1.14] and [42, Theorem A] the imageof the higher index under the connecting map is ∂ (cid:16) Ind π ( W ) ( e D W ) (cid:17) = ρ ( e D M , e g ) ∈ K n − ( C ∗ L, ( f W ) π ( W ) ) ∼ = K ( C ∗ L, ( f W ) π ( W ) ) (4.4.8)Coupling this identity with the main result of Section 4.3 we obtain − ch [ ϕ γ ] (cid:16) Ind π ( W ) ( e D W ) (cid:17) = τ [ ϕ γ ] (cid:16) ∂ (cid:16) Ind π ( W ) ( e D W ) (cid:17)(cid:17) = τ [ ϕ γ ] (cid:16) ρ ( e D M , e g ) (cid:17) = ( − m η [ ϕ γ ] ( e D M ) (4.4.9) References [1] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemanniangeometry. I. Math. Proc. Cambridge Philos. Soc , 77(1):43–69, 1975.[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemanniangeometry. II. Math. Proc. Cambridge Philos. Soc , 78(3):405–432, 1975.[3] M. F. Atiyah and I. M. Singer. 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