Delocalized eta invariants, cyclic cohomology and higher rho invariants
aa r X i v : . [ m a t h . K T ] M a y DELOCALIZED ETA INVARIANTS, CYCLIC COHOMOLOGYAND HIGHER RHO INVARIANTS
XIAOMAN CHEN, JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU
Abstract.
The first main result of this paper is to prove that the convergenceof Lott’s delocalized eta invariant holds for all differential operators with a suf-ficiently large spectral gap at zero. Furthermore, to each delocalized cycliccocycle, we define a higher analogue of Lott’s delocalized eta invariant andprove its convergence when the delocalized cyclic cocycle has at most expo-nential growth. As an application, for each cyclic cocycle of at most exponentialgrowth, we prove a formal higher Atiyah-Patodi-Singer index theorem on man-ifolds with boundary, under the condition that the operator on the boundaryhas a sufficiently large spectral gap at zero.Our second main result is to obtain an explicit formula of the delocalizedConnes-Chern character of all C ∗ -algebraic secondary invariants for word hy-perbolic groups. Equivalently, we give an explicit formula for the pairing be-tween C ∗ -algebraic secondary invariants and delocalized cyclic cocycles of thegroup algebra. When the C ∗ -algebraic secondary invariant is a K -theoretichigher rho invariant of an invertible differential operator, we show this pair-ing is precisely the higher analogue of Lott’s delocalized eta invariant alludedto above. Our work uses Puschnigg’s smooth dense subalgebra for word hy-perbolic groups in an essential way. We emphasize that our construction ofthe delocalized Connes-Chern character is at C ∗ -algebra K -theory level. Thisis of essential importance for applications to geometry and topology. As aconsequence, we compute the paring between delocalized cyclic cocycles and C ∗ -algebraic Atiyah-Patodi-Singer index classes for manifolds with boundary,when the fundamental group of the given manifold is hyperbolic. In particu-lar, this improves the formal delocalized higher Atiyah-Patodi-Singer theoremfrom above and removes the condition that the spectral gap of the operator onthe boundary is sufficiently large. Introduction
Higher index theory is a far-reaching generalization of the classic Fredholmindex theory by taking into consideration of the symmetries of the underlyingspace. Let X be a complete Riemannian manifold of dimension n with a discretegroup G acting on it properly and cocompactly by isometries. Each G -equivariantelliptic differential operator D on X gives rise to a higher index class Ind G ( D )in the K -group K n ( C ∗ r ( G )) of the reduced group C ∗ -algebra C ∗ r ( G ). This higher The first author is partially supported by NSFC 11420101001.The second author is partially supported by NSFC 11420101001.The third author is partially supported by NSF 1500823, NSF 1800737.The fourth author is partially supported by NSF 1700021, NSF 1564398. index is an obstruction to the invertibility of D . The higher index theory playsa fundamental role in the studies of many problems in geometry and topologysuch as the Novikov conjecture, the Baum-Connes conjecture and the Gromov-Lawson-Rosenberg conjecture. Higher index classes are invariant under homotopyand often referred to as primary invariants.When the higher index class of an operator is trivial and given a specific trivial-ization, a secondary index theoretic invariant naturally arises. One such exampleis the associated Dirac operator on the universal covering f M of a closed spinmanifold M equipped with a positive scalar curvature metric. It follows from theLichnerowicz formula that the Dirac operator on f M is invertible. In this case,there is a natural C ∗ -algebraic secondary invariant introduced by Higson and Roein [24, 25, 26, 51], called the higher rho invariant, which lies in K n ( C ∗ L, ( f M ) G ),where G is the fundamental group π ( M ) of M and C ∗ L, ( f M ) G is a certain geo-metric C ∗ -algebra. The precise definition of C ∗ L, ( f M ) G and that of the higher rhoinvariant are given in Section 2. This higher rho invariant is an obstruction tothe inverse of the Dirac operator being local and has important applications togeometry and topology.Parallel to the C ∗ -algebraic approach above, Lott developed a theory of sec-ondary invariants in the framework of noncommutative differential forms [39].Lott’s theory was in turn very much inspired by the work of Bismut and Cheegeron eta forms [7], which naturally arise in the index theory for families of mani-folds with boundary. Despite the fact that Lott’s higher eta invariant is definedby an explicit integral formula of noncommutative differential forms, it is difficultto compute in general. It is only after one pairs Lott’s higher eta invariant withcyclic cocycles of π ( M ) that it becomes more computable and more applicable toproblems in geometry and topology. However, due to certain convergence issues,the question when such a pairing can actually be rigorously defined is often verysubtle. We shall devote the first half of the current paper to these convergenceissues. We show that the pairing of Lott’s higher eta invariant and a (delocal-ized) cyclic cocycle is well-defined, under the condition that the operator on f M has a sufficiently large spectral gap at zero and the cyclic cocycle has at mostexponential growth. In particular, as a special case, if both π ( M ) and the cycliccocycle have sub-exponential growth, then the pairing is always well-defined forall invertible operators on f M .The second goal of our paper is to obtain an explicit formula for the delo-calized Connes-Chern character of all C ∗ -algebraic secondary invariants for hy-perbolic groups. Equivalently, this amounts to computing the pairing between C ∗ -algebraic secondary invariants and delocalized cyclic cocycles of the groupalgebra. In the case where the C ∗ -algebraic secondary invariant is a K -theoretichigher rho invariant, then the pairing is given explicitly in terms of Lott’s higher ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 3 eta invariant , or rather its periodic version . As mentioned above, one of themain technical difficulties is to resolve various convergence issues. In the caseof hyperbolic groups, we overcome these convergence issues with the help ofPuschnigg’s smooth dense subalgebra. As a consequence, we compute the paringbetween delocalized cyclic cocycles and Atiyah-Patodi-Singer higher index classesfor manifolds with boundary in terms of delocalized higher eta invariants, whenthe fundamental group of the given manifold is hyperbolic. The details of theseresults will occupy the second half of the paper.In the following, we shall give a more precise overview of some of the mainresults of this paper. Let us recall the definition of Lott’s delocalized eta invariant,which shall be thought of (at least formally) as a pairing between Lott’s higher etainvariant and traces (i.e. degree zero cyclic cocycles). Suppose h h i is a nontrivialconjugacy class in π ( M ) in the sense that the group element h is not equal to theidentity in π ( M ). If D is a self-adjoint elliptic differential operator on M and e D is the lifting of D to f M , then the delocalized eta invariant η h h i ( e D ) is defined bythe formula: η h h i ( e D ) := 2 √ π Z ∞ tr h h i ( e De − t e D ) dt. (1.1)Here tr h h i is the following trace maptr h h i ( A ) = X g ∈h h i Z x ∈F A ( x, gx ) dx on G -equivariant kernels A ∈ C ∞ ( f M × f M ), where F is a fundamental domainof f M under the action of G = π ( M ). As it stands, the above definition ofdelocalized eta invariant does not require a choice of a smooth dense subalgebraof C ∗ r ( G ). Of course, in the special event that e De − t e D lies in an appropriatesmooth dense subalgebra to which the trace map tr h h i continuously extends, thisdelocalized eta invariant indeed coincides with the pairing of tr h h i with Lott’shigher eta invariant.In [39], the convergence of the above formula is proved by Lott under theassumption that h h i has polynomial growth or is hyperbolic , and that e D isinvertible or more generally e D has a spectral gap at zero. Recall that e D is saidto have a spectral gap at zero if there exists an open interval ( − ε, ε ) ⊂ R suchthat spectrum( e D ) ∩ ( − ε, ε ) is either { } or empty. In general, the convergence of In the literature, the delocalized part of Lott’s noncommutative-differential higher eta in-variant sometimes is also referred to as higher rho invariant. To avoid confusion, we shall referthis noncommutative-differential higher rho invariant as
Lott’s higher eta invariant . See [38, Section 4.6] for a discussion of the periodic version of Lott’s higher eta invariant.As K -theory is 2-periodic, the periodic higher eta invariant is the correct version to used here. In [39, Proposition 8], Lott stated that the convergence of the formula (3.2) holds for bothgroups with polynomial growth and hyperbolic groups. However, his proof for hyperbolic groupscontained a technical problem which was later fixed by Puschnigg after constructing a differentsmooth dense subalgebra of the reduced group C ∗ -algebra for hyperbolic groups [48]. XIAOMAN CHEN, JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU (1.1) fails. For example, Piazza and Schick gave an explicit example where theconvergence of (1.1) fails when e D does not have a spectral gap at zero.As the first main result of this paper, we show that if e D has a sufficiently largespectral gap at zero, then Lott’s delocalized eta invariant η h h i ( e D ) in line (1.1)converges absolutely. We refer to Definition 3 . Theorem 1.1.
Let M be a closed manifold and f M the universal covering over M . Suppose D is a self-adjoint first-order elliptic differential operator over M and e D the lift of D to f M . If h h i is a nontrivial conjugacy class of π ( M ) and e D has a sufficiently large spectral gap at zero, then the delocalized eta invariant η h h i ( e D ) defined in line (1.1) converges absolutely. We would like to emphasize that the theorem above works for all fundamentalgroups. In the special case where the conjugacy class h h i has sub-exponentialgrowth, then any nonzero spectral gap is in fact sufficiently large, hence in thiscase η h h i ( e D ) converges absolutely as long as e D is invertible.Now a special feature of traces is that they always have uniformly bounded rep-resentatives, when viewed as degree zero cyclic cocycles. In fact, the techniquesused to prove Theorem 1 . h h i will be called a delocalized cyclic cocycle at h h i , cf. Defini-tion 3 .
16. Moreover, see Definition 3 .
22 in Section 3 for the precise definition ofexponential growth for cyclic cocycles.
Theorem 1.2.
Assume the same notation as in Theorem . . Let ϕ be a de-localized cyclic cocycle at a nontrivial conjugacy class h h i . If ϕ has exponentialgrowth and e D has a sufficiently large spectral gap at zero, then a higher analogue η ϕ ( e D ) ( cf. Definition . of the delocalized eta invariant converges absolutely. The explicit formula for η ϕ ( e D ) is described in terms of the transgression formulafor Connes-Chern character [12, 14] [27]. It is closely related to the periodicversion of Lott’s noncommutative-differential higher eta invariant. In the casewhere the fundamental group G has polynomial growth, we shall show that ourformula for η ϕ ( e D ) is equivalent to the periodic version of Lott’s noncommutative-differential higher eta invariant, cf. Section 8. As η ϕ ( e D ) is an analogue for higherdegree cyclic cocycles of Lott’s delocalized eta invariant, we shall call η ϕ ( e D ) a delocalized higher eta invariant from now on. Again, we refer to Definition 3 . G and ϕ have sub-exponential growth, then any nonzero spectral gap is in ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 5 fact sufficiently large, hence in this case η ϕ ( e D ) converges absolutely as long as e D is invertible.Formally speaking, just as Lott’s delocalized eta invariant η h h i ( e D ) can be in-terpreted as the pairing between the degree zero cyclic cocycle tr h h i and the K -theoretic higher rho invariant ρ ( e D ) (or the noncommutative differential highereta invariant), so can the delocalized higher eta invariant η ϕ ( e D ) be interpreted asthe pairing between the cyclic cocycle ϕ and the K -theoretic higher rho invariant ρ ( e D ) (or the noncommutative differential higher eta invariant). As pointed outin the discussion above, a key analytic difficulty here is to verify when such apairing is well-defined, or more ambitiously, to verify when one can extend thispairing to a pairing between the cyclic cohomology of C G and the K -theory group K ∗ ( C ∗ L, ( f M ) G ). The group K ∗ ( C ∗ L, ( f M ) G ) consists of C ∗ -algebraic secondary in-variants; in particular, it contains all higher rho invariants from the discussionabove. As pointed out above, such an extension of the pairing is important, of-ten necessary, for many interesting applications to geometry and topology (cf.[47, 57, 54]).In a previous paper [58], the third and fourth authors established a pairingbetween delocalized cyclic cocycles of degree zero (i.e. delocalized traces) and the K -theory group K ∗ ( C ∗ L, ( f M ) G ), under the assumption that the relevant conjugacyclass has polynomial growth. In this paper, we shall construct a pairing betweendelocalized cyclic cocycles of all degrees and the K -theory group K ∗ ( C ∗ L, ( f M ) G )for hyperbolic groups. Before we state the theorem, let us recall some notationthat will used in the statement of the next theorem. The cyclic cohomology ofa group algebra C G has a decomposition respect to the conjugacy classes of G ([9, 45]): HC ∗ ( C G ) ∼ = Y h h i HC ∗ ( C G, h h i ) , where HC ∗ ( C G, h h i ) denotes the component that corresponds to the conjugacyclass h h i , cf. Definition 3 .
16 below.
Theorem 1.3.
Let M be a closed manifold whose fundamental group G is hy-perbolic. Suppose h h i is a non-trivial conjugacy class of G . Then every element [ α ] ∈ HC k +1 − i ( C G, h h i ) induces a natural map τ [ α ] : K i ( C ∗ L, ( f M ) G ) → C such that the following are satisfied. ( i ) τ [ Sα ] = τ [ α ] , where S is Connes’ periodicity map S : HC ∗ ( C G, h h i ) → HC ∗ +2 ( C G, h h i ) . ( ii ) Suppose D is a first-order elliptic differential operator on M such that thelift e D of D to the universal cover f M of M is invertible. Then we have τ [ α ] ( ρ ( e D )) = − η α ( e D ) , XIAOMAN CHEN, JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU where ρ ( e D ) is the C ∗ -algebraic higher rho invariant of e D and η α ( e D ) is thedelocalized higher eta invariant defined in Definition . . In particular, inthis case, the delocalized higher eta invariant η α ( e D ) converges absolutely. The construction of the map τ [ α ] in the above theorem uses Puschnigg’s smoothdense subalgebra for hyperbolic groups [48] in an essential way. In more con-ceptual terms, the above theorem provides an explicit formula to compute thedelocalized Connes-Chern character of C ∗ -algebraic secondary invariants. Moreprecisely, the same techniques developed in this paper actually imply that thereis a well-defined delocalized Connes-Chern characterch deloc : K i ( C ∗ L, ( f M ) Γ ) → HC deloc ∗ ( B ( C G )) , where B ( C G ) is Puschnigg’s smooth dense subalgebra of the reduced group C ∗ -algebra of G and HC deloc ∗ ( B ( C G )) is the delocalized part of the cyclic homol-ogy of B ( C G ). Now for Gromov’s hyperbolic groups, every cyclic cohomologyclass of C Γ continuously extends to cyclic cohomology class of B ( C G ) (cf. [48]for the case of degree zero cyclic cocycles and Section 4 of this paper for thecase of higher degree cyclic cocycles). Thus the map τ [ α ] can be viewed as apairing between cyclic cohomology and delocalized Connes-Chern characters of C ∗ -algebraic secondary invariants. We point out that, although the spectral gapof e D is required to be sufficiently large in Theorem 1 . . η α ( e D )to converge, such a requirement is not needed in the case of hyperbolic groups.This is again a consequence of some essential properties of Puschnigg’s smoothdense subalgebra.As an application, we use this delocalized Connes-Chern character map toobtain a delocalized higher Atiyah-Patodi-Singer index theorem for manifoldswith boundary. More precisely, let W be a compact n -dimensional spin manifoldwith boundary ∂W . Suppose W is equipped with a Riemannian metric g W whichhas product structure near ∂W and in addition has positive scalar curvature on ∂W . Let f W be the universal covering of W and g f W the Riemannian metric on f W lifted from g W . With respect to the metric g f W , the associated Dirac operator e D W on f W naturally defines a higher index Ind G ( e D W ) in K n ( C ∗ ( f W ) G ) = K n ( C ∗ r ( G )),where G = π ( W ), cf. [56, Section 3]. Since the metric g f W has positive scalarcurvature on ∂ f W , it follows from the Lichnerowicz formula that the associatedDirac operator e D ∂ on ∂ f W is invertible, hence naturally defines a higher rho In fact, even more is true. The same techniques developed in the current paper imply thatif A is smooth dense subalgebra of C ∗ r (Γ) for any group Γ (not necessarily hyperbolic) andin addition A is a Fr´echet locally m -convex algebra, then there is a well-defined delocalizedConnes-Chern character Ch deloc : K i ( C ∗ L, ( f M ) Γ ) → HC deloc ∗ ( A ). Of course, in order to pairsuch a delocalized Connes-Chern character with a cyclic cocycle of C Γ, the key remainingchallenge is to continuously extend this cyclic cocycle of C Γ to a cyclic cocycle of A . Here the definition of cyclic homology of B ( C G ) takes the topology of B ( C G ) into account,cf. [12, Section II.5]. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 7 invariant ρ ( e D ∂ ) in K n − ( C ∗ L, ( f W ) Γ ). We have the following delocalized higherAtiyah-Patodi-Singer index theorem. Theorem 1.4.
With the same notations as above, if G = π ( W ) is hyperbolicand h h i is a nontrivial conjugacy class of G , then for any [ ϕ ] ∈ HC ∗ ( C G, h h i ) ,we have ch [ ϕ ] (Ind G ( e D W )) = 12 η [ ϕ ] ( e D ∂ ) , (1.2) where ch [ ϕ ] (Ind G ( e D W )) is the Connes-Chern pairing between the cyclic cohomol-ogy class [ ϕ ] and the C ∗ -algebraic index class Ind G ( e D W ) . There have been various versions of higher Atiyah-Patodi-Singer theorem inthe literature [37, 52, 22]. See the discussion after Theorem 7 . . . . not rely on theBaum-Connes isomorphism for hyperbolic groups [35, 44], although the theoremis closely connected to the Baum-Connes conjecture and the Novikov conjecture.On the other hand, if one is willing to use the full power of the Baum-Connesisomorphism for hyperbolic groups, there is in fact a different, but more indirect,approach to the delocalized Connes-Chern character map. First, observe that themap τ [ α ] factors through a map τ [ α ] : ( K i ( C ∗ L, ( EG ) G ) ⊗ C ) → C where EG is the universal space for proper G -actions. Now the Baum-Connesisomorphism µ : K G ∗ ( EG ) ∼ = −−→ K ∗ ( C ∗ r ( G )) for hyperbolic groups implies that onecan identify K i ( C ∗ L, ( EG ) G ) ⊗ C with L h h i6 =1 HC ∗ ( C G, h h i ), where HC ∗ ( C G, h h i )is the delocalized cyclic homology at h h i (a cyclic homology analogue of Defini-tion 3 .
16) and the direct sum is taken over all nontrivial conjugacy classes. Inparticular, after this identification, it follows that the map τ [ α ] becomes the usualcomponentwise pairing between cyclic cohomology and cyclic homology. How-ever, for a specific element, e.g. the higher rho invariant ρ ( e D ), in K i ( C ∗ L, ( EG ) G ),its identification with an element in L h h i6 =1 HC ∗ ( C G, h h i ) is rather abstract andimplicit. More precisely, the computation of the number τ [ α ] ( ρ ( e D )) essentiallyamounts to the following process. Observe that if a closed spin manifold M isequipped with a positive scalar curvature metric, then stably it bounds (more pre-cisely, the universal cover f M of M becomes the boundary of another G -manifold,after finitely many steps of cobordisms and vector bundle modifications). In prin-ciple, the number τ [ α ] ( ρ ( e D )) can be derived from a higher Atiyah-Patodi-Singerindex theorem for this bounding manifold. The drawback of such an approach isthat there is no explicit formula for τ [ α ] ( ρ ( e D )), since there is no explicit proce-dure for producing such a bounding manifold. In [18], Deeley and Goffeng also XIAOMAN CHEN, JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU constructed a delocalized Connes-Chern character for C ∗ -algebra secondary in-variants. Their approach is in spirit similar to the indirect method just describedabove (making use of the Baum-Connes isomorphism for hyperbolic groups), al-though their actual technical implementation is different. The key feature of ourapproach in this paper is that we obtain an explicit and intrinsic formula for thedelocalized Connes-Chern character of C ∗ -algebraic secondary invariants.The paper is organized as follows. In Section 2, we review some standard geo-metric C ∗ -algebras and a construction of Higson-Roe’s K -theoretic higher rhoinvariants. In Section 3, we prove the convergence of Lott’s delocalized eta in-variant holds for all operators with a sufficiently large spectral gap at zero. Moregenerally, for each higher degree delocalized cyclic cocycle, we define a higheranalogue of Lott’s delocalized eta invariant, and prove its convergence for all op-erators with a sufficiently large spectral gap at zero, provided that the given cycliccocycle has at most exponential growth. In Section 4, we review Puschnigg’sconstruction of smooth dense subalgebras of reduced group C ∗ -algebras for hy-perbolic groups. Puschnigg showed that any trace on the group algebra of ahyperbolic group extends continuously onto this smooth dense subalgebra. Weshall generalize this result to cyclic cocycles of all degrees, provided that thecyclic cocycles have polynomial growth (cf. Proposition 4.14 below). In Section5, we show that every cyclic cohomology class of a hyperbolic group has a repre-sentative of polynomial growth. Furthermore, if the cyclic cohomology class hasdegree >
2, then it admits a uniformly bounded representative. In Section 6, wegive an explicit formula for the pairing between C ∗ -algebraic secondary invariantsand delocalized cyclic cocycles of the group algebra for word hyperbolic groups.When the C ∗ -algebraic secondary invariant is a K -theoretic higher rho invari-ant of an invertible differential operator, we show this pairing is precisely thehigher delocalized eta invariant of the given operator. In Section 7, we computethe paring between delocalized cyclic cocycles and C ∗ -algebraic Atiyah-Patodi-Singer index classes for manifolds with boundary, when the fundamental groupof the given manifold is hyperbolic. In section 8, we identify our definition ofdelocalized higher eta invariant with Lott’s higher eta invariant.We would like to thank Denis Osin for providing us a proof of a useful resulton hyperbolic groups (Lemma 5 . Preliminaries
In this section, we review the construction of some geometric C ∗ -algebras andHigson-Roe’s higher rho invariants. We refer the reader to [50, 59, 24, 25, 26] formore details.Let X be a proper metric space, that is, every closed metric ball in X is com-pact. An X -module is a separable Hilbert space equipped with a ∗ -representationof C ( X ), the algebra of all continuous functions on X which vanish at infinity.An X -module is called nondegenerate if the ∗ -representation of C ( X ) is nonde-generate. An X -module is said to be standard if no nonzero function in C ( X )acts as a compact operator. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 9
Definition 2.1.
Let H X be an X -module and T a bounded linear operator actingon H X .(1) The propagation of T is defined to be sup { d ( x, y ) | ( x, y ) ∈ supp ( T ) } , where supp ( T ) is the complement (in X × X ) of the set of points ( x, y ) ∈ X × X for which there exist f, g ∈ C ( X ) such that gT f = 0 and f ( x ) g ( y ) = 0.(2) T is said to be locally compact if f T and T f are compact for all f ∈ C ( X ).(3) T is said to be pseudo-local if [ T, f ] is compact for all f ∈ C ( X ). Definition 2.2.
Let H X be a standard nondegenerate X -module and B ( H X ) theset of all bounded linear operators on H X .(1) The Roe algebra of X , denoted by C ∗ ( X ), is the C ∗ -algebra generated byall locally compact operators with finite propagations in B ( H X ).(2) D ∗ ( X ) is the C ∗ -algebra generated by all pseudo-local operators and withfinite propagations in B ( H X ). In particular, D ∗ ( X ) is a subalgebra of themultiplier algebra of C ∗ ( X ).(3) C ∗ L ( X ) (resp. D ∗ L ( X )) is the C ∗ -algebra generated by all bounded anduniformly-norm continuous functions f : [0 , ∞ ) → C ∗ ( X ) (resp. f :[0 , ∞ ) → D ∗ ( X )) such thatpropagation of f ( t ) → t → ∞ .D ∗ L ( X ) is a subalgebra of the multiplier algebra of C ∗ L ( X ).(4) The kernel of the following evaluation map ev : C ∗ L ( X ) → C ∗ ( X ) , f f (0)is defined to be C ∗ L, ( X ). In particular, C ∗ L, ( X ) is an ideal of C ∗ L ( X ).Similarly, we define D ∗ L, ( X ) as the kernel of the evaluation map from D ∗ L ( X ) to D ∗ ( X ).Now in addition we assume that there is a countable discrete group G , andacts properly on X by isometries. In particular, if the action of Γ is free, then X is simply a G -covering of the compact space X/G . Let H X be an X -moduleequipped with a covariant unitary representation of G . Let the representationof C ( X ) be φ and let the action of G be π . We call ( H X , G, φ, π ) a covariantsystem is to say π ( g ) φ ( f ) = φ ( g ∗ f ) π ( g ) , where g ∗ f ( x ) = f ( g − x ) for any f ∈ C ( X ), g ∈ G . Definition 2.3.
A covariant system ( H X , G, φ ) is called admissible if(1) The action of G is proper and cocompact;(2) H X is a nondegenerate standard X -module;(3) For each x ∈ X , the stabilizer group G x acts on H X regularly in the sensethat the action is isomorphic to the obvious action of G x on l ( G x ) ⊗ H for some infinite dimensional Hilbert space H . Here G x acts on ℓ ( G x ) by(left) translations and acts on H trivially. We remark that for each locally compact metric space X with a proper andcocompact isometric action of G , there exists an admissible covariant system( H X , G, φ ). Also, we point out that the condition 3 above is automatically sat-isfied if G acts freely on X . If no confusion arises, we will denote an admissiblecovariant system ( H X , G, φ ) by H X and call it an admissible ( X, G )-module.
Definition 2.4.
Let X be a locally compact metric space X with a proper andcocompact isometric action of G . If H X is an admissible ( X, G )-module, wedenote by C [ X ] G to be ∗ -algebra of all G -invariant locally compact operatorswith finite propagations in B ( H X ). We define C ∗ ( X ) G to be the completion of C [ X ] G in B ( H X ).Since the action of G on X is cocompact, it is known that C ∗ ( X ) G is ∗ -isomorphic to K ⊗ C ∗ r ( G ), where K is the algebra of all compact operators and C ∗ r ( G ) is the reduced group C ∗ -algebra.Similarly, we can also define D ∗ ( X ) G , C ∗ L ( X ) G , D ∗ L ( X ) G , C ∗ L, ( X ) G and D ∗ L, ( X ) G . Remark . Up to isomorphism, C ∗ ( X ) does not depend on the choice of thestandard nondegenerate X -module H X . The same holds for D ∗ ( X ), C ∗ L ( X ), D ∗ L ( X ), C ∗ L, ( X ), D ∗ L, ( X ) and their G -equivariant versions.Let M be a closed Riemannian manifold. Let G be a discrete finitely generatedcountable group. Suppose f M is a regular G -cover of M . For example, f M is theuniversal covering of M and G is the fundamental group of M . Let p be theassociated covering map from f M to M . Suppose E is an Hermitian vector bundleover M and e E the lifting of E to f M . Write H the collection of all L -sectionsof e E . The equivariant Roe algebra C ∗ ( f M ) G is defined to be the operator-normcompletion of all G -equivariant locally compact operators of finite propagationacting on H . The localization algebra C ∗ L ( f M ) G and C ∗ L, ( f M ) G can be definedsimilarly.Suppose that M is spin and E is the corresponding spinor bundle over M .Let D be the Dirac operator acting on E and e D its lifting to e E . If the scalarcurvature of the metric over M is strictly positive, the Dirac operator naturallydefines a K -theory class called the higher rho invariant in K ∗ ( C ∗ L, ( f M ) G ). Forsimplicity, we will only discuss the case where M is odd dimensional; the evendimensional case is completely similar, cf. [56].Define the following functions F t ( x ) = 1 √ π Z x/t −∞ e − s ds and U t ( x ) = exp(2 πiF t ( x )) . (2.1)Since the scalar curvature over f M is uniformly bounded below by a positivenumber, it follows from the Lichnerowicz formula that the Dirac operator e D isinvertible. This implies that F t ( e D ) converges to (1 + e D | e D | − ) in operator norm,as t →
0. Thus the path { U t ( e D ) } ≤ t< ∞ lies in C ∗ L, ( f M ) G . ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 11
Definition 2.6 ([24, 25, 26]) . The Higson-Roe higher rho invariant ρ ( e D ) of e D isdefined to be the K -theory class [ { U t ( e D ) } ≤ t< ∞ ] ∈ K ( C ∗ L, ( f M ) G ).3. Higher eta invariants
In this section, we show that Lott’s delocalized eta invariant converges abso-lutely for operators with a sufficiently large spectral gap at zero. More generally,we prove the convergence of a higher analogue of Lott’s delocalized eta invariantfor operators with a sufficiently large spectral gap at zero and higher degree cycliccocycles that have at most exponential growth.3.1.
Convergence of delocalized eta invariants with a large enough spec-tral gap.
Let M be a closed Riemannian manifold. Let G be a finitely generateddiscrete group. Suppose f M is a regular G -covering space of M . Let p be the as-sociated covering map from f M to M . Choose any fundamental domain F of G -action on f M . Suppose D is a first-order self-adjoint elliptic differential oper-ator acting on some Hermitian vector bundle E over M and e E (resp. e D ) is thelifting of E (resp. D ) to f M . Assume G acts on e E as well. Let H be the space of L -sections of e E . Then the operator e De − t e D lies in B ( H ), the algebra of boundedoperators on H . Moreover, the associated Schwartz kernel k t ( x, y ) of e De − t e D issmooth.Any conjugacy class h h i of G naturally induces a trace map tr h h i : C G → C , by X g ∈ G a g g X g ∈h h i a g . In particular, when h h i is the trivial conjugacy class, tr h e i is the canonical trace on C G . The trace map tr h h i generalizes (formally) to a trace map on G -equivariantintegral operators T with a smooth Schwartz kernel T ( x, y ) as follows:tr h h i ( T ) = X g ∈h h i Z x ∈F tr ( T ( x, gx )) dx, (3.1)provided that the right hand side converges. Definition 3.1 ([39]) . For any nontrivial conjugacy class h h i of G , Lott’s delo-calized eta invariant η h h i ( e D ) of e D is defined to be η h h i ( e D ) := 2 √ π Z ∞ tr h h i ( e De − t e D ) dt. (3.2)The terminology “delocalized” refers to the fact that h h i is a nontrivial con-jugacy class. If we were to take the trivial conjugacy class in Definition 3 .
1, wewould recover the L -eta invariant of Cheeger and Gromov [10].Lott proved the convergence of the integral in line (3.2) under the assumptionthat G has polynomial growth or is hyperbolic, and e D is invertible (or moregenerally has a spectral gap at zero) [39]. Piazza and Schick gave an examplewhere the formula (3.2) diverges for non-invertible e D [46, Section 3]. They then raised the question of whether a divergent example still exists if one assumes theinvertibility of e D [46, Remark 3.2].Our first main result states that the answer to this question of Piazza andSchick is negative, as long as the spectral gap of e D is sufficiently large. Beforewe give the precise statement of the theorem, we first fix some notation.Fix a finite generating set S of G . Let ℓ be the corresponding word lengthfunction on G determined by S . Since S is finite, there exist C > K h h i > { g ∈ h h i : ℓ ( g ) = n } Ce K h h i · n . (3.3)We define τ h h i to be τ h h i = lim inf g ∈h h i ℓ ( g ) →∞ (cid:16) inf x ∈F dist( x, gx ) ℓ ( g ) (cid:17) . (3.4)Since the action of G on f M is free and cocompact, we have τ h h i > D on M , we denote the principalsymbol of D by σ D ( x, v ), for x ∈ M and cotangent vector v ∈ T ∗ x M . We definethe propagation speed of D to be the positive number c D = sup {k σ D ( x, v ) k : x ∈ M, v ∈ T ∗ x M, k v k = 1 } . (3.5)When D is the Dirac operator on a spin manifold, we have c D = 1. Definition 3.2.
With the above notation, let us define σ h h i := 2 K h h i · c D τ h h i . (3.6)We say the spectral gap of e D is sufficiently large if the spectral gap of e D at zerois larger than σ h h i , i.e. spectrum( e D ) ∩ [ − σ h h i , σ h h i ] is either { } or empty.In the following, we shall show that the convergence of the formula (3.2) holdsif e D has a spectral gap at zero larger than σ h h i . In fact, the convergence of theformula (3.2) for the case where e D has a spectral gap at zero can be deducedfrom the invertible case by replacing e D with its restriction to the orthogonalcomplement of the kernel of e D . Without loss of generality, we will only give thedetails of the proof for the case where e D is invertible and its spectral gap at zerois larger than σ h h i . Theorem 3.3.
With the same notation as above, for any nontrivial conjugacyclass h h i , suppose that e D is invertible and its spectral gap at zero is larger than σ h h i . Then the delocalized eta invariant η h h i ( e D ) given in line (3.2) convergesabsolutely.Proof. The proof is divided into three steps. In the first step, we show thattr h h i ( e De − t e D ) is finite for any fixed t > t (Proposition 3.12). In thelast step, we show the convergence of the integral for large t (Proposition 3.13). ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 13
In fact, only the last step requires the assumption that the spectral gap of e D islarger than σ h h i . (cid:3) We say that h h i has sub-exponential growth if we can choose K h h i in line (3.3)to be arbitrarily small. In this case, we have the following corollary. Corollary 3.4. If h h i has sub-exponential growth, then η h h i ( e D ) given in line (3.2) converges absolutely, as long as e D has a spectral gap at zero. In Theorem 3 .
3, the condition that the spectral gap of e D is larger than σ h h i might first appear to be rather ad hoc. In the following, we shall show that sucha condition in fact holds for an abundance of natural examples whose higher rhoinvariant is nontrivial.Suppose that N is a closed spin manifold equipped with a positive scalar cur-vature metric g N , whose fundamental group F = π ( N ) is finite and its higherrho invariant ρ ( e D N ) is nontrivial. Here e D N is the Dirac operator on the universalcovering e N of N . For instance, let N to be a lens space, that is, the quotient ofthe 3-dimensional sphere by a free action of a finite cyclic group. In this case, theclassical equivariant Atiyah-Patodi-Singer index theorem implies that the higherrho invariant of N is nontrivial, cf. [19].Now let V be an even dimensional closed spin manifold, whose Dirac operator D V has nontrivial higher index in K ( C ∗ r (Γ)), where Γ = π ( V ). In particular,it follows that D V defines a nonzero element in the equivariant K -homology K ( C ∗ L ( E Γ) Γ ) of the universal space E Γ for free Γ actions. Consider the productspace M = V × N equipped with a metric g M = g V + ε · g N , where g V isan arbitrary Riemannian metric on V and the metric g N on N is scaled by apositive number ε . Denote the Dirac operator on the universal covering f M of M by e D M . The product formula for secondary invariants (cf. [56, Claim 2.19], [60,Corollary 4.15]) shows that the higher rho invariant ρ ( e D M ) is the product of the K -homology class of D V and the higher rho invariant ρ ( e D N ). By the assumptionsabove, we see that ρ ( e D M ) is nonzero in K ( C ∗ L, ( E (Γ × F )) Γ × F ).Now let us return to the proof of Theorem 3 .
3. First, we need a few technicallemmas. Let M be a closed Riemannian manifold. Suppose T is an integraloperator on the space L ( M ) of L functions on M , and assume the Schwartzkernel of T is a continuous function on M × M . In particular, we have T ( f )( x ) = Z M T ( x, y ) f ( y ) dy, for all f ∈ L ( M ). We denote the operator norm of an operator A on L ( M ) by k A k op . Lemma 3.5.
Let M be a closed Riemannian manifold and D a first-order self-adjoint elliptic differential operator on M . Suppose T is a bounded linear operatoron L ( M ) such that sup k + j dim M +3 k D k T D j k op < ∞ . Then T is an integral operator with a continuous Schwartz kernel K T ( x, y ) , andthere exists a positive number C ( independent of T ) such that sup x,y ∈ M | K T ( x, y ) | C · sup k + j dim M +3 k D k T D j k op . (3.7) Proof.
Let p be the smallest even integer that is greater than dim M . Then(1 + D p ) − is a Hilbert-Schmidt operator on L ( M ). Denote the Hilbert-Schmidtnorm of (1 + D p ) − by k (1 + D p ) − k HS ,By assumption, (1+ D p ) T is bounded. It follows that T = (1+ D p ) − ◦ (1+ D p ) T is also a Hilbert-Schmidt operator, and furthermore k T k HS ≤ k (1 + D p ) − k HS · k (1 + D p ) T k op . It follows that T is an integral operator whose Schwartz kernel K T ( x, y ) is an L -function on M × M .To see that K T is continuous on M × M , we consider the elliptic differentialoperator D = D ⊗ ⊗ D on M × M . Observe that k D n K T k L ( M × M ) = (cid:13)(cid:13)(cid:13) n X r =0 (cid:18) nr (cid:19) D r T D n − r (cid:13)(cid:13)(cid:13) HS ≤ n X r =0 (cid:18) nr (cid:19) k (1 + D p ) − k HS · k (1 + D p ) D r T D n − r k op . Therefore, our assumption sup k + j dim M +3 k D k T D j k op < ∞ implies that k D n K T k L ( M × M ) is finite for all n ≤ dim M + 1. It follows from theSobolev embedding theorem that there exists C > x,y ∈ M | K T ( x, y ) | C · sup n ≤ dim M +1 k D n K T k L ( M × M ) where the right hand side is dominated by C · sup k + j dim M +3 k D k T D j k op for some constant C >
0. This finishes the proof. (cid:3)
Remark . Let E be a Hermitian vector bundle over M and f M a regular G -covering space of M . Denote the lift of E to f M by e E . The above lemmaadmits an obvious analogue for G -equivariant operators T acting on L ( f M , e E ). Remark . As an immediate consequence of the above lemma, we see that fora closed Riemannian manifold M and any bounded linear operator T on L ( M ),if sup k + j dim M +3 k D k T D k ′ k op < ∞ , ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 15 then T is of trace class. In fact, in this case, we havetr( T ) = Z M T ( x, x ) dx, cf. [6, Chapter V, Proposition 3.1.1].Suppose D is a first-order self-adjoint elliptic differential operator acting on avector bundle E over M , and e E (resp. e D ) is the lift of E (resp. D ) to f M . If f is a function on R such that k x m f ( x ) k L ∞ < ∞ (3.8)for all m dim M + 3, then the corresponding Schwartz kernel of the operator f ( e D ) is continuous. Denote the Schwartz kernel of f ( e D ) by K f . Lemma 3.8.
With the notations above, for any µ > and r > , there exists aconstant C > such that k K f ( x, y ) k C · F f (cid:18) dist( x, y ) µ · c D (cid:19) , for ∀ x, y ∈ f M with dist( x, y ) > r and any f satisfying line (3.8) . Here dist( x, y ) stands for the distance between x and y , the notation k · k denotes a matrix normof elements in End( e E y , e E x ) , and the function F f is defined by F f ( s ) := sup n dim M +3 Z | ξ | >s (cid:12)(cid:12)(cid:12)(cid:12) d n dξ n b f ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) dξ, where b f is the Fourier transform of f .Proof. The condition on f implies that F f ( s ) < ∞ for any s ∈ R and f ( e D ) is anintegral operator with continuous Schwartz kernel (cf. Lemma 3.5).By the Fourier inverse transform formula, we have f ( e D ) = 12 π Z + ∞−∞ b f ( ξ ) e iξ e D dξ. Fix r >
0. Let x , y ∈ f M such that λ := dist( x , y ) > r . Choose a smoothfunction ϕ over R such that ϕ ( ξ ) = 1 for | ξ | > ϕ ( ξ ) = 0 for | ξ | /µ . Let ϕ λ ( ξ ) = ϕ ( ξ · c D /λ ). Let g be the function with Fourier transform b g ( ξ ) = ϕ λ ( ξ ) b f ( ξ ) , and L ∈ C ( f M × f M ) be the Schwartz kernel corresponding to the operator g ( e D ). It follows from standard finite propagation estimates of wave operatorsthat L ( x, y ) = K f ( x, y ) for all x, y ∈ f M with dist( x, y ) > λ . In particular, wehave L ( x , y ) = K f ( x , y ).By Lemma 3.5, it suffice to estimate the operator norm of e D k g ( e D ) e D j = e D k + j g ( e D ) for all k + j dim M + 3. Now for a given n dim M + 3, define ψ n ( x ) = x n g ( x ). We have b ψ n ( ξ ) = (cid:18) i ddξ (cid:19) n ( ϕ λ b f )( ξ ) . Since ϕ is supported on | ξ | > λ/ ( µc D ), there exist positive numbers C and C such that k ψ n ( e D ) k op π Z | ξ | > λ/ ( µc D ) | b ψ n ( ξ ) | dξ C n X j =0 (cid:16) c D r (cid:17) j Z | ξ | > λ/ ( µc D ) | b f ( n − j ) ( ξ ) | dξ C · F f (cid:18) λµ · c D (cid:19) By Lemma 3 .
5, we see that k K f ( x , y ) k = k L ( x , y ) k C · F f (cid:18) dist( x , y ) µ · c D (cid:19) . This finishes the proof. (cid:3)
To streamline our estimates later, we consider the following class of functionsin C ( R ) whose Fourier transform has exponential decay. Definition 3.9.
Let A Λ ,N be the subspace of C ( R ) consisting of functions f satisfying the following conditions:(1) f admits an analytic continuation e f on the strip {| Im( z ) | < Λ } ,(2) for any n N , | z n e f ( z ) | is uniformly bounded on the strip.Equip A Λ ,N with the norm k · k A defined by k f k A = sup n N sup | Im( z ) | < Λ | z n e f ( z ) | For any fixed Λ and N , it is easy to verify that A Λ ,N is closed under multipli-cation and conjugation. In fact, A Λ ,N is a Banach ∗ -subalgebra of C ( R ) underthe norm k · k A . Clearly, e − x and xe − x lie in A Λ ,N . Lemma 3.10.
Suppose f ∈ A Λ ,N for some N > . Let b f be its Fourier transform.Then for any < λ < Λ and n N − there exists some constant C = C λ,n such that Z | ξ | >s | d n dξ n ˆ f ( ξ ) | dξ C · k f k A · e − λs . (3.9) Proof.
For notational simplicity, let us denote the analytic continuation of f onthe strip {| Im( z ) | < Λ } still by f . For any | y | < Λ, f ( x − iy ) is a smooth ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 17 L -integrable function since | z f ( z ) | is uniformly bounded. Denote the Fouriertransform of f ( x − iy ) with respect to x by b f y , namely b f y ( ξ ) = Z + ∞−∞ f ( x − iy ) e − iξx dx Since the right-hand side is differentiable in y , and uniformly differentiable in x and y by Cauchy inequality, the left-hand side is also differentiable in y with ∂∂y b f y ( ξ ) = Z + ∞−∞ ∂∂y [ f ( x − iy )] e − iξx dx Since f is holomorphic, we have that ∂∂y f = i ∂∂x f by the Cauchy-Riemann equa-tion. Thus ∂∂y b f y ( ξ ) = Z + ∞−∞ i ∂∂x [ f ( x − iy )] e − iξx dx = ξ b f y ( ξ )It follows that b f y ( ξ ) = b f ( ξ ) e yξ Therefore by our assumption, for any n N − < λ < Λ there existssome constant C > | d n dξ n ( b f ( ξ ) e λξ ) | = | d n dξ n b f y ( ξ ) | = Z + ∞−∞ e ixξ ( ix ) n f ( x − iλ ) dx Z ∞−∞ | x n f ( x − iλ ) | dx C k f k A Thus by induction on n , for any n N − < λ < Λ, there exists a constant C > | d n dξ n ( b f ( ξ )) | C k f k A e − λξ . Hence for ξ > s , we have Z ξ>s | d n dξ n b f ( ξ ) | dξ Ce − λs . The estimates for the part R ξ< − s | d n dξ n b f ( ξ ) | dξ are completely similar. This finishesthe proof. (cid:3) If we fix a fundamental domain
F ⊂ f M for the action of G , then one naturallyidentifies L ( f M ) with L ( F ) ⊗ ℓ ( G ) through the mapping ˜ h h by h ( x, γ ) =˜ h ( γx ) for x ∈ F and γ ∈ G . In particular, every G -equivariant Schwartz kernel A on f M × f M becomes a formal sum A = X g ∈ G A g R g where A g ( x, y ) = A ( x, gy ) for x, y ∈ F and R g denotes the right translation of g on ℓ ( G ) corresponding to the right regular representation of G on ℓ ( G ). Now suppose A = f ( e D ) as in Lemma 3 . A g is a trace classoperator and tr( A g ) = Z F A g ( x, x ) dx = Z F A ( x, gx ) dx. Now we are ready to proceed with the first step of the proof for Theorem 3 . Proposition 3.11.
Suppose h h i is a nontrivial conjugacy class of G . If f ∈ A Λ ,N with N > dim M + 5 and Λ sufficiently large, then tr h h i ( f ( e D )) is finite.Proof. Fix a symmetric generating set S of G . Let ℓ be the length function on G determined by S . Since G acts freely and cocompactly on f M , there exists ε > x, gx ) > ε for any x ∈ f M and g ∈ h h i . Let k ( x, y ) be the Schwartzkernel of f ( e D ). Since the action of G on f M is free and cocompact, there exist C , C > x, gx ) > C · ℓ ( g ) − C . It follows from Lemma 3.8 that there exists C > | tr h h i ( f ( e D )) | X g ∈h h i Z F | tr( k ( x, gx )) | dx C X g ∈h h i F f (max { ε, C · ℓ ( g ) − C } ) C ∞ X n =1 | S | n F f (max { ε, C · n − C } ) , where | S | is the cardinality of the generating set S . By Lemma 3.10, when N > dim M + 5, for any λ < Λ, there exists
C > F f ( x ) Ce − λx . Observe that the summation ∞ X n =1 | S | n e − λ · max { ε, C · n − C } converges absolutely, as long as λ is sufficiently large. This finishes the proof. (cid:3) Now let us prove the convergence of the integral (3.2) for small t . Proposition 3.12.
Suppose h h i is a nontrivial conjugacy class of G . If f ∈ A Λ ,N with N > dim M + 5 and Λ sufficiently large, then the following integral Z t k · tr h h i ( f ( t e D )) dt is absolutely convergent for any k ∈ R . ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 19
Proof.
Let us denote f t ( x ) = f ( tx ). Clearly, f t ∈ A Λ ,N since f ∈ A Λ ,N . Similarto the proof of Proposition 3 .
11, we have | t k tr h h i ( f t ( e D )) | C ∞ X n =1 | S | n t k F f t (max { ε, C · n − C } ) . Recall that b f t ( ξ ) = t − · b f ( ξ/t ). In particular, we have d n dξ n b f t ( ξ ) = 1 t n +1 b f ( n ) ( ξ/t ) , where b f ( n ) is the n -th derivative of b f . It follows from Lemma 3 .
10 that thereexists C Λ > F f t ( s ) = sup n dim M +3 t n +1 Z | ξ | >s (cid:12)(cid:12)(cid:12) b f ( n ) ( ξ/t ) (cid:12)(cid:12)(cid:12) dξ F f ( s · t − ) t dim M +3 C Λ t dim M +3 · k f k A · exp( − Λ s t ) (3.10)for all t ∈ (0 , ∞ X n =1 | S | n t k − ( dim M +3) exp (cid:16) − Λ · max { ε, C · n − C } t (cid:17) is integrable on (0 , (cid:3) Proposition 3.13.
Let σ be the infimum of the spectrum of e D . If σ > σ h h i defined in line (3.6) , then the following integral Z + ∞ tr h h i ( e De − t e D ) dt is absolutely convergent.Proof. View e De − t e D as an element in K ⊗ C ∗ r ( G ) and write e De − t e D = X g ∈ G A g,t g. Note that A g,t are compact operators for all g ∈ G and t >
1. By Lemma 3.5, A g,t is of trace class and there exist C , N > | A g,t | C t N · e − t σ , where | · | stands for the trace norm.By the definition of τ h h i in line (3.4), for any ε > L > x, gx ) > ( τ h h i − ε ) ℓ ( g ) , for all x ∈ F and g ∈ h h i satisfying ℓ ( g ) > L . By Lemma 3.8 and Proposition3.11, if ℓ ( g ) > L , then there exist C , N > | A g,t | C ( t · ℓ ( g )) N · exp (cid:18) − ( τ h h i − ε ) ℓ ( g )4 t ( µ · c D ) (cid:19) . (3.11) Write N = max { N , N } and C = max { C , C } . Then for δ > | A g,t | C ( t · ℓ ( g )) N · exp (cid:18) −
12 ( τ h h i − ε ) ℓ ( g )4 t ( µ · c D ) − t σ (cid:19) C ( t · ℓ ( g )) N · exp − ( τ h h i − ε )( √ σ − δ ) ℓ ( g )2 µ · c D ! · e − t δ , as long as ℓ ( g ) > L . By the assumption that σ > σ h h i , we may find suitable µ, ε, δ such that ( τ h h i − ε )( √ σ − δ ) > µ · c D · K h h i . Therefore, by line (3.3), there exist C , C such that | tr h h i ( e De − t e D ) | X g ∈h h i ,ℓ ( g ) L | A g,t | + X g ∈h h i ,ℓ ( g ) >L | A g,t | C e K h h i L · C ( tL ) N e − t σ + X n>L C e K h h i n · C ( nt ) N exp − ( τ h h i − ε )( √ σ − δ ) n µc D ! e − t δ C t N e − δ t . Hence the integral Z ∞ tr h h i ( e De − t e D ) dt converges absolutely. This finishes the proof. (cid:3) Delocalized higher eta invariants.
In this subsection, we shall gener-alize the results of the previous subsection to higher degree cyclic cocyles. Forsimplicity, we only give details for the odd case; the even case is similar.Let us first recall the definition of cyclic cocycles.
Definition 3.14.
Let C n ( C G ) be the space spanned by all ( n + 1)-linear func-tionals ϕ on C G such that ϕ ( g , g , · · · , g n , g ) = ( − n ϕ ( g , g , · · · , g n ) . The coboundary map b : C n ( C G ) → C n +1 ( C G ) is defined to be bϕ ( g , g , · · · , g n +1 ) = n X j =0 ( − j ϕ ( g , g , · · · , g j g j +1 , · · · , g n +1 )+ ( − n +1 ϕ ( g n +1 g , g , · · · , g n ) . The cohomology of this cochain complex ( C n ( C G ) , b ) is the cyclic cohomologyof C G , denoted by HC ∗ ( C G ). Definition 3.15.
Fix any conjugacy class h h i of G . Let C n ( C G, h h i ) be thespace spanned by all elements ϕ ∈ C n ( C G ) satisfying the condition: g g · · · g n / ∈ h h i = ⇒ ϕ ( g , g , · · · , g n ) = 0 . ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 21
It is easy to verify that that ( C n ( C G, h h i ) , b ) is a subcomplex of ( C n ( C G ) , b ).We denote the cohomology of ( C n ( C G, h h i ) , b ) by HC ∗ ( C G, h h i ). Definition 3.16.
If the conjugacy class h h i is nontrivial, then a cyclic cocyclein C n ( C G, h h i ) is called a delocalized cyclic cocycle at h h i , and HC ∗ ( C G, h h i ) iscalled the delocalized cyclic cohomology of C G at h h i .Recall that (cf. [45]) HC ∗ ( C G ) ∼ = Y h h i HC ∗ ( C G, h h i ) . Moreover, it is easy to verify that HC ( C G, h h i ) is a one dimensional vector spacegenerated by tr h h i .Let us write w = P g w g g for an element w ∈ C ∗ ( f M ) G . Given ϕ ∈ C n ( C G, h h i )and w = w ⊗ w ⊗ · · · ⊗ w n in ( C ∗ ( f M ) G ) ⊗ n +1 , we define the following map( ϕ w ) = X g , ··· ,g n ∈ G tr( w g · · · w g n n ) ϕ ( g , · · · , g n ) (3.12)whenever the above formula converges. Here tr( w g · · · w g n n ) stands for the traceof the operator w g · · · w g n n .Following the definition of higher rho invariant in line (2.1), we define u t ( x ) = U /t ( x ) = exp (cid:16) πi √ π Z xt −∞ e − s ds (cid:17) . (3.13)Note that the functions u t ( x ) − u − t ( x ) − u t ( x ) u − t ( x ) = 2 √ πixe − t x areSchwartz functions. We define the delocalized higher eta invariant as follows. Definition 3.17.
For any ϕ ∈ C m ( C G, h h i ) with h h i nontrivial, we define thedelocalized higher eta invariant of e D with respect to ϕ to be η ϕ ( e D ) := m ! πi Z ∞ η ϕ ( e D, t ) dt, (3.14)where η ϕ ( e D, t ) = ϕ u t ( e D ) u − t ( e D ) ⊗ (( u t ( e D ) − ⊗ ( u − t ( e D ) − ⊗ m ) . (3.15)More precisely, we have η ϕ ( e D, t ) := X g i ∈ G h ϕ ( g , g , · · · , g m ) Z F m +1 tr (cid:0) k ,t ( x , g x ) k ,t ( x , g x ) · · · k m,t ( x m , g m x ) (cid:1) dx · · · dx m i . (3.16)where k i,t ( x, y ) is the corresponding Schwartz kernel of ˙ u t ( e D ) u − t ( e D ), u t ( e D ) − u − t ( e D ) − Remark . Clearly, if m = 0, then η tr h h i ( e D ) = η h h i ( e D ). Hence the delocalizedhigher eta invariant is indeed a natural generalization of Lott’s delocalized etainvariant. Remark . In fact, the delocalized higher eta invariant can be defined for amore general class of representatives besides u t ( e D ) above. We shall deal with thegeneral case in Section 6. Remark . The map ϕ C ∗ ( f M ) G ) + of C ∗ ( f M ) G by defining ] ϕ C ∗ ( f M ) G ) + . With this notation, the formula of η ϕ ( e D ) becomes η ϕ ( e D ) = m ! πi Z ∞ ] ϕ u t ( e D ) u − t ( e D ) ⊗ ( u t ( e D ) ⊗ u − t ( e D ) ⊗ m ) dt. Remark . We have only discussed the odd case so far. The even case iscompletely analogous. In this case, in the construction of the higher rho invariant ρ ( e D ), the path of invertibles { u t ( e D ) } t< ∞ is replaced by a path of projections { p t ( e D ) } t< ∞ (see for example [56]). Given ϕ ∈ C m +1 ( C G, h h i ), the delocalizedhigher eta invariant of e D with respect to ϕ is defined to be η ϕ ( e D ) = 1 πi (2 m )! m ! Z ∞ η ϕ ( e D, t ) dt, where η ϕ ( e D, t ) = ] ϕ p t ( e D ) , p t ( e D )] ⊗ p t ( e D ) ⊗ m +1 ) . The integral formula in line (3.14) does not converge in general. In the follow-ing, we shall show that the convergence holds whenever e D has a sufficiently largespectral gap at zero. Definition 3.22.
Let h h i be a nontrivial conjugacy class of G . An element ϕ ∈ C n ( C G, h h i ) is said to have exponential growth if there exist C and K ϕ > | ϕ ( g , g , · · · , g n ) | Ce K ϕ · ( ℓ ( g )+ ℓ ( g )+ ··· + ℓ ( g n )) (3.17)for ∀ ( g , g , · · · , g n ) ∈ G n +1 .Similar to the definition of τ h h i in line (3.4), we define τ to be the followingpositive number τ = lim inf ℓ ( g ) →∞ (cid:16) inf x ∈F dist( x, gx ) ℓ ( g ) (cid:17) . (3.18)Since G is finitely generated, there exist C and K G > { g ∈ G : ℓ ( g ) = n } Ce K G · n . (3.19) Definition 3.23.
We define σ ϕ =: 2( K G + K ϕ ) · c D τ . (3.20)where c D is the propagation speed of D as defined in line (3.5). Theorem 3.24.
Suppose that ϕ ∈ C n ( C G, h h i ) has exponential growth. Withthe same notation from above, if the spectral gap of e D at zero is larger than σ ϕ given in line (3.20) above, then the delocalized higher eta invariant η ϕ ( e D ) givenin line (3.14) converges absolutely. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 23
Proof.
The proof is divided into three steps. First we show that η ϕ ( e D, t ) is well-defined for any fixed t > t converges absolutely (Proposition 3.27). The last step is to show theconvergence of the integral for large t (Proposition 3.30). In fact, only the laststep requires the spectral gap of e D to be larger than σ ϕ . (cid:3) We say G (resp. ϕ ) has sub-exponential growth if we may choose K G in line(3.19) (resp. K ϕ in line (3.17)) to be arbitrarily small. The following corollary isan immediate consequence of Theorem 3 .
24 above.
Corollary 3.25.
With the same notation from above, if both G and ϕ havesub-exponential growth, then the delocalized higher eta invariant η ϕ ( e D ) given inline (3.14) converges absolutely, as long as e D has a spectral gap at zero. Proposition 3.26.
Suppose that h h i be a nontrivial conjugacy class of G and ϕ ∈ C n ( C G, h h i ) has exponential growth. If N > dim M +5 and Λ is sufficientlylarge, then there exists C > such that for any f i ∈ A Λ ,N ( i = 0 , , · · · , n ), ϕ f ( e D ) ⊗ · · · ⊗ f n ( e D )) C k f k A · · · k f n k A . Proof.
Fix a symmetric generating set S of G . Let ℓ be the length function on G determined by S . Denote the cardinality of S by | S | .For each 0 i n , let w i = f i ( e D ) for some function f i ∈ A Λ ,N . Let us write w i as a formal sum w i = P g w gi g . If we denote by k i ( x, y ) the Schwartz kernel of w i , thentr( w g · · · w g n n ) = Z F n +1 tr( k ( x , g x ) · · · k n ( x n , g n x )) dx · · · dx n . (3.21)It follows from Lemma 3.5 and the definition of k · k A (cf. Definition 3 .
9) thatthere exists a constant C such that k k i ( x, y ) k C k f i k A for 0 i n. For any ( g , g , · · · , g n ) ∈ G n +1 , we divide F n +1 into ( n + 1) disjoint (possiblyempty) Borel sets F n +1( j ) , ( g ,g , ··· ,g n ) such that dist( x j , g j x j +1 ) is the maximum of theset { dist( x i , g i x i +1 ) } ≤ i ≤ n , where x n +1 = x . In other words,dist( x j , g j x j +1 ) > dist( x i , g i x i +1 ) on F n +1( j ) , ( g ,g , ··· ,g n ) for all 0 i n. If no confusion is likely to arise, we shall write F n +1( j ) in place of F n +1( j ) , ( g ,g , ··· ,g n ) .Since the action of G on f M is free and cocompact, there exist C , C such thatdist( x, gy ) > C ℓ ( g ) − C , for ∀ x, y ∈ F . It follows that dist( x j , g j x j +1 ) > C P ni =0 ℓ ( g i ) n + 1 − C on F n +1( j ) . Again, as the action of G on f M is free and cocompact, there exists ε > x, gx ) > ε for all x ∈ f M and all g = e . Note that, since the metric on f M is G -equivariant, we have n X i =0 dist( x i , g i x i +1 ) > dist( x , g g · · · g n x ) . To summarize, we havedist( x j , g j x j +1 ) > max (cid:8) εn + 1 , C P ni =0 ℓ ( g i ) n + 1 − C (cid:9) , on F n +1( j ) . It follows from Lemma 3.8 that, for any g g · · · g n ∈ h h i , there exist C > C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z F n +1( j ) tr( k ( x , g x ) · · · k n ( x n , g n x )) dx · · · dx n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C · F f j (cid:16) C · max (cid:8) εn + 1 , C P ni =0 ℓ ( g i ) n + 1 − C (cid:9)(cid:17) · Y i = j k f i k A . Since ϕ has exponential growth, there exist C, C ′ , K ϕ > | ϕ w ⊗ w ⊗ · · · ⊗ w n ) | C X g g ··· g n ∈h h i e K ϕ P ni =0 ℓ ( g i ) · | tr( w g · · · w g n n ) | C X g g ··· g n ∈h h i e K ϕ P ni =0 ℓ ( g i ) · (cid:16) n X j =0 Z F n +1( j ) | tr( k ( x , g x ) · · · k n ( x n , g n x )) | dx · · · dx n (cid:17) CC ∞ X m =1 e K ϕ · m · | S | ( n +1) m n X j =0 h F f j (cid:16) C · max (cid:8) εn + 1 , mC n + 1 − C (cid:9)(cid:17) · Y i = j k f i k A i! C ′ C ∞ X m =1 e K ϕ · m · | S | ( n +1) m exp (cid:16) − Λ2 C · max (cid:8) εn + 1 , mC n + 1 − C (cid:9)(cid:17) n Y i =0 k f i k A , where the last summation converges for sufficiently large Λ by Lemma 3 .
10. Thisfinishes the proof. (cid:3)
Let us now prove the convergence of the integral in line (3.14) for small t . Proposition 3.27.
Suppose that h h i be a nontrivial conjugacy class of G and ϕ ∈ C n ( C G, h h i ) has exponential growth. If N > dim M +5 and Λ is sufficientlylarge, then the following integral Z t k ϕ f ( t e D ) ⊗ · · · ⊗ f n ( t e D )) dt is absolutely convergent for all f i ∈ A Λ ,N and any k ∈ R . ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 25
Proof.
Define f j,t ( x ) = f j ( tx ) for j = 0 , , · · · , n . From the definition of k · k A (cf. Definition 3.9), we have k f j,t k A t − N k f j k A for all t ∈ (0 ,
1] and f j ∈ A Λ ,N . Recall the inequality in line (3.10): F f j,t ( s ) F f j ( s · t − ) t dim M +3 C Λ t dim M +3 · k f j k A · exp (cid:16) − Λ s t (cid:17) . Similar to the proof of Proposition 3 .
26, there exist positive constants K ϕ , C , C ,C , C , C , C and C such that | t k ϕ f ( t e D ) ⊗ · · · ⊗ f n ( t e D )) | t k C ∞ X m =1 e K ϕ · m · | S | ( n +1) m n X j =0 h F f j,t (cid:16) C · max (cid:8) εn + 1 , mC n + 1 − C (cid:9)(cid:17) · Y i = j k f i,t k A i! t k − ( dim M +3) n − nN C n Y i =0 k f i k A ∞ X m =1 e mC exp (cid:18) − mC · Λ2 t (cid:19) , which proves the proposition as long as Λ is sufficiently large. (cid:3) To prove the convergence of the integral in line (3.14) for large t , we need tofix some notation. Definition 3.28.
For any
K >
0, let L K be the subspace of K ⊗ C ∗ r ( G ) consistingof operators A = P A g g such that(1) for any g ∈ G , A g is of trace class;(2) and we have X g ∈ G e K · ℓ ( g ) | A g | < ∞ , where | · | is the trace norm.Equip L K with the following norm k A k L = X g ∈ G e K · ℓ ( g ) | A g | . Lemma 3.29.
The space L K is a Banach algebra with the norm k · k L .Proof. It is not difficult to see that L K is a Banach space under the norm k · k L . Itremains to show that k · k L is sub-multiplicative. Indeed, given A = P g ∈ G A ,g g and A = P g ∈ G A ,g g in L K , we have k A A k L X g ∈ G e K · ℓ ( g ) X g ∈ G | A ,gg | · | A ,g − | = X g ∈ G X g ∈ G e K · ( ℓ ( g g ) − ℓ ( g ) − ℓ ( g )) (cid:0) e K · ℓ ( g ) | A ,g | (cid:1) (cid:0) e K · ℓ ( g ) | A ,g | (cid:1) k A k L · k A k L . (cid:3) Now we prove the convergence of the integral in (3.14) for large t . Proposition 3.30.
Suppose that ϕ ∈ C m ( C G ) has exponential growth. If thespectral gap of e D at zero is larger than σ ϕ given in line (3.20) , then the integral Z ∞ η ϕ ( e D, t ) dt is absolutely convergent.Proof. The case where e D has a spectral gap at zero can be deduced from theinvertible case by replacing e D with its restriction to the orthogonal complementof the kernel of e D . Without loss of generality, let us assume e D is invertible.Since ϕ has exponential growth, there exist C and K ϕ such that | ϕ ( g , g , · · · , g n ) | Ce K ϕ ( ℓ ( g )+ ℓ ( g )+ ··· + ℓ ( g n )) for ∀ ( g , g , · · · , g n ) ∈ G n +1 . We first show that ˙ u t ( e D ) u − t ( e D ), u t ( e D ) − u − t ( e D ) − L K ϕ , where L K ϕ is given in Definition 3.28 above.Let σ be the spectral gap of e D at zero. Note that˙ u t ( e D ) u − t ( e D ) = 2 √ πi e De − t e D . Since σ > σ ϕ = K G + K ϕ ) · c D τ , the same proof of Proposition 3.13 shows that˙ u t ( e D ) u − t ( e D ) lies in L K ϕ and furthermore there exists sufficiently small ω > k ˙ u t ( e D ) u − t ( e D ) k L √ πe − ωt . (3.22)Observe that u t ( e D ) − (cid:18) − Z + ∞ t ˙ u s ( e D ) u − s ( e D ) ds (cid:19) − . By the inequality in line (3.22) above, we have Z + ∞ t k ˙ u s ( e D ) u − s ( e D ) k L ds πe − ωt √ ω . Thus for any t >
1, we have Z + ∞ t ˙ u s ( e D ) u − s ( e D ) ds ∈ L K ϕ and (cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ t ˙ u s ( e D ) u − s ( e D ) ds (cid:13)(cid:13)(cid:13)(cid:13) L πe − ωt √ ω . By Lemma 3 . L K ϕ is a Banach algebra under the norm k · k L . It follows that u t ( e D ) − (cid:18) − Z + ∞ t ˙ u s ( e D ) u − s ( e D ) ds (cid:19) − ∈ L K ϕ . Here ϕ is not necessarily delocalized. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 27
Moreover, we have k u t ( e D ) − k L ∞ X n =1 n ! (cid:18)(cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ t ˙ u s ( e D ) u − s ( e D ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (cid:19) n e C · e − ωt − , where C = π √ ω . A similar estimate holds for u − t ( e D ) − | η ϕ ( e D, t ) | C · k ˙ u t ( e D ) u − t ( e D ) k L · k u t ( e D ) − k m L · k u − t ( e D ) − k m L . It follows that | η ϕ ( e D, t ) | √ πe − ωt ( e C · e − ωt − m , where the right hand side is clearly integrable over [1 , ∞ ). This finishes theproof. (cid:3) The following proposition shows that the delocalized higher eta invariant inDefinition 3 .
17 is independent of the choice of representative within the samecyclic cohomology class.
Proposition 3.31.
Let h h i be a nontrivial conjugacy class of G . Suppose that ϕ and ϕ are two cyclic cocycles in C m ( C G, h h i ) with exponential growth and ϕ and ϕ are cohomologous via a cochain with exponential growth. If e D has asufficiently large spectral gap at zero, then η ϕ ( e D ) = η ϕ ( e D ) .Proof. It suffices to show that if ϕ ∈ C m − ( C G, h h i ) has exponential growth,then η bϕ ( e D ) = 0. By the definition of ] ϕ w j = 1 for some j , then ] ϕ w ⊗ w ⊗ w ⊗ · · · ⊗ w n ) = 0 , For notational simplicity, let us write u t in place of u t ( e D ) in the following.Observe that ddt ] ϕ u t ⊗ u − t ) ⊗ m ) = m ] ϕ u t ⊗ u − t ⊗ ( u t ⊗ u − t ) ⊗ m − ) − m ] ϕ u t ⊗ u − t ˙ u t u − t ⊗ ( u t ⊗ u − t ) ⊗ m − ) , and( ^ bϕ u t u − t ⊗ ( u t ⊗ u − t ) ⊗ m ) = ] ϕ u t ⊗ u − t ⊗ ( u t ⊗ u − t ) ⊗ m − ) − ] ϕ u t ⊗ u − t ˙ u t u − t ⊗ ( u t ⊗ u − t ) ⊗ m − ) . It follows that ddt ] ϕ u t ⊗ u − t ) ⊗ m ) = m · η bϕ ( e D, t ) . (3.23)In particular, we have Z t t η bϕ ( e D, t ) dt = ] ϕ u t ⊗ u − t ) ⊗ m ) (cid:12)(cid:12) t = t − ] ϕ u t ⊗ u − t ) ⊗ m ) (cid:12)(cid:12) t = t = ϕ u t − ⊗ ( u − t − ⊗ m ) (cid:12)(cid:12) t = t − ϕ u t − ⊗ ( u − t − ⊗ m ) (cid:12)(cid:12) t = t . By the proof of Proposition 3.27, we havelim t → ϕ u t − ⊗ ( u − t − ⊗ m ) = 0 . Furthermore, it follows from the proof of Proposition 3.30 thatlim t →∞ ϕ u t − ⊗ ( u − t − ⊗ m ) = 0 , as long as the spectral gap of e D is sufficiently large. This finishes the proof. (cid:3) In the remaining part of this section, we show that the delocalized highereta invariant is stable under Connes’ periodicity map. First, let us recall thedefinition of Connes’ periodicity map (cf. [12, Page 121]): S : HC m ( C G ) → HC m +2 ( C G )First, let us fix some notation. For 0 i n , we define b i : L n ( C G ) → L n +1 ( C G )by b i f ( g , g , · · · , g n +1 ) = f ( g , g , · · · , g j g j +1 , · · · , g n +1 ) , and for i = n + 1, we define b n +1 : L n ( C G ) → L n +1 ( C G ) by b n +1 f ( g , g , · · · , g n +1 ) = ( − n +1 f ( g n +1 g , g , · · · , g n ); Definition 3.32.
Connes’ periodicity map S : C n ( C G ) → C n +2 ( C G ) is definedto be S := − n + 2)( n + 1) X i Let h h i be a nontrivial conjugacy class of G . If a cycliccocycle ϕ ∈ C m ( C G, h h i ) has exponential growth and the spectral gap of e D atzero is sufficiently large, then η Sϕ ( e D ) = η ϕ ( e D ) .Proof. We will prove the proposition by a direct computation. It is much easier tofollow Connes and carry out the computation in the context of universal gradeddifferential algebras [12, Part II]. Let us recall the construction of the universalgraded differential algebra Ω( A ) associated to an algebra A . Denote by e A thealgebra obtained from A by adjoining a unit: e A = { a + λI | a ∈ A , λ ∈ C } .Let Ω ( A ) = A and Ω n ( A ) = e A ⊗ ( A ) ⊗ n ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 29 for n ≥ 1. The differential d ¯: Ω n ( A ) → Ω n +1 ( A ) is given by d ¯ (cid:0) ( a + λI ) ⊗ a ⊗ · · · ⊗ a n ) = I ⊗ a ⊗ a ⊗ · · · ⊗ a n . Clearly, one has d = 0. The product structure (cf. [12, Part II, Proposition 1])on Ω ∗ ( A ) is defined so that the following are satisfied:(1) d ¯( ω ω ) = ( d ¯ ω ) ω +( − | ω | ω d ¯ ω for ω ∈ Ω i and ω ∈ Ω j , where | ω | = i is the degree of ω ;(2) ˜ a d ¯ a d ¯ a · · · d ¯ a n = ˜ a ⊗ a ⊗ a ⊗ · · · ⊗ a n in Ω n ( A ).An ( n + 1)-linear functional ϕ on A induces a linear functional b ϕ on Ω n ( A ) bysetting b ϕ (cid:0) ( a + λI ) da · · · da n (cid:1) = ϕ ( a , a , · · · , a n ) . By [12, Part II, Proposition 1], since ϕ is a cyclic cocycle, b ϕ is a closed gradedtrace on Ω n ( A ). In particular, we have b ϕ ( ω ω ) = ( − | ω |·| ω | b ϕ ( ω ω ) . By using the equality d ¯( ω ω ) = ( d ¯ ω ) ω +( − | ω | ω d ¯ ω above, a straightforwardcalculation shows that the formula for the periodicity operator S becomes thefollowing (compare with [12, Part II, Corollary 10]): c Sϕ ( a d ¯ a · · · d ¯ a n +2 ) = c n n +1 X j =1 b ϕ ( a d ¯ a · · · d ¯ a j − ( a j a j +1 ) d ¯ a j +2 · · · d ¯ a n +2 ) , (3.24)where c n = 1( n + 2)( n + 1) . Now we shall prove the proposition. Let ϕ ∈ C m ( C G, h h i ) be a cyclic cocyclewith exponential growth, then Sϕ also has exponential growth. Recall that thedelocalized higher eta invariant of e D with respect to ϕ to be η ϕ ( e D ) = m ! πi Z ∞ ϕ u t u − t ⊗ (( u t − I ) ⊗ ( u − t − I )) ⊗ m ) dt, where u t = u t ( e D ) as in line (3.13). For notational simplicity, let us write ϕ ( ˙ u t u − t ⊗ (( u t − I ) ⊗ ( u − t − I )) ⊗ m )in place of ϕ u t u − t ⊗ (( u t − I ) ⊗ ( u − t − I )) ⊗ m ), and furthermore write a = ˙ u t u − t and for j ≥ a j = ( u t − I if j is odd, u − t − I if j is even.By the above discussion, we have b ϕ ( a da · · · da m ) = ϕ ( ˙ u t u − t ⊗ (( u t − ⊗ ( u − t − ⊗ m ) . The following observations will be useful in the computation below.(1) ( u t − I )( u − t − I ) = 2 I − u t − u − t ; Here we us the notation u − t − I for the corresponding term u − t − . 17. Thisis to emphasis that I is the identity operator, which is the unit adjoined. (2) d ¯( u t − I ) = d ¯ u t and d ¯( u − t − I ) = d ¯ u − t ;(3) ( d ¯ u t ) u − t = − u t ( d ¯ u − t ) and ( d ¯ u − t ) u t = − u − t ( d ¯ u t ) . Observation (1) immediately implies that b ϕ ( a d ¯ a · · · d ¯ a j − ( a j a j +1 ) d ¯ a j +2 · · · d ¯ a m +2 )=2 b ϕ ( a d ¯ a · · · d ¯ a j − d ¯ a j +2 · · · d ¯ a m +2 ) − b ϕ ( a d ¯ a · · · d ¯ a j − ( u t ) d ¯ a j +2 · · · d ¯ a m +2 ) − b ϕ ( a d ¯ a · · · d ¯ a j − ( u − t ) d ¯ a j +2 · · · d ¯ a m +2 ) . Now observations (2) and (3) imply the following:(i) if j is odd, then b ϕ ( a d ¯ a · · · d ¯ a j − ( u t ) d ¯ a j +2 · · · d ¯ a m +2 )= b ϕ ( ˙ u t d ¯ u − t d ¯ u t · · · d ¯ u − t d ¯ u t | {z } ( j − 1) terms d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t | {z } (2 m − j + 1) terms ) , and b ϕ ( a d ¯ a · · · d ¯ a j − ( u − t ) d ¯ a j +2 · · · d ¯ a m +2 )= − b ϕ ( ˙ u − t d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t | {z } ( j − 1) terms d ¯ u − t d ¯ u t · · · d ¯ u − t d ¯ u t | {z } (2 m − j + 1) terms );(ii) if j is even, then b ϕ ( a d ¯ a · · · d ¯ a j − ( u t ) d ¯ a j +2 · · · d ¯ a m +2 )= b ϕ ( ˙ u − t d ¯ u t d ¯ u − t · · · d ¯ u − t d ¯ u t | {z } ( j − 1) terms d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t d ¯ u t | {z } (2 m − j + 1) terms ) , and b ϕ ( a d ¯ a · · · d ¯ a j − ( u − t ) d ¯ a j +2 · · · d ¯ a m +2 )= − b ϕ ( ˙ u t d ¯ u − t d ¯ u t · · · d ¯ u t d ¯ u − t | {z } ( j − 1) terms d ¯ u − t d ¯ u t · · · d ¯ u − t d ¯ u t d ¯ u − t | {z } (2 m − j + 1) terms ) . Since b ϕ is a closed graded trace, it follows that m +1 X j =1 b ϕ ( a d ¯ a · · · d ¯ a j − ( u t ) d ¯ a j +2 · · · d ¯ a m +2 ) = ddt b ϕ (cid:0) ( u t − d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t | {z } m terms (cid:1) and m +1 X j =1 b ϕ ( a d ¯ a · · · d ¯ a j − ( u − t ) d ¯ a j +2 · · · d ¯ a m +2 ) = ddt b ϕ (cid:0) ( u − t − d ¯ u − t d ¯ u t · · · d ¯ u − t d ¯ u t | {z } m terms (cid:1) ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 31 On the other hand, by the proof of Proposition 3.27 and Proposition 3.30, wehave Z ∞ ddt b ϕ (cid:0) ( u t − d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t | {z } m terms (cid:1) dt = lim t →∞ ϕ (cid:0) ( u t − ⊗ (( u t − ⊗ ( u − t − ⊗ m (cid:1) − lim t → ϕ (cid:0) ( u t − ⊗ (( u t − ⊗ ( u − t − ⊗ m (cid:1) =0Similarly, we have Z ∞ ddt b ϕ (cid:0) ( u − t − d ¯ u − t d ¯ u t · · · d ¯ u − t d ¯ u t | {z } m terms (cid:1) dt = 0 . To summarize, we have Z ∞ c Sϕ ( ˙ u t u − t d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t | {z } (2 m + 2) terms ) dt = 1 m + 1 Z ∞ b ϕ ( ˙ u t u − t d ¯ u t d ¯ u − t · · · d ¯ u t d ¯ u − t | {z } m terms ) dt, which implies that η Sϕ ( e D ) = η ϕ ( e D ) . This finishes the proof. (cid:3) A higher Atiyah-Patodi-Singer index formula. In this subsection, foreach cyclic cocycle of at most exponential growth, we prove a formal higherAtiyah-Patodi-Singer index theorem (abbr. higher APS index theorem) on man-ifolds with boundary, under the condition that the operator on the boundary hasa sufficiently large spectral gap at zero. We point out that there is no conditionon the fundamental group in this formal higher APS index theorem.Leichtnam and Piazza proved a higher APS index theorem in terms of noncom-mutative differential forms on a certain smooth dense subalgebra of the reduced C ∗ -algebra of the fundamental group [37, Theorem 4.1]. Heuristically speaking,our version of higher APS theorem is the pairing between their version of higherAPS theorem and cyclic cocycles of the fundamental group. However, this is notthe approach we take in this section. In fact, in general it is rather difficult tomake this heuristic argument rigorous. A main difficulty here is whether cycliccocycles of a group algebra extends continuously to cyclic cocycles on a givensmooth dense subalgebra of the reduced group C ∗ -algebra. In this section, ourapproach is based on the convergence results from the previous sections, andavoids the subtle issue of continuous extension of cyclic cocylces. On the otherhand, in order to apply the higher APS index theorem in this subsection to prob-lems in geometry and topology (cf. [47, 57, 54]), one actually needs to extend thepairing to be defined at the level of (periodic) cyclic cohomology and K -theoryof C ∗ -algebras. In later sections, we shall use Puschnigg’s smooth dense subal-gebra to define such a pairing at the level of (periodic) cyclic cohomology and K -theory of C ∗ -algebras for all hyperbolic groups. As a consequence, in the caseof hyperbolic groups, we shall prove a higher APS index theorem without theassumption that the operator on the boundary has a sufficiently large spectralgap at zero (cf. Section 7).For notational simplicity, we shall only discuss the case of even dimensionalspin manifolds. The exact same strategy clearly works for the more general caseof Dirac-type operators acting on Clifford modules over Riemannian manifolds ofall dimensions.Let W be an even dimensional compact spin manifold with boundary M , and D the Dirac operator on W . Suppose the metric of W is a product metric whenrestricted to the boundary M . Let G be a finitely presented discrete group and f W a regular G -covering space of W . Let e D be the lift of D to f W and e D ∂ therestriction e D to the boundary of f W .Let us briefly review Lott’s noncommutative differential higher eta invariant.We shall follow closely the notation in Lott’s paper [38]. For each q > 0, wedefine B ωq to be the following dense subalgebra of C ∗ r ( G ): B ωq = (cid:8) f : G → C | X g ∈ G e q · ℓ ( g ) | f ( g ) | < ∞ (cid:9) , where ℓ is a word-length function on G . Note that B ωq is generally not closedunder holomorphic functional calculus in C ∗ r ( G ). The universal graded differentialalgebra of B ωq is Ω ∗ ( B ωq ) = ∞ M k =0 Ω k ( B ωq )where as a vector space, Ω k ( B ωq ) = B ωq ⊗ ( B ωq / C ) ⊗ k . As B ωq is a Banach algebra(cf. Lemma 3 . 29 above), we consider the Banach completion of Ω ∗ ( B ωq ), whichwill still be denoted by Ω ∗ ( B ωq ).Let S be the restriction of spinor bundle of W on M . We denote the corre-sponding B ωq -vector bundle by S = ( f M × G B ωq ) ⊗ S and the space of smoothsections by C ∞ ( M ; S ). Now suppose ψ is a smooth function on f M with comapctsupport such that X g ∈ G g ∗ ψ = 1 . Then we have a superconnection ∇ : C ∞ ( M ; S ) → C ∞ ( M ; S ⊗ B ωq Ω ( B ωq )) givenby ∇ ( f ) = X g ∈ G ( ψ · g ∗ f ) ⊗ B ωq dg. See [49] for more details of the superconnection formalism. Definition 3.34 ([38, Section 4.4 & 4.6]) . Lott’s higher eta invariant e η ( e D ∂ ) isdefined by the formula e η ( e D ∂ ) = Z ∞ STR( e De − ( t e D ∂ + ∇ ) ) dt, ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 33 where STR is the corresponding supertrace, cf. [38, Proposition 22]. Remark . The above integral formula for e η ( e D ∂ ) generally does not converge todefine an element in Ω ∗ ( B ωq ). On the other hand, the estimates in the previoussubsections show that the above integral converges absolutely and defines anelement in Ω ∗ ( B ωq ), provided the spectral gap of e D ∂ at zero is sufficiently large.Now let W ∞ be the complete Riemannian manifold obtained by attaching aninfinite cylinder M × [0 , ∞ ) to W . We still denote the associated Dirac operatoron W ∞ by D and its lift to f W ∞ by e D . Combining Quillen’s superconnectionformalism with Melrose’s b -calculus formalism [41], for each t > 0, one definesthe b -Connes-Chern character of e D on f W ∞ to be b -Ch t ( e D ) = b -STR( e − ( t e D + ∇ ) ) ∈ Ω ∗ ( B ωq ) , where b -STR is the corresponding b -supertrace in this b -calculus setting. See forexample [36] for more details. Theorem 3.36. Assume that ϕ ∈ C m ( C G ) is a cyclic cocycle with exponentialgrowth. Let σ ϕ be the positive number from Definition . . If the spectral gapof e D ∂ at zero is larger than σ ϕ , then the pairing (cid:10) ϕ, e η ( e D ∂ ) (cid:11) converges, the limit lim t →∞ (cid:10) ϕ, b -Ch t ( e D )) (cid:11) exists, and furthermore lim t →∞ (cid:10) ϕ, b -Ch t ( e D )) (cid:11) = (cid:10) ϕ, Z W ˆ A ∧ ω (cid:11) − (cid:10) ϕ, e η ( e D ∂ ) (cid:11) . (3.25) where ˆ A is the associated ˆ A -form on W and ω is an element in Ω ∗ ( W ) ⊗ Ω ∗ ( B ωq ) for some q > , ( cf. [36, Theorem 13.6]) . In particular, if both G and ϕ havesub-exponential growth, then the equality in line (3.25) holds as long as e D ∂ isinvertible.Proof. By Proposition 3.26 and Proposition 3.27, we have that, for any ε > µ > k ∈ N , there exist C, N , N such that k ( e D k e − t e D )( x, y ) k C d ( x, y ) N t N exp (cid:16) − d ( x, y ) µ c D t (cid:17) , (3.26)for all x, y ∈ f W ∞ with d ( x, y ) > ε , and k ( e D k∂ e − t e D ∂ )( x, y ) k C d ( x, y ) N t N exp (cid:16) − d ( x, y ) µ c D ∂ t (cid:17) . (3.27)for all x, y ∈ f M with d ( x, y ) > ε , where c D (resp. c D ∂ ) is the propagation speedof D (resp. D ∂ ), cf. line (3.5).Since the spectral gap of e D ∂ is larger than σ ϕ = 2( K G + K ϕ ) · c D ∂ τ , the proofof Proposition 3 . 13 shows that for each k ∈ N , the operator e D k∂ e − t e D ∂ lies in L K ϕ It is not difficult to adapt the estimates from the previous subsections to the b -calculussetting and show that b -tr s ( e − ( t e D + ∇ ) ) indeed defines an element in Ω ∗ ( B ωq ). (cf. Definition 3 . 28) and furthermore there exists sufficiently small ω > k e D k∂ e − t e D ∂ k L √ πe − ωt . Now apply the commutator formula for b -trace (cf. [41, (In.22) on Page 8]),and a straightforward calculation shows that for any 0 < t < t , the equality b -Ch t ( e D ) − b -Ch t ( e D )= − Z t t STR( e D ∂ e − ( s e D ∂ + ∇ ) ) ds + d Z t t b -STR( e De − ( ∇ + s e D ) ) ds (3.28)holds in Ω ∗ ( B ωq ) with q = K ϕ , where d : Ω ∗ ( B ωq ) → Ω ∗ +1 ( B ωq ) is the differentialon Ω ∗ ( B ωq ), cf. [21, Section 6][36, Proposition 14.2]. In particular, by pairingboth sides of (3.28) with ϕ , we have (cid:10) ϕ, b -Ch t ( e D ) (cid:11) − (cid:10) ϕ, , b -Ch t ( e D ) (cid:11) = − Z t t (cid:10) ϕ, STR( e D ∂ e − ( s e D ∂ + ∇ ) ) (cid:11) ds. Let us write ϕ = ϕ e + ϕ d , where ϕ d is the delocalized part of ϕ , i.e., ϕ d ( g , g , · · · , g m ) = ( ϕ ( g , g , · · · , g m ) if g g · · · g m = e, . 26, combined with Getzler’ssymbol calculus (cf. [20]), shows thatlim t → (cid:10) ϕ e , b -Ch t ( e D ) (cid:11) = Z W ˆ A ∧ ω, where ω is an element in Ω ∗ ( W ) ⊗ Ω ∗ ( B ωq ), cf. [36, Theorem 13.6]. Moreover, itfollows from the inequality in line (3.26) thatlim t → (cid:10) ϕ d , b -Ch t ( e D ) (cid:11) = 0 . Therefore, as t → 0, we obtain the following formula: (cid:10) ϕ, b -Ch t ( e D ) (cid:11) − (cid:10) ϕ, Z W ˆ A ∧ ω (cid:11) = − Z t (cid:10) ϕ, STR( e D ∂ e − ( s e D ∂ + ∇ ) ) (cid:11) ds. Now it follows from the discussion above that the integral on the right hand sideconverges absolutely as t → ∞ , under the condition that the spectral gap of e D ∂ is larger than σ ϕ . This finishes the proof. (cid:3) Remark . Formally speaking, the term lim t →∞ (cid:10) ϕ, b -Ch t ( e D )) (cid:11) represents thepairing between the higher index class of e D and the cyclic cocycle ϕ . However,to make this formal assertion rigorous, one needs to extend the the pairing in(3.25) from B ωq to a smooth dense subalgebra of C ∗ r ( G ), which is a rather subtle In general, there is no natural way to define the higher index class of a Dirac operator on amanifold with boundary. However, in our setup above, due to the invertibility of the operator e D ∂ on the boundary, there is a natural higher index class associated to e D . ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 35 issue in general. In the remaining sections below, we will show the existence ofsuch an extension of the pairing, in the special case where G is hyperbolic.4. Puschnigg smooth dense subalgebra for hyperbolic Groups In this section, we review the construction of Puschnigg’s smooth dense algebraof C ∗ r ( G ) for hyperbolic groups [48]. One particular feature is that every trace on C G admits a continuous extension to of this Puschnigg smooth dense subalgebra,cf. [48, Theorem 5.2]. We shall generalize this extension result to cyclic cocyclesof all degrees.4.1. Unconditional seminorms and tensor products. In this subsection,we review the construction of Puschnigg’s smooth dense subalgebra of C ∗ r ( G ) forhyperbolic groups G [48].Let X be a set and R a normed algebra equipped with a sub-multiplicativenorm | · | . We denote by RX the algebra consisting of all finitely supportedfunctions on X with values in R . For each element A = P A x x ∈ RX , we defineits absolute value to be | A | = X | A x | x ∈ C X. Define a partial order on elements in RX by A A ′ ⇐⇒ | A x | | A ′ x | for ∀ x ∈ X. Recall the following notion of unconditional seminorm due to Bost and Lafforgue(cf. [34]). Definition 4.1. A seminorm k · k on RX is called unconditional if | A | | A ′ | = ⇒ k A k k A ′ k , for ∀ A, A ′ ∈ RX. Any seminorm k · k on RX naturally determines an unconditional seminorm k · k + by k A k + := inf | A ′ | > | A | k| A ′ |k . (4.1) Lemma 4.2 ([48, Lemma 2.3]) . Let X, Y be two sets and k·k X , k·k Y be seminormson RX and RY respectively. Let ϕ : ( RX, k · k X ) → ( RY, k · k Y ) be a boundedlinear map. Assume that ϕ is expressed by a positive integral kernel, that is ϕ ( X A y y )( x ) = X y ∈ Y ϕ x,y A y y, (4.2) where ϕ x,y ∈ R > , for ∀ x ∈ X, ∀ y ∈ Y . Then ϕ is also bounded with respect tothe corresponding unconditional seminorms k · k + X , k · k + Y , and k ϕ k + k ϕ k . (4.3)Now we recall the notion of unconditional tensor product seminorm. Definition 4.3. Let X, Y be sets and k · k X , k · k Y be unconditional seminormson RX and R ′ Y respectively. Let R ⊗ R ′ be the algebraic tensor product of R and R ′ equipped with the projective seminorm. The unconditional tensor productseminorm k·k uc on RX ⊗ R ′ Y ∼ = ( R ⊗ R ′ )( X × Y ) is defined to be the unconditionalnorm determined by this projective seminorm. More precisely, k · k uc is given by k A k uc := inf | A | P | A ′ i |⊗| A ′′ i | X i k A ′ i k X k A ′′ i k Y , for ∀ A ∈ RX ⊗ R ′ Y. This norm is less than or equal to projective seminorm over RX ⊗ R ′ Y . Anexample where these two are not equal is given in [48, Example 2.4].Given a finitely generated group G , we fix a symmetric generating set S of G . Let ℓ be the corresponding word metric on G . In the following, let S be thecollection of all trace class operators equipped with the trace norm | · | . Let S G be the subalgebra of K ⊗ C ∗ r ( G ) consisting of all finite sums P A g g with A g ∈ S . Definition 4.4. For any fixed p ≥ 1, we define an unconditional norm k · k RD,p on S G by k A k RD,p = X g | A g | (1 + ℓ ( g )) p , (4.4)for A = P g A g g ∈ S G .We denote the completion of S G with respect to k · k RD,p by RD p ( S G ). Simi-larly, the same formula also defines an unconditional norm k · k RD,p on C G . Wedenote the completion of C G under this norm by RD p ( G ). In the following, if noconfusion is likely to arise, we shall omit p from the notation.Let us assume G is hyperbolic for the rest of this section. In this case, itis known that RD ( G ) is a smooth dense subalgebra of C ∗ r ( G ), cf. [29, 17, 33].Similarly, RD ( S G ) is a smooth dense algebra of K ⊗ C ∗ r ( G ).Recall the following quasiderivation map defined by Puschnigg:∆ : S G → S G ⊗ C G ∼ = S ( G × G ) , A g g X g g = gℓ ( g )+ ℓ ( g )= ℓ ( g ) A g g ⊗ g . Definition 4.5. Let k · k B,p be the norm on S G given by k A k B,p := k A k RD,p + k ∆ A k uc , ∀ A ∈ S G. (4.5)Here k · k uc is the unconditional tensor product norm on S G ⊗ C G ∼ = S ( G × G )determined by the unconditional norm k · k RD,p on both S G and C G .Let B p ( S G ) be the completion of S G with respect to k · k B,p . Apply the sameconstruction to C G and we obtain B p ( C G ). If no confusion is likely to arise, weshall omit p from the notation.We define a more flexible quasiderivation as follows. Definition 4.6. For any g ∈ G and q > 0, let C ( q, g ) be the collection of allpairs ( g , g ) ∈ G × G satisfying the following conditions:(1) g g = g , ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 37 (2) there exists a geodesic [ e, g ] connecting the identity e and g in the Cayleygraph of G such that g lies in the q -neighborhood of [ e, g ].We define ∆ q : C G → C G ⊗ C G ∼ = C ( G × G ) , g X ( g ,g ) ∈ C ( q,g ) g ⊗ g . If q = 0, then ∆ agrees with ∆. By definition, for any ( g , g ) ∈ C ( q, g ), thereexists a group element v ∈ G with ℓ ( v ) q such that ℓ ( g v − ) + ℓ ( vg ) = ℓ ( g ).For each v ∈ G , define a map s v : C G ⊗ C G → C G ⊗ C G, g ⊗ g g v ⊗ v − g . Then we have for A ∈ C G ∆ q | A | X ℓ ( v ) q s v ∆ | A | . By Lemma 4.2, the operator norm of s v (with respect to the unconditional norm k · k uc ) does not exceed (1 + ℓ ( v )) . Since the number of elements in G of length q is finite, we see that there exists a constant K q such that k (∆ q | A | ) k uc K q · k (∆ | A | ) k uc for all ∈ C G . Proposition 4.7. [48, Proposition 3.5] If G is a hyperbolic group whose Cayleygraph is δ -hyperbolic, then there exists C > such that k ∆( AA ′ ) k uc C ( k ∆( A ) k uc k A ′ k RD + k A k RD k ∆( A ′ ) k uc ) , for A, A ′ ∈ S G Proof. By the discussion above, it suffices to prove the following pointwise in-equality ∆ | AA ′ | ∆ δ ( | A | )(1 ⊗ | A ′ | ) + ( | A | ⊗ δ ( | A ′ | ) , (4.6)for A, A ′ ∈ S G . Without loss of generality, it suffices to consider the case where A = g and A ′ = g ′ , for g, g ′ ∈ G .Let k = gg ′ . If a term k ⊗ k appears in the summation expression of ∆ | AA ′ | ,then k is a point on the geodesic [ e, k ]. Since the Cayley graph of G is δ -hyperbolic, k lies in the δ -neighborhood of the union of [ e, g ] and [ g, k ]. Eitherthere is a group element g ∈ [ e, g ] such that dist( k , g ) < δ , or there is a groupelement g ∈ [ g, k ] such that dist( k , g ) < δ . We prove the former case; thelatter case is similar. In the former case, we see that the term k ⊗ k − g appearsin the summation expression of ∆ δ ( g ). This implies that the term k ⊗ k =( k ⊗ k − g )(1 ⊗ g ′ ) appears in ∆ δ ( | A | )(1 ⊗ | A ′ | ). This finishes the proof. (cid:3) Remark . The above proof in fact shows that the pointwise inequality in line(4.6) is equivalent to the hyperbolicity of the group. Proposition 4.9. B ( S G ) and B ( C G ) are smooth dense algebras of K ⊗ C ∗ r ( G ) and C ∗ r ( G ) respectively. Proof. We prove the case of B ( C G ); the other case is similar.Since RD ( G ) is a smooth dense subalgebra of C ∗ r ( G ), it suffices to show thatif an element T ∈ B ( C G ) + is invertible in RD ( G ) + , then T is invertible in B ( C G ) + . In fact, it suffices to show that there exists a constant ε > T ∈ B ( C G ) + satisfies k T − k RD < ε , then T is invertiblein B ( C G ) + . Indeed, let S be an element in B ( C G ) + such that S is invertiblein RD ( G ) + with inverse R . Since B ( C G ) + is dense in RD ( G ) + , there exists aninvertible element U of RD ( G ) + such that k U − R k RD < ε · k S k RD . It followsthat SU is invertible in RD ( G ) + and k SU − k RD < ε . Then by our assumption, SU is invertible in B ( C G ) + , which implies S is invertible in B ( C G ) + .Now suppose A ∈ B ( C G ) + such that k A − k RD < min { /C, } , where C isthe same constant as in Proposition 4.7. It follows from Proposition 4.7 that k ∆(( A − n ) k uc n ( C k A − k RD ) n − k ∆( A − k uc . This immediately implies that A − = (1 − (1 − A )) − = P ∞ n =0 (1 − A ) n lies in B ( C G ) + . Therefore B ( C G ) is a smooth dense subalgebra of RD ( G ). (cid:3) Continuous extension of traces. In this subsection, we review Puschnigg’sresult on continuous extension of traces from C G to B ( C G ) for hyperbolic groups[48, Theorem 5.2]. In fact, for the purposes of this paper, we only need a weakerversion of Pushnigg’s theorem, to which we give a slightly different proof. Lemma 4.10 ([48, Lemms 4.1]) . Let G be a group whose Cayley graph is δ -hyperbolic. Given h ∈ G , if g lies in the conjugacy class h h i , then there exist g , g ∈ G such that g g = g , ℓ ( g ) + ℓ ( g ) = ℓ ( g ) and ℓ ( g g ) δ + 6 + 3 ℓ ( h ) .Proof. Suppose h = ugu − for some u ∈ G . In the following, we denote by[ a, b ] a geodesics connecting a, b ∈ G . By hyperbolicity, there exist vertices w ∈ [ e, ug ], u ∈ [ e, u ] and ug ∈ [ u, ug ] such that dist( w, u ) < δ + 1 and d ( w, ug ) < δ + 1. Moreover, there exists v ∈ G such that ugv ∈ [ ug, h ]and dist( w, ugv ) < δ + 1 + ℓ ( h ), since the [ e, ug ] lies entirely in the δ + ℓ ( h )neighborhood of [ ug, h ]. Let u , g , v be elements in G such that u = u u , g = g g and u − = v v . Let us write g g = ( g v )( v − u − )( u g ) . Clearly, we have that ℓ ( g v ) = dist( ug , ug g v ) = dist( ug , ugv ) < δ + 2 + ℓ ( h ) , and ℓ ( u g ) = dist( u , u u g ) = dist( u , ug ) < δ + 2 . Furthermore, observe that v − is a vertex on the geodesic [ e, u ]. It follows that ℓ ( v − u − ) = ℓ ( v u ) = dist( v − , u ) = | ℓ ( v − ) − ℓ ( u ) | . ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 39 Therefore, we have ℓ ( v − u − ) = | ℓ ( v − ) − ℓ ( u ) | = | dist( h, hv − ) − dist( e, u ) | | dist( e, h ) + dist( e, hv − ) − dist( e, u ) | | dist( e, h ) | + | dist( u , hv − ) | < δ + 2 + 2 ℓ ( h ) . This finishes the proof. (cid:3) Theorem 4.11 ([48, Theorem 5.2]) . Let G be a hyperbolic group and B ( C G ) thePuschnigg smooth dense subalgebra of C ∗ r ( G ) . For any conjugacy class h h i of G ,the map tr h h i : C G → C , X g ∈ G a g g X g ∈h h i a g . admits a continuous extension to B ( C G ) .Proof. Define a map µ : C G → C ( G × G ) as follows: if g ∈ h h i , then µ ( g ) := g ⊗ g where ( g , g ) ∈ G × G is a pair of elements as given in Lemma 4.10; if g / ∈ h h i ,define µ ( g ) = 0. Clearly, µ ( | A | ) ∆ | A | . Thus µ admits a continuous extensionfrom B ( C G ) to RD ( G ) ⊗ uc RD ( G ), which we will still denote by µ .By Lemma 4.2, the mapsT : RD ( G ) ⊗ uc RD ( G ) → RD ( G ) ⊗ uc RD ( G ) , g ⊗ g g ⊗ g , M : RD ( G ) ⊗ uc RD ( G ) → RD ( G ) , g ⊗ g g g , are continuous.We define an evaluation map E : RD ( G ) → C as follows:E( g ) = (cid:26) ℓ ( g ) δ + 6 + 3 ℓ ( h ),0 otherwise.Clearly, E is also well-defined and continuous. It follows that the composition B ( C G ) µ −→ RD ( G ) ⊗ uc RD ( G ) T −−→ RD ( G ) ⊗ uc RD ( G ) M −−→ RD ( G ) E −→ C is a continuous extension of tr h h i . This finishes the proof. (cid:3) The previous theorem has the following obvious analogue where the coefficient C is replaced by the algebra of trace class operators S . Proposition 4.12. Let B ( S G ) be the smooth dense subalgebra of K ⊗ C ∗ r ( G ) defined above. For any conjugacy class h h i in G , let tr h h i : S G → C be the tracemap defined by tr h h i ( A ) = X g ∈h h i tr( A g ) , for A = X A g g ∈ S G. Then tr h h i extends to a continuous trace map on B ( S G ) . Continuous extension of higher degree cyclic cochains. In this sub-section, we generalize the continuous extension result for traces to higher degreecyclic cochains. Definition 4.13. Fix a length function ℓ on G . For any ϕ ∈ C n ( C G, h h i ), wesay ϕ has polynomial growth if there exist constants C and k such that | ϕ ( g , g , · · · , g n ) | C n Y i =0 (1 + ℓ ( g i )) k . In Section 5 below, we will show that, when G is hyperbolic, every element in HC n ( C G, h h i ) have a representative with polynomial growth.Denote by ( S G ) ⊗ n +1 the algebraic tensor product of ( n + 1) copies of S G .Recall the unconditional tensor product defined in Definition 4.3. We constructthe unconditional tensor product norms k · k RD and k · k B over ( S G ) n +1 , anddenote their completions by RD ( S G ) ⊗ n +1 and B ( S G ) ⊗ n +1 respectively. Proposition 4.14. Let G be a hyperbolic group whose Cayley graph is δ -hyperbolic.If ϕ ∈ C n ( C G, h h i ) has polynomial growth, then the map ϕ S G ) ⊗ n +1 → C given by A g g ⊗ A g g ⊗ · · · ⊗ A g n g n tr( A g A g · · · A g n ) ϕ ( g , g , · · · , g n ) (4.7) extends continuously to B ( S G ) ⊗ n +1 .Remark . Before we prove the proposition, let us point out that the con-struction of B p ( S G ) involves a choice of some sufficiently large p . In order toextend ϕ B p ( S G ) ⊗ n +1 , we assume that p is sufficiently largeso that it “dominates” the growth rate of ϕ . Hence, strictly speaking, the algebra B ( S G ) ⊗ n +1 may vary for different cyclic cochains. Proof. Suppose that | ϕ ( g , g , · · · , g n ) | C n Y i =0 (1 + ℓ ( g i )) k for ( g , g , · · · , g n ) ∈ G n +1 .Define the following maps:(1) π ϕ : ( S G ) ⊗ n +1 → ( S G ) ⊗ n +1 by A g g ⊗ A g g ⊗ · · · ⊗ A g n g n ϕ ( g , g , · · · , g n ) A g g ⊗ A g g ⊗ · · · ⊗ A g n g n ;(2) M : ( S G ) ⊗ n +1 → S G by A g g ⊗ A g g ⊗ · · · ⊗ A g n g n A g A g · · · A g n g g · · · g n . Clearly, the composition( S G ) ⊗ n +1 π ϕ −−→ ( S G ) ⊗ n +1 M −−→ S G tr h h i −−→ C is exactly the map ϕ ◦ π ϕ extends to acontinuous map B p ( S G ) ⊗ n +1 to B p − k ( S G ). It follows from Lemma 4.2 that themap M ◦ π ϕ extends to a continuous map from RD p ( S G ) ⊗ n +1 to RD p − k ( S G ). ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 41 In the following, let us prove the case where k = 0, that is, ϕ is uniformlybounded over G n +1 . The general case is similar. If ϕ is uniformly bounded, thenfor any A ∈ ( S G ) ⊗ n +1 , we observe that | ∆(M ◦ π ϕ A ) | = ∆( | M ◦ π ϕ A | ) ∆(M | π ϕ A | ) C · ∆(M | A | ) . Define another multiplication map M ′ : C G n +1) → C G by g ⊗ g ′ ⊗ g ⊗ g ′ ⊗ · · · ⊗ g n ⊗ g ′ n g g · · · g n ⊗ g ′ g ′ · · · g ′ n . This is also bounded with respect to the unconditional norm. We claim that∆(M | A | ) M ′ ∆ ⊗ ( n +1) nδ ( | A | ) , ∀ A ∈ C G n +1 . (4.8)Here ∆ nδ is the quasiderivation from Definition 4.6, and ∆ ⊗ ( n +1) nδ stands for thetensor product of ( n + 1)-copies of ∆ nδ from C G n +1 to C ( G ) ( n +1) ∼ = C G n +1) .Assume the claim holds for the moment. Clearly, the map M ′ is boundedwith respect to the unconditional norm. Moreover, by the discussion beforeProposition 4 . 7, there exists a constant K such that k ∆ ⊗ ( n +1) nδ ( | A | ) k uc K k ∆ ⊗ ( n +1) ( | A | ) k uc for all A ∈ C G n +1 . This proves the proposition.Now let us prove the claim. It suffices to prove the inequality (4.8) when | A | = g ⊗ · · · ⊗ g n . Denote g g · · · g n by g . Suppose g ′ ⊗ g ′′ appears on theleft-hand side of the inequality (4.8), where by definition g ′ is a point on thegeodesic [ e, g ]. We will show that g ′ ⊗ g ′′ also appears on the right-hand sideof the inequality (4.8). Indeed, by hyperbolicity, there exists a point x on thepath [ e, g ] , [ g , g g ] , · · · , [ g g · · · g n − , g ] such that the distance from x to g ′ isless than nδ . More precisely, there exist j > v, v ′ ∈ G such that vv ′ = g j , ℓ ( v ) + ℓ ( v ′ ) = ℓ ( g j ) and d ( g ′ , g g · · · g j − v ) = d ( v, ( g g · · · g j − ) − g ′ ) < nδ . Thusthe following element( g ⊗ ⊗ · · · ⊗ ( g j − ⊗ ⊗ (cid:0) ( g g · · · g j − ) − g ′ ⊗ g ′′ ( g j +1 g j +2 · · · g n ) − (cid:1) ⊗ (1 ⊗ g j +1 ) ⊗ · · · ⊗ (1 ⊗ g n )appears in the summation expression of ∆ ⊗ n +1 nδ ( g ⊗ g ⊗ · · · ⊗ g n ). After applyingthe map M ′ , we see that g ′ ⊗ g ′′ indeed appears on the right-hand side of theinequality (4.8). This proves the claim, hence finishes the proof of the proposition. (cid:3) Cyclic gohomology of hyperbolic Groups In this section, we show that every cyclic cohomology class of a hyperbolicgroup has a uniformly bounded representative if its degree is ≥ 2. Since forany group, the equivalence class of a cyclic cocycle of degree ≤ G is a word hyperbolic group. For each h ∈ G , let G h be the centralizerof h in G , and N h the quotient of G h by the cyclic group generated by h . ( G 1) If h ∈ G has infinite order, then N h is finite.( G 2) For any h ∈ G , the centralizer G h is a quasi-convex subspace of G , thatis, there exists some K > G connecting a pairof points in G h lies in a K -neighborhood of G h .( G 3) For any h ∈ G , its centralizer G h is also word hyperbolic, and the inclusion G h ֒ → G is a quasi-isometry.Moreover, we will use the following result of Mineyev [43, Theorem 11] in anessential way.( M 1) Suppose G is a word hyperbolic group with a given length function ℓ .If n > 2, then every element in H n ( G ; C ) — the group cohomology of G — admits a uniformly bounded representative. Here a cocycle ele-ment ϕ is said to be uniformly bounded if there exists C > | ϕ ( g , g , · · · , g n ) | C for g i ∈ G .We will also need the following lemma, the proof of which is communicated tous by Denis Osin. Lemma 5.1. Let G be a δ hyperbolic group with a word length function ℓ . Foreach element h ∈ G , there exists a constant K h > such that min { ℓ ( γ ) | γ − hγ = g } ℓ ( g ) + K h . Proof. Let β be a group element of minimal length such that β − hβ = g . Considerthe geodesic quadrilateral [ e, β − ] , [ β − , β − h ] , [ β − h, g ] and [ g, e ] in the Cayleygraph of G (see Figure 1 below). eg = β − hβ β − β − hββ h Figure 1. Geodesic quadrilateral.By continuity, there exists a point x on [ e, g ] such that x is equidistant fromthe two sides [ e, β − ] and [ β − h, g ]. Let y (resp. z ) be a closest point on [ e, β − ](resp. [ β − h, g ]) to x , that is, d ( x, y ) = d ( x, z ) equals the distance between x andthe geodesic [ e, β − ]. By hyperbolicity, it is not difficult to see that d ( x, y ) = d ( x, z ) δ + ℓ ( h ) , cf. Figure 2 below. It follows immediately that both d ( e, y ) and d ( z, g ) are lessthan 2 δ + ℓ ( h ) + ℓ ( g ). ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 43 eg = β − hβ β − β − hββ hx y z Figure 2. d ( x, y ) = d ( x, z ) δ + ℓ ( h ).It remains to estimate the length of either [ y, β − ] or [ z, β − h ]. Suppose β β · · · β m is a word of minimal length that represents β . Let t i be a geodesicconnecting the points labeled by β i on [ e, β − ] and [ β − h, g ] (see Figure 3 be-low). If t i = t j for some i < j , we can cut the shaded region in Figure 4 and eg = β − hβ β − β − hhx y z β β β β β k β k β β · · ·· · · t k t t t Figure 3. Estimates for the length of [ y, β − ].it follows that the element α = β · · · β i β j +1 · · · β m satisfies that α − hα = g and ℓ ( α ) < ℓ ( β ). But this contradicts the assumption that β is a group element ofminimal length such that β − hβ = g . Thus all t i ’s are pairwise distinct.Using hyperbolicity on the quadrilateral with vertices { y, z, β − h, β − } (cf. [8,Chapter III.H, Lemma 1.15]), it is not difficult to see that there exists a constant C > β i is to the right of both y and z asshown in Figure 3, then ℓ ( t i ) ≤ C . Here C only depends on ℓ ( h ) and δ , and inparticular is independent of g . It follows immediately the length of either [ y, β − ]or [ z, β − h ] is C + 1, where C is the number of elements of G of length atmost C .Combining the above estimates together, we see that there exists a constant K h (only dependent on h and δ ) such that ℓ ( β ) ≤ ℓ ( g ) + K h . This finishes the proof. eg = β − hβ β − β − hhx y z β β β β β j β j β i β i · · · t i t t t j Figure 4. If t i = t j for some i < j , we can shorten β . (cid:3) Theorem 5.2. Suppose G is a word hyperbolic group. Fix a conjugacy class h h i of G . Then every element in HC n ( C G, h h i ) has a representative ϕ : G n +1 → C suchthat ϕ is of polynomial growth. Furthermore, when n > , such representativecan be chosen to be uniformly bounded over G n +1 .Proof. Elements of HC n ( C G, h h i ) have the following description, cf. [38, Section4.1]. Let C n ( G, G h , h ) be the space spanned by all ( n + 1)-linear maps on C G satisfying the following conditions: φ ( g σ (0) , g σ (1) , · · · , g σ ( n ) ) = ( − σ φ ( g , g , · · · , g n ) , ∀ σ ∈ S n ; (5.1) φ ( zg , zg , · · · , zg n ) = φ ( g , g , · · · , g n ) , ∀ z ∈ G h ; (5.2) φ ( hg , g , · · · , g n ) = φ ( g , g , · · · , g n ) . (5.3)Define a coboundary map ∂ : C n ( G, G h , h ) → C n +1 ( G, G h , h ) by ∂φ ( g , g , · · · , g n +1 ) = n +1 X j =0 ( − j φ ( g , g , · · · , g j − , g j +1 , · · · , g n +1 . )Denote the resulting cohomology groups by H ∗ ( G, G h , h ). For each cocycle φ in C n ( G, G h , h ), there is a cyclic cocycle T φ ∈ C n ( C G, h h i ) given by T φ ( g , g , · · · , g n ) = ( g g · · · g n / ∈ h h i ,φ ( γ, γg , · · · , γg · · · g n − ) if g g · · · g n = γ − hγ. (5.4)Let us first consider the case of cyclic cocycles with degree ≥ 2. Observe that if φ is uniformly bounded, then T φ is also uniformly bounded. Therefore, it sufficesto show that φ is uniformly bounded. It is not difficult to see that H ∗ ( G, G h , h )is isomorphic to H ∗ ( N h ; C ) . If the order of h is infinite, then H n ( N h ; C ) vanishesfor n > 0, since N h is a finite group by item ( G 1) above. Thus in the case, H n ( G, G h , h ) with n > C G .Let us assume h has finite order for the rest of the proof. In fact, it is moreconvenient for us to work with the group cohomology H n ( G h ; C ) of G h . Byapplying the transfer map, we immediately see that H n ( G h ; C ) surjects onto ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 45 H n ( N h ; C ). More precisely, consider the chain complex ( E n ( G, G h ) , b ) by re-moving the condition in line (5.3). There are two natural chain morphisms:the inclusion map ι : ( C n ( G, G h , h ) , b ) → ( E n ( G, G h ) , b ), and the transfer map τ : ( E n ( G, G h ) , b ) → ( C n ( G, G h , h ) , b ) defined by τ ( ψ )( g , g , · · · , g n ) = ord( h ) X j =1 ψ ( h j g , g , · · · , g n ) . Since τ ◦ ι = ord( h ) · Id, it follows that τ induces a surjection on cohomologygroups. Clearly, if ψ is uniformly bounded, then τ ( ψ ) is also uniformly bounded.Therefore it suffices to show that for n ≥ 2, every element H n ( E ∗ ( G, G h )) admitsa uniformly bounded representative.Let Y (resp. Y h ) be the ∆-complex consisting of all simplices of ordered ( n +1)-tuple { g , g , · · · , g n } , where g i ∈ G (resp. g i ∈ G h ). Observe that G h acts freelyon both Y and Y h . Moreover, we see that the cochain complex of G h -equivariantsimplicial cochain on Y is essentially E ∗ ( G, G h ), and the cochain complex of G h -equivariant simplicial cochain on Y h gives the standard resolution cochaincomplex for the group cohomology of G h .Let π : Y → Y h be any G h -equivariant projection, that is, π ◦ i = Id on Y h ,where i : Y h → Y is the inclusion map; such map always exists (see the discussionbelow for a specific construction of such a map). Then π induces a chain mapfrom the standard resolution cochain complex for the group cohomology of G h to E ∗ ( G, G h ), which is an isomorphism π ∗ : H n ( G h ; C ) ∼ = −→ H n ( E ∗ ( G, G h )) at thelevel of cohomology. In particular, any uniformly bounded group cocycle of G h pulls back to a uniformly bounded cocycle of the complex E ∗ ( G, G h ). On theother hand, by the item ( G 3) above, G h is hyperbolic. Therefore, by the item( M 1) above, every element of H n ( G h ; C ) has a uniformly bounded representative,when n > 2. This finishes the proof for cyclic cocycles of degree n > n = 0, HC ( C G, h h i ) is a one dimensional linear space spanned by tr h h i ; andtr h h i is clearly uniformly bounded on G .The only remaining case is when n = 1. We divide the proof of this caseas follows. First, we shall show that every element of H ( E ∗ ( G, G h )) has arepresentative of polynomial growth. Second, we shall construct a G h -equivariantprojection π : Y → Y h such that π is simplicial and furthermore Lipschitz withrespect to the word length metric on G and the corresponding subspace metricon G h . More precisely, we say π is Lipschitz if there exists a constant C > d ( π ( g ) , π ( g )) Cd ( g , g ) , for all g , g ∈ G , where d is the given word length metric on G . Now by the item( G G h is quasi-isometric to the subspace metric inherited from G . It follows that if a degree 1 group cocycle ϕ of G h has polynomial growth, thenthe pullback π ∗ ( ϕ ) of ϕ to the complex E ∗ ( G, G h ) also has polynomial growth. To be precise, elements of E ∗ ( G, G h ) are assumed to be skew-symmetric (i.e. the conditionin line (5.1)). But this can be easily fixed by applying a standard anti-symmetrization map. In this case, by applying Lemma 5 . T π ∗ ( ϕ ) ∈ C n ( C G, h h i ) has polynomial growth.Let us now show that every element of H ( E ∗ ( G, G h )) has a representativeof polynomial growth. By definition, a degree 1 group cocycle ϕ of G h is a G h -equivariant function ϕ : G h × G h → C , such that ϕ ( g , g ) − ϕ ( g , g ) + ϕ ( g , g ) =0. In particular, it follows that ϕ is determined by the function ψ : G h → C , where ψ ( g ) = ϕ (1 , g ). The cocycle condition implies that ψ ( g g ) = ψ ( g ) + ψ ( g ), i.e., ψ is a group homomorphism from G h to C . As any homomorphism from G h to C factors through the abelianization of G h , it follows that ψ has polynomialgrowth, so does ϕ .Now let us construct a G h -equivariant Lipschitz projection π : Y → Y h , whichwill finish the proof of the theorem by the above discussion. Fix an element,say α i ∈ G , for each coset of G h in G . Let α ′ i be an element of G h such that d ( α i , α ′ i ) = d ( α i , G h ). For simplicity, if α i = e ∈ G , then we map e to itself.Define a G h -equivariant map π : G → G h by mapping β · α i β · α ′ i , where β ∈ G h . Clearly, π extends by linear combination to a G h -equivariant map from Y to Y h , which will still be denoted by π .Let us show that π : Y → Y h is Lipschitz. For two distinct points g , g ∈ G ,choose a geodesic [ π ( g ) , π ( g )] in the Cayley graph of G . By the quasi-convexityof G h from item ( G 2) above, [ π ( g ) , π ( g )] lies in a K -neighborhood of G h . Byassumption, G is hyperbolic. More specifically, let us assume the Cayley graphof G is δ -hyperbolic. Then the geodesic [ π ( g ) , π ( g )] lies in the 2 δ -neighborhoodof the union of geodesics [ π ( g ) , g ], [ g , g ] and [ g , π ( g )]. Choose γ ∈ G to be a“midpoint” of [ π ( g ) , π ( g )], that is, | d ( π ( g ) , γ ) − d ( π ( g ) , γ ) | . Then there exists a point β on one of the geodesics [ π ( g ) , g ], [ g , g ] or [ g , π ( g )]such that d ( γ, β ) δ .(1) we claim that, if there exists a point β on [ π ( g ) , g ] such that d ( γ, β ) δ ,then we have d ( β, π ( g )) ≤ δ + K . Indeed, otherwise, we could find anelement h ∈ G h such that d ( g , h ) < d ( g , π ( g )), which contradicts thefact d ( g , π ( g )) = d ( g , G h ). Therefore, in this case, we see that d ( π ( g ) , π ( g )) δ + K ) + 1 . Things are similar for p ′ lies on [ g , π ( g )].(2) Similarly, if there exists β on [ π ( g ) , g ] such that d ( γ, β ) δ , then wealso have d ( π ( g ) , π ( g )) δ + K ) + 1 . (3) If there exists β on [ g , g ] such that d ( γ, β ) δ , then both d ( g , G h )and d ( g , G h ) are d ( g , g ) + 2 δ + K . It follows immediately that d ( π ( g ) , π ( g )) d ( g , g ) + 4 δ + 2 K. To summarize, we see that there exists a constant C > d ( π ( g ) , π ( g )) Cd ( g , g ) , ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 47 for all g , g ∈ G . This finishes the proof. Remark . By a theorem of Meyer [42, Theorem 5.2 & Corollary 5.3], the samestrategy in the above proof can be used to show the following: given a cycliccohomology class [ α ] of a hyperbolic group G , if ϕ and ϕ are two representativeswith polynomial growth of [ α ], then there exists a cyclic cocycle ψ such that ϕ − ϕ = bψ and ψ has polynomial growth. Here b : C n ( C G ) → C n +1 ( C G ) isthe coboundary map of the cyclic cochain complex. (cid:3) Delocalized Connes-Chern character of secondary invariants In this section, we construct a delocalized Connes-Chern character map for C ∗ -algebraic secondary invariants and prove the second main theorem (Theorem6 . 1) of the paper. We will only give the details for the odd dimensional case; theeven dimensional case is completely similar. Theorem 6.1. Let M be a closed manifold whose fundamental group G is hy-perbolic. Suppose h h i is non-trivial conjugacy class of G . Then every element [ α ] ∈ HC k +1 − i ( C G, h h i ) induces a natural map τ [ α ] : K i ( C ∗ L, ( f M ) G ) → C such that the following are satisfied. ( a ) τ [ Sα ] = τ [ α ] , where S is Connes’ periodicity map S : HC ∗ ( C G, h h i ) → HC ∗ +2 ( C G, h h i ) . ( b ) Suppose D is an elliptic operator on M such that the lift e D of D to theuniversal cover f M of M is invertible. Let ϕ be a representative of [ α ] withpolynomial growth. Then the delocalized higher eta invariant η ϕ ( e D ) ( cf.Definition . converges absolutely. Moreover, we have τ [ α ] ( ρ ( e D )) = − η ϕ ( e D ) , where ρ ( e D ) is the higher rho invariant of e D . In more conceptual terms, the above theorem provides a formula to computethe Connes-Chern character of elements of K i ( C ∗ L, ( f M ) G ). Moreover, the theoremestablishes a precise connection between Higson-Roe’s K -theoretic higher rhoinvariants and Lott’s higher eta invariants. Remark . Note that, in part ( b ) of the theorem, η ϕ ( e D ) converges absolutelyfor all invertible e D . In particular, it is not necessary for the spectral gap of e D tobe sufficiently large.This section is organized as follows. First, we show that each element in K i ( C ∗ L, ( f M ) G ) has a particular type of nice representatives. Second, we con-struct an explicit formula for the map τ [ α ] by using such nice representatives, and prove that the formula is well-defined. We shall only give the details for the caseof K ( C ∗ L, ( f M ) G ); the even case is completely similar. Definition 6.3. Let B L ( f M ) G to be the subalgebra of C ∗ L ( f M ) G consisting ofelements f ( t ) ∈ C ∗ L ( f M ) G such that f ( t ) ∈ B ( S G ) for all t ∈ [0 , ∞ ) and f ( t ) ispiecewise smooth with respect to k · k B . B L ( f M ) G is a smooth dense subalgebra of C ∗ L ( f M ) G . Similarly, we define B L, ( f M ) G to be the kernel of the evaluation map ev : B L ( f M ) G → B ( S G ) , f f (0) . Note that B L, ( f M ) G is a smooth dense subalgebra of C ∗ L, ( f M ) G . In particular, itfollows that K ∗ ( B L, ( f M ) G ) ∼ = K ∗ ( C ∗ L, ( f M ) G ) . Definition 6.4. Let SC ∗ ( f M ) G be the suspension of C ∗ ( f M ) G , and ϕ ∈ be an in-vertible element in SC ∗ ( f M ) G , that is, a loop ϕ : S = [0 , / { , } → ( C ∗ ( f M ) G ) + of invertible elements such that ϕ (1) = 1, where ( C ∗ ( f M ) G ) + is the unitization C ∗ ( f M ) G . We say ϕ is local if it is the image of an invertible element ψ ∈ SC ∗ L ( f M ) G under the evaluation map SC ∗ L ( f M ) G → SC ∗ ( f M ) G . Similarly, an invertible ele-ment ϕ ∈ SB ( S G ) is called local if it is the image of an invertible element of ψ ∈ SB L ( f M ) G under the evaluation map. Definition 6.5 ([32, Definition 3.3]) . A path ζ ∈ B L ( f M ) G is said to have poly-nomial B-norm control if(1) the propagation of ζ ( t ) is finite and goes to zero as t → ∞ ;(2) there exists some polynomial q such that k ζ ( t ) k B q (cid:16) prop ζ ( t ) (cid:17) for suffi-ciently large t ≫ 0. Here prop ζ ( t ) stands for the propagation of ζ ( t ).In the following, we shall prove a sharpened version of [58, Proposition 3.6].We show that every element of K ( B L, ( f M ) G ) has nice representatives that sat-isfy certain regularity properties, in particular, the polynomial control propertyabove.Let us first prove the following technical lemma. Lemma 6.6. Suppose D is a self-adjoint first order elliptic differential operatorover M and e D is the lifting of D to the universal cover f M of M . If G = π ( M ) is hyperbolic and f ∈ A Λ ,N ( cf. Definition . with N > dim M + 5 and Λ sufficiently large, then f ( e D ) ∈ B ( S G ) .Proof. Fix a symmetric generating set S of G . Let ℓ be the corresponding wordlength function of G determined by S and | S | the cardinality of S . Suppose f ( e D ) = P g ∈ G A g g . By lemma 3 . C , C > | A g | C · e − C Λ ℓ ( g ) , ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 49 for all but finitely many g ∈ G , where | A g | stands for the trace norm of A g . Letus denote A ( n ) = X ℓ ( g ) n A g g. It suffices to show that { A ( n ) } is a Cauchy sequence under the norm k · k B,p (cf.Definition 4 . m < n , we have k A ( n ) − A ( m ) k RD,p = X m<ℓ ( g ) n | A g | (1 + ℓ ( g )) p C · n X j = m +1 e − C Λ j (1 + j ) p | S | j , and k ∆( A ( n ) − A ( m ) ) k uc X m<ℓ ( g ) n | A g | X g g = gℓ ( g )+ ℓ ( g )= ℓ ( g ) (1 + ℓ ( g )) p (1 + ℓ ( g )) p X m<ℓ ( g ) n | A g | (1 + ℓ ( g )) p · { ( g , g ) : g g = g, ℓ ( g ) + ℓ ( g ) = ℓ ( g ) } C · C · X m<ℓ ( g ) n e − C Λ ℓ ( g ) (1 + ℓ ( g )) p +1 C · C · n X j = m +1 e − C Λ j (1 + j ) p +1 | S | j , where we have used the fact that there exists C > { ( g , g ) : g g = g, ℓ ( g ) | + ℓ ( g ) = ℓ ( g ) } C · ℓ ( g ) , since G is hyperbolic. It follows that, as long as Λ is sufficiently large, both k A ( n ) − A ( m ) k RD,p and k ∆( A ( n ) − A ( m ) ) k uc go to zero, as m, n → ∞ . (cid:3) Now let us show that every element of K ( B L, ( f M ) G ) has nice representativesthat satisfy certain regularity properties, in particular, the polynomial controlproperty above. The main motivation for choosing such nice representatives isto justify the explicit construction of τ [ α ] : K ( B L, ( f M ) G ) → C below (Definition6 . K ( B L, ( f M ) Γ ), two differentsuch regularized representatives can be connected by a family of representatives ofthe same kind. This allows us to show that the integral in line (6.1) in Definition6 . 10 is independent of the choice of such representatives. Proposition 6.7. Every element [ u ] ∈ K ( B L, ( f M ) G ) has a representative w ∈ ( B L, ( f M ) G ) + such that w ( t ) = u ( t ) if t ,h ( t ) if t ,e πi F ( t − if t > , where h is a path of invertible elements connecting u (1) and exp(2 πi F ( t − ) ,and F is a piecewise smooth map F : [1 , ∞ ) → D ∗ ( f M ) G satisfying (1) F ( t ) − ∈ B ( S G ) and F ( t ) ∗ = F ( t ) , (2) its derivative F ′ ( t ) ∈ B ( S G ) , (3) k F ( t ) k op , where k · k op stands for operator norm, (4) propagation of F ( t ) is finite, and goes to zero as t → ∞ . (5) both F ( t ) − and F ′ ( t ) have polynomial B-norm control in the sense ofDefinition . above.Moreover, if v is another such representative, then there exists a piecewise smoothpath of invertibles u s ∈ ( B L, ( f M ) G ) + and piecewise smooth maps F s : [0 , ∞ ) → D ∗ ( f M ) G satisfying conditions above, with s ∈ [0 , , such that (I) u = w , u ( t ) = v ( t ) for t / ∈ (1 , , (II) u s ( t ) = exp(2 πi F s ( t − ) for all t > , (III) u v − : [1 , → ( B ( S G )) + is a local loop of invertible elements, (IV) ∂ s ( F s ) has polynomial B -norm control, (V) the operator norm of F s ( t ) is uniformly bounded, and the degrees of poly-nomials used for the polynomial B -norm control of F s and ∂ s F s are uni-formly bounded, and the propagation of F s ( t ) goes to zero uniformly in s ,as t → ∞ .Remark . We shall call a representative appearing in the proposition above a regularized representative from now on. Proof. View the invertible element u ∈ ( B L, ( f M ) G ) + as an invertible element in( B L ( f M ) G ) + . Consider the element ˆ u = u : [1 , ∞ ) → ( B ( f M ) G ) + in K ( B L ( f M ) G ).Since the K -theory of B L ( f M ) G is the K -homology of M , it follows from theBaum-Douglas geometric description of K -homology [4] that ˆ u can be representedby a twisted Dirac operator over a spin c manifold. More precisely, let X be a spin c manifold together with a vector bundle E over X and a continuous map ψ : X → M . Suppose D E is the associated twisted Dirac operator on X . Let e X be the G -covering space of X induced by ψ , and e D be the lift of D E to e X .Choose an odd continuous function χ : R → [ − , 1] such that χ ( x ) → ± x → ±∞ and its distributional Fourier transform b χ has compact support. Wedefine F ( t ) = ψ ∗ ( χ ( e D/t )), where ψ ∗ : D ∗ ( e X ) G → D ∗ ( f M ) G is the natural mapinduced by ψ . It is not difficult to see that F satisfies the properties (1)-(5) listed ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 51 above . Moreover, we have [ˆ u ] = [ e πi F ( t )+12 ] ∈ K ( B L ( f M ) G ) . In particular, u ishomotopic to the invertible element w defined by w ( t ) = u ( t ) if 0 t ,h ( t ) if 1 t ,e πi F ( t − if t > , where h is a path of invertible elements connecting u (1) and e πi F (1)+12 .Now suppose v is another representative of [ u ] such that v ( t ) = u ( t ) if 0 t ,g ( t ) if 1 t ,e πi G ( t − if t > , where g is a path of invertible elements connecting u (1) and e πi G (1)+12 , and G t isa piecewise smooth map from [1 , ∞ ) to D ∗ ( f M ) G satisfying the properties (1)-(5)above.By Theorem 3.8 in [32], there exists a piecewise smooth family F s : [1 , ∞ ) → D ∗ ( f M ) G with s ∈ [0 , 1] such that F = F and F = G ; F s ( t ) ∗ = F s ( t ); propagationof F s ( t ) goes to zero, as t → ∞ ; and all F s ( t ) − ∂ t F s ( t ), and ∂ s F s ( t ) lie in B ( S G ). Furthermore, since the propagation of ∂ s F s ( t ) (resp. F s ( t )) is finiteand the propagation is bounded uniformly in t , it is not difficult to see that ∂ s F s has polynomial B -norm control and the degrees of polynomials used for thepolynomial B -norm control of F s and ∂ s F s are uniformly bounded.Let ̟ : [0 , ∞ ) → ( B ( S G )) + be the path of invertibles defined as ̟ ( t ) = u ( t ) if 0 t ,h ( t ) if 1 t ,e πi Fs (1)+12 if 2 t = s + 2 ,e πi G ( t − t > . Clearly, w is homotopic to ̟ . On the other hand, after a re-parameterization,it is not difficult to see that ̟ differs from v by the loop f : [0 , → ( B ( S G )) + defined by f ( t ) = g ( t ) − h (2 t ) if 0 t / ,g ( t ) − e πi F t − if 1 / t . Since by construction the propagation of F ( t ) is uniformly bounded (in particular finite)for all t ∈ [1 , ∞ ), the polynomial B -norm control in property (5) follows from the work of [32,Section 4]. Roughly speaking, the polynomial B -norm control is a consequence of the existenceof partition of unity { ψ n,j } on the manifold X for each n ∈ N such that the diameter of ψ n,j is /n and the norm of dψ n,j is bounded by q (1 /n ) for some polynomial q . Moreover, f is a local loop in the sense of Definition 6.4. This finishes theproof. (cid:3) The following lemma will be useful in the proof of Theorem 6 . Lemma 6.9. Let u = u s ( t ) be the family of invertible elements from Proposition . above. Then for any delocalized cyclic cocycle ϕ with polynomial growth, lim t → + ∞ ] ϕ u s ( t ) − ∂ s ( u s ( t )) ⊗ ( u s ( t ) ⊗ u s ( t ) − ) ⊗ m ) = 0 uniformly in s ∈ [0 , .Proof. By definition, u s ( t ) = e πi Fs ( t − , where F s and ∂ s F s have polynomial B -norm control. Denote P = P s ( t ) = F s ( t )+12 .Then P s − P s = ( F s − / 4. Let f n ( x ) = n X k =0 (2 πix ) k k ! . Note that we have f n ( P ) = n X k =0 (2 πi ) k k ! P k = 1 + (cid:16) n X k =1 (2 πi ) k k ! (cid:17) P + (cid:16) n X k =1 (2 πi ) k k ! k − X j =0 P j (cid:17) ( P − P )Define A n = f n ( P ) − (cid:16) n X k =1 (2 πi ) k k ! (cid:17) P . Clearly, A n ∈ B ( S G ) + for all n > 1, and u − A n = (cid:16) ∞ X k = n +1 (2 πi ) k k ! (cid:17) ( P − P ) + (cid:16) ∞ X k = n +1 (2 πi ) k k ! k − X j =0 P j (cid:17) ( P − P ) . Recall that, by construction, there exists a polynomial q such that k P ( t ) − P ( t ) k B ≤ q (cid:0) / prop F ( t ) (cid:1) . Since the operator norm of P ( t ) is uniformly bounded and the propagation of P ( t ) goes to zero as t → ∞ , a routine calculation shows that there exists C > k P j ( t )( P ( t ) − P ( t )) k B ≤ C j · q (cid:0) / prop F ( t ) (cid:1) for all j ∈ N and sufficiently large t ≫ 0. It follows that there exists K > k u ( t ) − A n ( t ) k B < K C n ( n + 1)! · q (cid:0) / prop F ( t ) (cid:1) , ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 53 and k u ( t ) − k B < Ke C · q (cid:0) / prop F ( t ) (cid:1) for sufficiently large t ≫ 0. The same type of estimates also apply to u − and ∂ s u .Now fix ε > a i ∈ B ( S G ) + has propagation ε for all 0 i m ,then ] ϕ a ⊗ a ⊗ · · · ⊗ a m ) = 0 . By the proof of Proposition 3 . 26, such an ε exits. For each n ∈ N , there exists t n > prop F s ( t ) < εn , for all t > t n . In particular, we have that(1) prop A n ( t ) < ε ,(2) k u ( t ) − A n ( t ) k B < K C n ( n +1)! · q (cid:0) n/ε ),(3) k u ( t ) − k B < Ke C · q (cid:0) n/ε )for all t > t n . Similarly, the same type of estimates hold for u − and ∂ s u . Thelemma easily follows from these estimates. This finishes the proof. (cid:3) Now for each class [ α ] ∈ HC m ( C G, h h i ). we define a map τ [ α ] : K ( B L, ( f M ) G ) → C as follows. Definition 6.10. Let ϕ be a representative of [ α ] with polynomial growth. Foreach [ u ] ∈ K ( B L, ( f M ) G ), let w be a regularized representative of [ u ]. We define τ [ α ] ([ u ]) := τ ϕ ( w ) = m ! πi Z ∞ ] ϕ w ( t ) w ( t ) − ⊗ ( w ( t ) ⊗ w ( t ) − ) ⊗ m ) dt. (6.1)The convergence of the integral in line (6.1) follows from the following twoobservations.(1) By Proposition 4.14, the integrand ] ϕ w ( t ) w ( t ) − ⊗ ( w ( t ) ⊗ w ( t ) − ) ⊗ m )is a piecewise smooth function with respect to t on [0 , ∞ ). In particular,this implies that the integral in line (6.1) converges absolutely for small t .(2) By the proof of Proposition 6.7, when t > 2, we have that w ( t ) = e πi F ( t )+12 = e πi χ ( e D/t )+12 Set s = 1 /t , and then we have Z ∞ ] ϕ w ( t ) w ( t ) − ⊗ ( w ( t ) ⊗ w ( t ) − ) ⊗ m ) dt = πi Z / ] ϕ (cid:16) ˙ χ ( s e D ) e D ⊗ (cid:0) e πi χ ( s e D )+12 ⊗ e − πi χ ( s e D )+12 (cid:1) ⊗ m (cid:17) ds. Since the Fourier transform of χ has compact support, it follows that xχ ′ ( x ) and exp( ± πi χ ( x )+12 ) − A Λ ,N for any Λ , N . Thereforeby Proposition 3.26 and 3.27, the integral with respect to s converges absolutely for small s . Consequently the integral from line (6.1) convergesfor large t . Proof of Theorem . . Let ϕ be a representative of [ α ] with polynomial growth.Let us first show that τ ϕ ([ u ]) is independent of the choice of regularized rep-resentative of [ u ]. Suppose w and v are two regularized representatives of [ u ].By Proposition 6.7, there exists a piecewise smooth family of invertibles u s ∈ B L, ( f M ) G with the stated properties (I)-(V).Now a straightforward calculation shows that ∂ s (cid:16) ] ϕ u − ∂ t u ⊗ ( u ⊗ u − ) ⊗ m ) (cid:17) = ∂ t (cid:16) ] ϕ u − ∂ s u ⊗ ( u ⊗ u − ) ⊗ m ) (cid:17) . (6.2)It follows that Z T ] ϕ u u − ⊗ ( u ⊗ u − ) ⊗ m ) dt − Z T ] ϕ u u − ⊗ ( u ⊗ u − ) ⊗ m ) dt = Z ] ϕ u − ∂ s u ⊗ ( u ⊗ u − ) ⊗ m ) (cid:12)(cid:12)(cid:12) t = Tt =0 ds By Lemma 6.9 below, we have ] ϕ u − ∂ s u ⊗ ( u ⊗ u − ) ⊗ m ) → t → ∞ . Also, note that u s (0) ≡ s . It follows that τ ϕ ( u ) = τ ϕ ( u ) . On the other hand, u differs from v by a local loop f : S → B ( S G ) + . By[58, Lemma 3.4], for ∀ ε > 0, there exists an idempotent p ∈ B ( S G ) + such thatthe propagation of p is ε and f is homotopic, in the algebra SB ( S G ), to theelement β ( t ) = e πit p + (1 − p ) , where 0 t . It follows that Z ] ϕ f ( t ) f ( t ) − ⊗ ( f ( t ) ⊗ f ( t ) − ) ⊗ m ) dt = Z ] ϕ β ( t ) β ( t ) − ⊗ ( β ( t ) ⊗ β ( t ) − ) ⊗ m ) dt = Z ϕ πip ⊗ (( e πit − p ⊗ ( e − πit − p ) ⊗ m ) dt where the last integral is clearly zero, as long as ε is sufficiently small. Therefore, τ [ α ] ([ u ]) is independent of the choice of regularized representative of [ u ].For a given regularized representative w of [ u ], the same proof from Propo-sition 3.31 show that τ [ α ] ( w ) is independent of the choice of polynomial growthrepresentative ϕ of [ α ] (cf. Remark 5 . τ [ α ] : K ( B L, ( f M ) G ) → C ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 55 is well-defined. Furthermore, the same proof from Proposition 3.33 shows that τ ϕ ( w ) = τ Sϕ ( w ) . This proves part ( a ) of the theorem.We shall prove part ( b ) of the theorem in three steps.(i) Recall that the definition of η ϕ ( e D ) (cf. Definition 3 . 17) uses the represen-tative u t = U /t ( e D ) of the higher rho invariant ρ ( e D ), where U t = e πiF t ( e D ) with F t ( x ) = 1 √ π Z x/t ∞ e − s ds for t > U = 1 . We first prove that the path U t is an element of ( B L, ( f M ) G ) + .(ii) Second, we prove that the integral η ϕ ( e D ) = ( − m ! πi Z ∞ ] ϕ U t U − t ⊗ ( U t ⊗ U − t ) ⊗ m ) dt absolutely converges, where the minus sign is due to the change of vari-ables 1 /t → t . Note that here we do not require the spectral gap of e D tobe sufficiently large. In other words, the convergence of the integral holdsas long as e D is invertible.(iii) Recall that we defined τ [ α ] ( ρ ( e D )) by using a regularized representative of ρ ( e D ). In the third step, we use a transgression formula as in line (6.2) toprove that τ [ α ] ( ρ ( e D )) = − η ϕ ( e D ).The first and second steps are proved in Proposition 6 . 11 below. Let us now turnto the third step. Let χ be a normalizing function from the proof of Proposition6.7, that is, an odd continuous function χ : R → [ − , 1] such that χ ( x ) → ± x → ±∞ and its distributional Fourier transform b χ has compact support.Furthermore, without loss of generality, we can assume in addition x · b χ ( x ) isa smooth function. Denote E t ( e D ) = χ ( e D )+12 . It follows from Lemma 3.10 andLemma 6.6 that e πiE t ( e D ) is a smooth path in ( B ( S G )) + . Let us define V t = U t if 0 t ,e πi ((2 − t ) F ( e D )+( t − E ( e D )) if 1 t ,e πiE t − ( e D ) if t > . Then the path V t is a regularized representative of ρ ( e D ) in B L, ( S G ) + . Further-more, V t and U t are homotopic in B L, ( S G ) + by the following family of elements H s , with 0 ≤ s ≤ H s ( t ) = U t if 0 t ,e πi ((2 − t ) F +( t − sE +(1 − s ) F )) if 1 t s,e πi ((1 − s ) F t − + sE t − ) if t > s. Now the same transgression formula in line (6.2) can be applied to show that τ ϕ ( V ) = τ ϕ ( U ) = − η ϕ ( e D ) . This finishes the proof. (cid:3) Proposition 6.11. Under the same assumptions of Theorem . , let U t be therepresentative of ρ ( e D ) given by U t = e πiF t ( e D ) with F t ( x ) = 1 √ π Z x/t ∞ e − s ds for t > U = 1 . Then the path U t defines an invertible element of B L, ( S G ) + . Furthermore, if ϕ in C m ( C G, h h i )) has polynomial growth, then the following integral Z ∞ ] ϕ U t U − t ⊗ ( U t ⊗ U − t ) ⊗ m ) dt converges absolutely.Proof. Since U t ( x ) − . U t = U t ( e D ) ∈ B ( S G ) + for each t > U t is smooth with respect to the norm k · k B on (0 , ∞ ). It remains to show that U t is continuous at t = 0 with respect to the norm k · k B .Since e D is invertible, let σ > e D at zero. Then thespectral radius of e − e D as an element in B ( S G ) is e − σ , since B ( S G ) is a smoothdense subalgebra of C ∗ ( f M ) G . Recall that B ( S G ) is a Banach algebra with respectto the norm k · k B (cf. Proposition 4 . k A A k B k A k B k A k B , for any A , A ∈ B ( S G ) . By the spectral radius formulalim n →∞ ( k ( e − e D ) n k B ) n = e − σ , there exists C > k e − t e D k B C e − t σ for all sufficiently small t > 0. It follows that there exists C > k ˙ U t U − t k B = (cid:13)(cid:13) − √ πi e Dt e − e D /t (cid:13)(cid:13) B t (cid:13)(cid:13) √ π e De − e D (cid:13)(cid:13) B · (cid:13)(cid:13) e − (1 /t − e D (cid:13)(cid:13) B C t e − σ /t . (6.3)By the definition of U t , we have U t − (cid:18)Z t ˙ U s U − s ds (cid:19) − . Rescale the norm k · k B if necessary. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 57 In fact, the integral on the right hand side converges in B ( S G ), thanks to theinequality in line (6.3). In particular, it follows that k U t − k B = (cid:13)(cid:13)(cid:13) ∞ X n =1 n ! (cid:18)Z t ˙ U s U − s ds (cid:19) n (cid:13)(cid:13)(cid:13) B ∞ X n =1 n ! (cid:18) Ct e − σ t (cid:19) n = exp (cid:18) Ct e − σ t (cid:19) − , (6.4)which goes to zero as t goes to zero. It follows that the path U t , with t ∈ [0 , ∞ ),gives an invertible element in B L, ( S G ) + . Furthermore, by Proposition 3.27 andProposition 4.14, the following integral Z ∞ ] ϕ U t U − t ⊗ ( U t ⊗ U − t ) ⊗ m ) dt converges absolutely. This finishes the proof. (cid:3) Delocalized higher Atiyah-Patodi-Singer index theorem In this section, we apply the results from previous sections to prove a delocal-ized higher Atiyah-Patodi-Singer index theorem.Let us first review the Connes-Chern character map in our context. We shallonly discuss the even dimensional case; the odd case is similar. Let G be adiscrete group, and [ α ] ∈ HC m ( C G ). The [ α ]-component of the Connes-Cherncharacter of an idempotent p ∈ S G is given bych [ α ] ( p ) = (2 m )! m ! ϕ p ⊗ m +1 ) , (7.1)where ϕ is a cyclic cocycle representative of [ α ]. It has been implied that ch [ ϕ ] ( p ) isindependent of the choice of representative of [ α ]. Indeed, for a cyclic coboundary bψ , we have bψ p ⊗ m +1 ) = ψ ( p ⊗ m ) = 0 . The last equality follows from the fact ψ is cyclic, which in particular impliesthat ψ ( p ⊗ m ) = − ψ ( p ⊗ m ).If G is hyperbolic, then by Proposition 4.14 the formula in line (7.1) continuesto make sense for idempotents in B ( S G ), as long as ϕ has polynomial growth.In fact, in this case, ch [ α ] defines a Connes-Chern character map at the level of K -theory: ch [ α ] : K ( B ( S G )) → C . Indeed, suppose [ p ] = [ p ] ∈ K ( B ( S G )). Let p t be a piecewise smooth path ofidempotents in B ( S G ) + connecting p and p . Suppose ϕ has polynomial growth.Then a routine calculation shows that ddt ϕ (cid:0) p ⊗ m +1 t (cid:1) = (2 m + 1)( bϕ (cid:16) ( ˙ p t p t − p t ˙ p t ) ⊗ p ⊗ m +1 t (cid:17) = 0 , since ϕ is a cyclic cocycle. It follows immediately that ϕ p ⊗ m +10 ) = ϕ p ⊗ m +11 ) . Therefore, the map ch [ α ] : K ( B ( S G )) → C is well-defined.By Theorem 5 . 2, every cyclic cohomology class of a hyperbolic group has arepresentative with polynomial growth. Hence, to summarize the above, we havethe following proposition. Proposition 7.1. Suppose that G is a hyperbolic group and h h i is a conjugacyclass of G . For each class [ α ] ∈ HC m ( C , h h i ) , the Connes-Chern character map ch [ α ] : K ( C ∗ r ( G )) → C (7.2) given by the formula (7.1) is well-defined. Moreover, we have ch S [ α ] = ch [ α ] , where S : HC m ( C G, h h i ) → HC m +2 ( C G, h h i ) is Connes’ periodicity map.Proof. The formula for Connes’s periodicity map is given in Definition 3 . 32. Astraightforward computation shows that( Sϕ p ⊗ m +3 ) = 12(2 m + 1) ϕ p ⊗ m +1 ) , from which the second statement of the proposition immediately follows. (cid:3) The Connes-Chern character ch [ α ] : K ( C ∗ r ( G )) → C above and the delocalizedConnes-Chern character map τ [ α ] : K i ( C ∗ L, ( f M ) G ) → C from Theorem 6 . Proposition 7.2. Suppose G is hyperbolic and h h i is a nontrivial conjugacy classof G . Given [ α ] ∈ HC m ( C G, h h i ) , we have the following commutative diagram: K ( C ∗ r ( G )) ch [ α ] −−−→ C ∂ y y × ( − K ( C ∗ L, ( f M ) G ) τ [ α ] −−−→ C where ∂ : K ( C ∗ r ( G )) → K ( C ∗ L, ( f M ) G ) is the connecting map in the six-termK-theoretical exact sequence for the short exact sequence: → C ∗ L, ( f M ) G → C ∗ L ( f M ) G → C ∗ ( f M ) G → Proof. Each element of K ( C ∗ r ( G )) is represented by the formal difference of twoidempotents in B ( S G ) + . For notational simplicity, let us carry out the compu-tation for an idempotent p in B ( S G ).Recall that ∂ [ p ] is defined as follows: let { a t } t ∈ [0 , ∞ ) be the following lift of p in B L ( f M ) G : a t = ( (1 − t ) p if 0 t , t > . Then we have ∂p := u with u ( t ) = e πia t for t ∈ [0 , ∞ ) . Note that for 0 t 1, we have u t = e πi (1 − t ) p = 1 + ( e πi (1 − t ) − p. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 59 It follows that Z ∞ ] ϕ u t u − t ⊗ ( u t ⊗ u − t ) ⊗ m ) dt = Z ] ϕ u t u − t ⊗ ( u t ⊗ u − t ) ⊗ m ) dt = − ϕ p ⊗ m +1 ) Z (2 πi )( e πi (1 − t ) − m ( e − πi (1 − t ) − m dt = − πi (2 m )!( m !) ϕ p ⊗ m +1 ) = − πim ! ch [ α ] ([ p ]) . It follows that ch [ α ] ([ p ]) = − τ [ α ] ( ∂ [ p ]) . This finishes the proof. (cid:3) Let W be a compact n -dimensional spin manifold with boundary M = ∂W .Suppose W is equipped with a Riemannian metric which has product structurenear M and in additional has positive scalar curvature on M . Let f W be theuniversal covering of W equipped with the metric lift from W . Denote π ( W )by G . The associated Dirac operator e D W naturally defines a higher index in K n ( C ∗ ( f W ) G ), denoted by Ind G ( e D W ) as in [56, Section 3]. Denote the lift of M with respect to the covering map by f M = ∂ f W . The associated Dirac operator e D M naturally defines a higher rho invariant ρ ( e D M ) in K n − ( C ∗ L, ( f M ) G ). The image of ρ ( e D M ) under the natural homomorphism K n − ( C ∗ L, ( f M ) G ) → K n − ( C ∗ L, ( f W ) G )will still be denoted by ρ ( e D M ).We denote by ∂ : K n ( C ∗ ( f W ) G ) → K n − ( C ∗ L, ( f W ) G ) the connecting map in thesix-term K-theoretical exact sequence for the short exact sequence:0 → C ∗ L, ( f W ) G → C ∗ L ( f W ) G → C ∗ ( f W ) G → . By [47, Theorem 1.14] and [56, Theorem A], we have ∂ (Ind G ( e D W )) = ρ ( e D M ) in K n − ( C ∗ L, ( f W ) G ) . (7.3)This together with Proposition 7 . Theorem 7.3. Let W be a compact even-dimensional spin manifold with bound-ary M . Suppose W is equipped with a Riemannian metric which has productstructure near M and in additional has positive scalar curvature on M . Suppose G = π ( W ) is hyperbolic and h h i is a non-trivial conjugacy class of G . Then forany [ α ] ∈ HC m ( C G, h h i ) , we have ch [ α ] (Ind G ( e D W )) = 12 η [ α ] ( e D M ) . (7.4) Proof. Observe that Proposition 7 . f M by f W . In partic-ular, we have the following commutative diagram: K ( C ∗ r ( G )) ch [ α ] −−−→ C ∂ y y × ( − K ( C ∗ L, ( f W ) G ) τ [ α ] −−−→ C Now the theorem follows immediately from Theorem 6 . ∂ (Ind G ( e D W )) = ρ ( e D M ) in K n − ( C ∗ L, ( f W ) G ) . (cid:3) By using Theorem 6 . 1, we have derived Theorem 7 . K -theoretic counterpart. This is possible only because we have realized η [ α ] ( e D M )as the pairing between the cyclic cocycle [ α ] and the C ∗ -algebraic secondaryinvariant ρ ( e D M ) in K ( C ∗ L, ( f W ) G ).Alternatively, one can also derive Theorem 7 . C ∗ r ( G ); or noncommutative differential forms on a certainclass of smooth dense subalgebras (if exist) of general C ∗ -algebras (not just group C ∗ -algebras) in Wahl’s version. In the case of Gromov’s hyperbolic groups, onecan choose such a smooth dense subalgebra to be Puschnigg’s smooth densesubalgebra B ( C G ). For hyperbolic groups, every cyclic cohomology class of C G continuously extends to a cyclic cohomology class of B ( C G ), cf. Section 4 andSection 5. Now Theorem 7 . C Γ.One can also try to pair the higher Atiyah-Patodi-Singer index formula ofLeichtnam-Piazza and Wahl with group cocycles of Γ, or equivalently cyclic cocy-cles in HC ∗ ( C Γ , h i ), where h i stands for the conjugacy class of the identity ele-ment of Γ. In this case, for fundamental groups with property RD, Gorokhovsky,Moriyoshi and Piazza proved a higher Atiyah-Patodi-Singer index theorem forgroup cocycles with polynomial growth [22, Theorem 7.2].8. Delocalized higher eta invariant and its relation to Lott’shigher eta invariant In this section, we shall establish the relation between our definition of the de-localized higher eta invariant (cf. Definition 3 . 17) and Lott’s higher eta invariant[38, Section 4.4 & 4.6]. In particular, we prove that our definition of the delocal-ized higher eta invariant is equal to Lott’s higher eta invariant up to a constant1 √ π . The main techniques used in this section are from Connes’ papers [11, 13]. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 61 Let M be a closed manifold and f M the universal covering over M . Suppose D is a first order self-adjoint elliptic differential operator acting on a vector bundle E over M and e D the lift of D to f M . Suppose that h h i is a nontrivial conjugacyclass of G = π ( M ) and e D is invertible. Throughout this section, we assume that G has polynomial growth.Let B be the following dense subalgebra of C ∗ r ( G ): B = (cid:8) f : G → C | X g ∈ G (1 + ℓ ( g )) k | f ( g ) | < ∞ for all k ∈ N } , where ℓ is a word-length function on G . B is Fr´echet locally m -convex algebra.Moreover, since G has polynomial growth, B is a smooth dense subalgebra of C ∗ r ( G ). The universal graded differential algebra of B isΩ ∗ ( B ) = ∞ M k =0 Ω k ( B )where as a vector space, Ω k ( B ) = B ⊗ ( B ) ⊗ k . As B is a Fr´echet algebra, weconsider the completion of Ω ∗ ( B ), which will still be denoted by Ω ∗ ( B ).Let E = ( f M × G B ) ⊗ E be the associated B -vector bundle and C ∞ ( M ; E ) beits space of smooth sections. Now suppose ψ is a smooth function on f M withcomapct support such that X g ∈ G g ∗ ψ = 1 . Then we have a superconnection ∇ : C ∞ ( M ; E ) → C ∞ ( M ; E ⊗ B Ω ( B )) given by ∇ ( f ) = X g ∈ G ( ψ · g ∗ f ) ⊗ B dg. Definition 8.1 ([38, Section 4.4 & 4.6]) . For each β > 0, Lott’s higher etainvariant e η ( e D ) is defined by the formula e η ( e D, β ) = β / Z ∞ STR( e De − β ( t e D + ∇ ) ) dt. Here we follow the superconnection formalism, and STR is the correspondingsupertrace, cf. [38, Proposition 22].We recall the following periodic version of Lott’s higher eta invariant. Definition 8.2 ([38, Section 4.6]) . Define ˜ η ( e D ) ∈ Ω ∗ ( B ) to be˜ η ( e D ) = Z ∞ e − β ˜ η ( e D, β ) dβ. Similar estimates as those in Section 3 show that, under the assumption G has polynomial growth, the above integral converges in Ω ∗ ( B ), hence ˜ η ( e D ) iswell-defined. Let us write ˜ η ( e D ) = X m > ˜ η k ( e D ) = X m > Z ∞ e − β ˜ η m ( e D, β ) dβ, where ˜ η m ( e D ) and ˜ η m ( e D, β ) are the 2 m -th components of ˜ η ( e D ) and ˜ η ( e D, β )in Ω m ( B ) respectively.For each m ≥ 0, only a finite number of terms in Duhamel expansion of e De − β ( t e D + ∇ ) will contribute to ˜ η m ( e D, β ). Suppose ϕ ∈ C m ( G, h h i ) is a cycliccocycle with polynomial growth. Without loss of generality, we assume that ϕ isnormalized, that is, ϕ ( g , g , · · · , g k ) = 0 if g i = 1 for some i ≥ . Let us consider the paring h ϕ, ˜ η m ( e D, β ) i . Observe that, since ˜ η m ( e D, β ) ispaired with ϕ , we can relax the smoothness condition on ψ in the definitionof the connection ∇ above and choose ψ to be the characteristic function of afundamental domain of f M under the action G . More precisely, for such a choiceof ψ , we should treat the summands t ∇ e D and t e D ∇ in the supercommutator[ ∇ , t e D ] = t ( ∇ e D + e D ∇ ) separately so that we avoid taking the differential of ψ .As a consequence, the term ∇ does not contribute to the pairing h ϕ, ˜ η m ( e D, β ) i ,since ϕ is normalized. To summarize, we have h ϕ, ˜ η m ( e D, β ) i = Z ∞ β m +1 Z ( P mj =0 s j )= β D ϕ, STR (cid:0) e De − s βt e D [ ∇ , t e D ] e − s βt e D [ ∇ , t e D ] · · ·× [ ∇ , t e D ] e − s m βt e D (cid:1)E ds ds · · · ds m dt = Z ∞ β m + Z ( P mj =0 s j )= β D ϕ, STR (cid:0) e De − s t e D [ ∇ , t e D ] e − s t e D [ ∇ , t e D ] · · ·× [ ∇ , t e D ] e − s m t e D (cid:1)E ds ds · · · ds m dt, where the second equality follows from the change of variables t 7→ √ βt . Here[ ∇ , t e D ] is the supercommutator (i.e. graded-commutator) of ∇ and t e D .Let f ( x ) = e De − xt e D and f j ( x ) = [ ∇ , t e D ] e − xt e D for j > 0. From the abovecalculation, we see that h ϕ, ˜ η m ( e D, β ) i = β m + Z ∞ (cid:10) ϕ, STR (cid:0) f ∗ f ∗ · · · ∗ f m ( β ) (cid:1)(cid:11) dt, where ∗ stands for the convolution:( f ∗ h )( β ) = Z β f ( x ) h ( β − x ) dx. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 63 Recall that the Laplace transform f 7→ L ( f )( s ) = Z ∞ e − sβ f ( β ) dβ converts convolutions of functions into pointwise products of functions. Let usdefine H ( s ) := (cid:10) ϕ, STR (cid:0) Q mj =0 L ( f j )( s ) (cid:1)(cid:11) , which is the Laplace transform of Z ∞ (cid:10) ϕ, STR (cid:0) f ∗ f ∗ · · · ∗ f m ( β ) (cid:1)(cid:11) dt. Recall that L ( e − aβ )( s ) = 1 a + s . Therefore, we have H ( s ) = Z ∞ (cid:10) ϕ, STR (cid:0) e Dt e D + s [ ∇ , t e D ] t e D + s · · · [ ∇ , t e D ] t e D + s (cid:1)(cid:11) dt = s − m − Z ∞ (cid:10) ϕ, STR (cid:0) e Dt e D +1 [ ∇ , t e D ] t e D +1 · · · [ ∇ , t e D ] t e D +1 (cid:1)(cid:11) dt = s − m − H (1)where the second equality follows from the change of variables t → t/ √ s . Applythe inverse Laplace transform to H ( s ) and we obtain Z ∞ (cid:10) ϕ, STR (cid:0) f ∗ f ∗ · · · ∗ f m ( β ) (cid:1)(cid:11) dt = β m − Γ( m + ) H (1)where Γ( m + ) = m (2 m )! m ! √ π . It follows that h ϕ, ˜ η m ( e D, β ) i = β m Γ( m + ) H (1) . Hence we have h ϕ, ˜ η m ( e D ) i = Z ∞ h ϕ, e − β ˜ η m ( e D, β ) i dβ = 4 m m ! √ π Z ∞ (cid:10) ϕ, STR (cid:0) e Dt e D +1 [ ∇ , t e D ] t e D +1 · · · [ ∇ , t e D ] t e D +1 (cid:1)(cid:11) dt We shall identify this formula with our formula for delocalized higher eta invariantin Definition 3 . 17. To this end, let us first define w t ( x ) := tx − itx + i . Note that the path w /t ( e D ) is a representative of the higher rho invariant ρ ( e D ).A direct computation shows that˙ w t ( e D ) w t ( e D ) − = 2 i e Dt e D + 1 and[ w t ( e D ) , ∇ ] · [ w t ( e D ) − , ∇ ] = 4( t e D + i ) − [ ∇ , t e D ]( t e D + 1) − [ ∇ , t e D ]( t e D − i ) − , where [ w t ( e D ) , ∇ ] and [ w t ( e D ) − , ∇ ] are the usual ungraded commutator . Fornotational simplicity, let us write w t in place of w t ( e D ). The above computationimplies that tr (cid:0) ˙ w t w − t ([ w t , ∇ ][ w − t , ∇ ]) m (cid:1) =4 m tr (cid:0) i e Dt e D +1 (cid:0) t e D + i [ ∇ , t e D ] t e D +1 [ ∇ , t e D ] t e D − i (cid:1) m (cid:1) =4 m tr (cid:0) i e Dt e D +1 [ ∇ , t e D ] t e D +1 · · · [ ∇ , t e D ] t e D +1 (cid:1) . Since ϕ is normalized, again ∇ does not contribute to the pairing h ϕ, tr (cid:0) ˙ w t w − t ([ w t , ∇ ][ w − t , ∇ ]) m (cid:1) i . It follows that h ϕ, tr (cid:0) ˙ w t w − t ([ w t , ∇ ][ w − t , ∇ ]) m (cid:1) i = h ϕ, tr (cid:0) ˙ w t w − t ( ∇ w t ∇ w − t ) m (cid:1) i + h ϕ, tr (cid:0) ˙ w t w − t ( w t ∇ w − t ∇ ) m (cid:1) i , when m > 1. By the definition of the connection ∇ and the definition of thetrace, we have h ϕ, tr (cid:0) ˙ w t w − t ( ∇ w t ∇ w − t ) m (cid:1) i = ] ϕ (cid:0) ˙ w t w − t ⊗ ( w t ⊗ w − t ) ⊗ m (cid:1) . On the other hand, we have h ϕ, tr (cid:0) ˙ w t w − t ( w t ∇ w − t ∇ ) m (cid:1) i = h ϕ, tr (cid:0) ˙ w t ( ∇ w − t ∇ w t ) m − ∇ w − t ∇ (cid:1) i = 0 . Therefore, for all m > 0, we have Z ∞ h ϕ, tr (cid:0) ˙ w t w − t ([ w t , ∇ ][ w − t , ∇ ]) m (cid:1) i dt = Z ∞ ] ϕ (cid:0) ˙ w t w − t ⊗ ( w t ⊗ w − t ) ⊗ m (cid:1) dt = πim ! τ ϕ ( w ) . where τ ϕ is the map from Definition 6.10. Now Lemma 8 . τ ϕ ( w ). To summarize, we have established the following preciserelation between our definition of the delocalized higher eta invariant (cf. Defi-nition 3 . 17) and Lott’s higher eta invariant [38, Section 4.4 & 4.6]. In the superconnection formalism of Definition 8 . Z / w t ( e D ) is defined on the original bundle, instead of the superbundle. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 65 Proposition 8.3. Suppose D is a first order self-adjoint elliptic differential oper-ator acting on a vector bundle E over M and e D the lift of D to f M . Assume that h h i is a nontrivial conjugacy class of G = π ( M ) , e D is invertible and G = π ( M ) has polynomial growth. Then we have τ ϕ ( ρ ( e D )) = 1 √ π h ϕ, ˜ η m ( e D ) i . In the remaining part of this section, we prove Lemma 8 . . G = π ( M ) has polynomial growth. It remains anopen question how to identify our formulation of higher eta invariants with Lott’shigher eta invariant in general. Lemma 8.4. With the same notation as above, the following integral Z ∞ ] ϕ (cid:0) ˙ w t w − t ⊗ ( w t ⊗ w − t ) ⊗ m (cid:1) dt (8.1) converges absolutely.Proof. let a t ( x ) = w t ( x ) − − i ( tx + i ) − . For each t > 0, the Fourier transformof a t is b a t ( ξ ) = − π t e − ξ/t θ ( ξ ) , where θ is the characteristic function of the interval (0 , ∞ ). The function b a t and all its derivatives are smooth away from ξ = 0 and decay exponentially as | ξ | → ∞ . It follows from the proof of Lemma 3.8 that the Schwartz kernel of w t ( e D ) − f M × f M . The same holds for w − t − w t w − t . Similar arguments as in the proof of Proposition 3.26 andProposition 3.27 show that ] ϕ w t w − t ⊗ ( w t ⊗ w − t ) ⊗ m ) is finite for each t > t .Now we prove the integral in line (8.1) converges absolutely for large t . Since G acts freely and cocompactly on f M , there exists a constant ε > x, gx ) > ε for all x ∈ f M and all g = e ∈ G . Fix a point x ∈ f M . For x ∈ f M ,let ν ( x ) be a smooth approximation of the distance from x to x . More precisely,let ν be a smooth function on f M satisfying the following:(1) dist( x, x ) ν ( x ) x, x ) if dist( x, x ) > ε ,(2) ν has uniformly bounded derivatives up to order N with N sufficientlylarge.Let δ be the unbound derivation on C ∗ ( f M ) G defined by δ ( T ) := [ T, ν ] = T ◦ ν − ν ◦ T for T ∈ C ∗ ( f M ) G . If T admits a distributional Schwartz kernel which is smoothaway from the diagonal of f M × f M , then we have δ k ( T )( x, y ) = T ( x, y )( ν ( x ) − ν ( y )) k , for all ( x, y ) ∈ f M × f M .Denote A t = w t ( e D ) − 1. We have δ ( A t ) = 2 it t e D + i [ e D, ν ] 1 t e D + i . Since ν has uniformly bounded derivatives and e D is invertible, there exists C > k e Dδ ( A t ) e D k op C t , where k · k op denotes the operator norm. By induction, we see that there exists C > k + j N +1 k e D k δ N ( A t ) e D j k op C t . Let K t ( x, y ) be the distributional Schwartz kernel of A t . Then the Schwartzkernel of δ N ( A t ) is K t ( x, y )( ν ( y ) − ν ( x )) N for all x, y ∈ f M . It follows from Lemma 3.5 that there exists C > | K t ( x, y ) | · | ν ( y ) − ν ( x ) | N C t for all x, y ∈ f M , where | K t ( x, y ) | is the norm of the matrix K t ( x, y ). In particular,if ( x, y ) = ( x , gx ) with g = e , then we have | K t ( x , gx ) | C t · (dist( x , gx )) N . Now for each x ∈ f M , use a smooth approximation of the distance functioncentered at x and apply the same estimates above. Since the action of G on f M is cocompact, we may choose C so that | K t ( x, gx ) | C t · (dist( x, gx )) N for all x ∈ F and all g = e ∈ G , where F is a fundamental domain of f M underthe action of G . Similar estimates hold for the Schwartz kernels of w − t − w t w − t .By assumption, G has polynomial growth and ϕ is a delocalized cyclic cocyclewith polynomial growth. A straightforward computation shows that there existsa constant C > (cid:12)(cid:12)(cid:12) ] ϕ w t w − t ⊗ ( w t ⊗ w − t ) ⊗ m ) (cid:12)(cid:12)(cid:12) Ct m +2 , which implies that the integral in line (8.1) converges absolutely for large t . Thisfinishes the proof. (cid:3) ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 67 References [1] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry. I. Math. Proc. Cambridge Philos. Soc. , 77:43–69, 1975.[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry. II. Math. Proc. Cambridge Philos. Soc. , 78(3):405–432, 1975.[3] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry. III. Math. Proc. Cambridge Philos. Soc. , 79(1):71–99, 1976.[4] Paul Baum and Ronald G. Douglas. K homology and index theory. In Operator algebrasand applications, Part I (Kingston, Ont., 1980) , volume 38 of Proc. Sympos. Pure Math. ,pages 117–173. Amer. Math. Soc., Providence, R.I., 1982.[5] Moulay-Tahar Benameur and Indrava Roy. The Higson-Roe exact sequence and ℓ etainvariants. J. Funct. Anal. , 268(4):974–1031, 2015.[6] P. Bernat, N. Conze, M. Duflo, M. L´evy-Nahas, M. Ra¨ıs, P. Renouard, and M. Vergne. Repr´esentations des groupes de Lie r´esolubles . Dunod, Paris, 1972. Monographies de laSoci´et´e Math´ematique de France, No. 4.[7] Jean-Michel Bismut and Jeff Cheeger. η -invariants and their adiabatic limits. J. Amer.Math. Soc. , 2(1):33–70, 1989.[8] Martin R. Bridson and Andr´e Haefliger. Metric spaces of non-positive curvature , volume319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences] . Springer-Verlag, Berlin, 1999.[9] Dan Burghelea. The cyclic homology of the group rings. Comment. Math. Helv. , 60(3):354–365, 1985.[10] Jeff Cheeger and Mikhael Gromov. Bounds on the von Neumann dimension of L -cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differential Geom. ,21(1):1–34, 1985.[11] A. Connes. Entire cyclic cohomology of Banach algebras and characters of θ -summableFredholm modules. K -Theory , 1(6):519–548, 1988.[12] Alain Connes. Noncommutative differential geometry. Inst. Hautes ´Etudes Sci. Publ.Math. , (62):257–360, 1985.[13] Alain Connes. On the Chern character of θ summable Fredholm modules. Comm. Math.Phys. , 139(1):171–181, 1991.[14] Alain Connes. Noncommutative geometry . Academic Press, Inc., San Diego, CA, 1994.[15] Alain Connes, Mikha¨ıl Gromov, and Henri Moscovici. Conjecture de Novikov et fibr´espresque plats. C. R. Acad. Sci. Paris S´er. I Math. , 310(5):273–277, 1990.[16] Alain Connes and Henri Moscovici. Cyclic cohomology, the Novikov conjecture and hyper-bolic groups. Topology , 29(3):345–388, 1990.[17] Pierre de la Harpe. Groupes hyperboliques, alg`ebres d’op´erateurs et un th´eor`eme de Jolis-saint. C. R. Acad. Sci. Paris S´er. I Math. , 307(14):771–774, 1988.[18] Robin J. Deeley and Magnus Goffeng. Realizing the analytic surgery group of Higson andRoe geometrically part III: higher invariants. Math. Ann. , 366(3-4):1513–1559, 2016.[19] Harold Donnelly. Eta invariants for G -spaces. Indiana Univ. Math. J. , 27(6):889–918, 1978.[20] Ezra Getzler. A short proof of the local Atiyah-Singer index theorem. Topology , 25(1):111–117, 1986.[21] Ezra Getzler. Cyclic homology and the Atiyah-Patodi-Singer index theorem. In Indextheory and operator algebras (Boulder, CO, 1991) , volume 148 of Contemp. Math. , pages19–45. Amer. Math. Soc., Providence, RI, 1993.[22] Alexander Gorokhovsky, Hitoshi Moriyoshi, and Paolo Piazza. A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings. J. Noncommut. Geom. , 10(1):265–306,2016.[23] M. Gromov. Hyperbolic groups. In Essays in group theory , volume 8 of Math. Sci. Res.Inst. Publ. , pages 75–263. Springer, New York, 1987. [24] Nigel Higson and John Roe. Mapping surgery to analysis. I. Analytic signatures. K -Theory ,33(4):277–299, 2005.[25] Nigel Higson and John Roe. Mapping surgery to analysis. II. Geometric signatures. K -Theory , 33(4):301–324, 2005.[26] Nigel Higson and John Roe. Mapping surgery to analysis. III. Exact sequences. K -Theory ,33(4):325–346, 2005.[27] Arthur Jaffe, Andrzej Lesniewski, and Konrad Osterwalder. Quantum K -theory. I. TheChern character. Comm. Math. Phys. , 118(1):1–14, 1988.[28] Sheagan John. Higher rho invariants, cyclic cohomology and groups of polynomial growth.thesis in prepration.[29] Paul Jolissaint. Rapidly decreasing functions in reduced C ∗ -algebras of groups. Trans.Amer. Math. Soc. , 317(1):167–196, 1990.[30] Gennadi Kasparov. Equivariant KK -theory and the Novikov conjecture. Invent. Math. ,91(1):147–201, 1988.[31] Gennadi Kasparov and Georges Skandalis. Groups acting properly on “bolic” spaces andthe Novikov conjecture. Ann. of Math. (2) , 158(1):165–206, 2003.[32] Navin Keswani. Geometric K -homology and controlled paths. New York J. Math. , 5:53–81,1999.[33] Vincent Lafforgue. A proof of property (RD) for cocompact lattices of SL(3 , R ) andSL(3 , C ). J. Lie Theory , 10(2):255–267, 2000.[34] Vincent Lafforgue. K -th´eorie bivariante pour les alg`ebres de Banach et conjecture deBaum-Connes. Invent. Math. , 149(1):1–95, 2002.[35] Vincent Lafforgue. La conjecture de Baum-Connes `a coefficients pour les groupes hyper-boliques. J. Noncommut. Geom. , 6(1):1–197, 2012.[36] Eric Leichtnam and Paolo Piazza. The b -pseudodifferential calculus on Galois coverings anda higher Atiyah-Patodi-Singer index theorem. M´em. Soc. Math. Fr. (N.S.) , (68):iv+121,1997.[37] ´Eric Leichtnam and Paolo Piazza. Homotopy invariance of twisted higher signatures onmanifolds with boundary. Bull. Soc. Math. France , 127(2):307–331, 1999.[38] John Lott. Higher eta-invariants. K -Theory , 6(3):191–233, 1992.[39] John Lott. Delocalized L -invariants. J. Funct. Anal. , 169(1):1–31, 1999.[40] John Lott. Diffeomorphisms and noncommutative analytic torsion. Mem. Amer. Math.Soc. , 141(673):viii+56, 1999.[41] Richard B. Melrose. The Atiyah-Patodi-Singer index theorem , volume 4 of Research Notesin Mathematics . A K Peters Ltd., Wellesley, MA, 1993.[42] Ralf Meyer. Combable groups have group cohomology of polynomial growth. Q. J. Math. ,57(2):241–261, 2006.[43] I. Mineyev. Straightening and bounded cohomology of hyperbolic groups. Geom. Funct.Anal. , 11(4):807–839, 2001.[44] Igor Mineyev and Guoliang Yu. The Baum-Connes conjecture for hyperbolic groups. In-vent. Math. , 149(1):97–122, 2002.[45] V. Nistor. Group cohomology and the cyclic cohomology of crossed products. Invent. Math. ,99(2):411–424, 1990.[46] Paolo Piazza and Thomas Schick. Groups with torsion, bordism and rho invariants. PacificJ. Math. , 232(2):355–378, 2007.[47] Paolo Piazza and Thomas Schick. Rho-classes, index theory and Stolz’ positive scalarcurvature sequence. J. Topol. , 7(4):965–1004, 2014.[48] Michael Puschnigg. New holomorphically closed subalgebras of C ∗ -algebras of hyperbolicgroups. Geom. Funct. Anal. , 20(1):243–259, 2010.[49] Daniel Quillen. Superconnections and the Chern character. Topology , 24(1):89–95, 1985. ELOCALIZED ETA, CYCLIC COHOMOLOGY AND HIGHER RHO 69 [50] John Roe. Coarse cohomology and index theory on complete Riemannian manifolds. Mem.Amer. Math. Soc. , 104(497):x+90, 1993.[51] John Roe. Index theory, coarse geometry, and topology of manifolds , volume 90 of CBMSRegional Conference Series in Mathematics . Published for the Conference Board of theMathematical Sciences, Washington, DC; by the American Mathematical Society, Provi-dence, RI, 1996.[52] Charlotte Wahl. The Atiyah-Patodi-Singer index theorem for Dirac operators over C ∗ -algebras. Asian J. Math. , 17(2):265–319, 2013.[53] Shmuel Weinberger. Higher ρ -invariants. In Tel Aviv Topology Conference: RothenbergFestschrift (1998) , volume 231 of Contemp. Math. , pages 315–320. Amer. Math. Soc.,Providence, RI, 1999.[54] Shmuel Weinberger, Zhizhang Xie, and Guoliang Yu. Additivity of higher rho invariantsand nonrigidity of topological manifolds. arXiv:1608.03661, 2016.[55] Shmuel Weinberger and Guoliang Yu. Finite part of operator K -theory for groups finitelyembeddable into Hilbert space and the degree of nonrigidity of manifolds. Geom. Topol. ,19(5):2767–2799, 2015.[56] Zhizhang Xie and Guoliang Yu. Positive scalar curvature, higher rho invariants and local-ization algebras. Adv. Math. , 262:823–866, 2014.[57] Zhizhang Xie and Guoliang Yu. Higher rho invariants and the moduli space of positivescalar curvature metrics. Adv. Math. , 307:1046–1069, 2017.[58] Zhizhang Xie and Guoliang Yu. Delocalized eta invariants, algebraicity, and K -theory ofgroup C ∗ -algebras. arXiv:1805.07617 , 2018.[59] Guoliang Yu. Localization algebras and the coarse Baum-Connes conjecture. K -Theory ,11(4):307–318, 1997.[60] Rudolf Zeidler. Positive scalar curvature and product formulas for secondary index invari-ants. J. Topol. , 9(3):687–724, 2016.(Xiaoman Chen) School of Mathematical Sciences, Fudan University E-mail address : [email protected] (Jinmin Wang) Shanghai Center for Mathematical Sciences E-mail address : [email protected] (Zhizhang Xie) Department of Mathematics, Texas A&M University E-mail address : [email protected] (Guoliang Yu) Department of Mathematics, Texas A&M University E-mail address ::