Approximations of delocalized eta invariants by their finite analogues
aa r X i v : . [ m a t h . K T ] A ug APPROXIMATIONS OF DELOCALIZED ETAINVARIANTS BY THEIR FINITE ANALOGUES
JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU
Abstract.
For a given self-adjoint first order elliptic differentialoperator on a closed smooth manifold, we prove a list of results onwhen the delocalized eta invariant associated to a regular coveringspace can be approximated by the delocalized eta invariants asso-ciated to finite-sheeted covering spaces. One of our main results isthe following. Suppose M is a closed smooth spin manifold and f M is a Γ-regular covering space of M . Let h α i be the conjugacy classof a non-identity element α ∈ Γ. Suppose { Γ i } is a sequence offinite-index normal subgroups of Γ that distinguishes h α i . Let π Γ i be the quotient map from Γ to Γ / Γ i and h π Γ i ( α ) i the conjugacyclass of π Γ i ( α ) in Γ / Γ i . If the scalar curvature on M is everywherebounded below by a sufficiently large positive number, then the de-localized eta invariant for the Dirac operator of f M at the conjugacyclass h α i is equal to the limit of the delocalized eta invariants forthe Dirac operators of M Γ i at the conjugacy class h π Γ i ( α ) i , where M Γ i = f M / Γ i is the finite-sheeted covering space of M determinedby Γ i . In another main result of the paper, we prove that the limitof the delocalized eta invariants for the Dirac operators of M Γ i atthe conjugacy class h π Γ i ( α ) i converges, under the assumption thatthe rational maximal Baum-Connes conjecture holds for Γ. Introduction
The delocalized eta invariant for self-adjoint elliptic operators wasfirst introduced by Lott [22] as a natural extension of the classical etainvariant of Atiyah-Patodi-Singer [1, 2, 3]. It is a fundamental invari-ant in the studies of higher index theory on manifolds with boundary,positive scalar curvature metrics on spin manfolds and rigidity prob-lems in topology. More precisely, the delocalized eta invariant can beused to detect different connected components of the space of posi-tive scalar curvature metrics on a given closed spin manifold [9, 21].
The first author is partially supported by NSFC 11420101001.The second author is partially supported by NSF 1800737.The third author is partially supported by NSF 1700021, NSF 1564398 andSimons Fellows Program.
Furthermore, it can be used to give an estimate of the connected com-ponents of the moduli space of positive scalar curvature metrics on agiven closed spin manifold [36]. Here the moduli space is obtained bytaking the quotient of the space of positive scalar curvature metricsunder the action of self-diffeomorphisms of the underlying manifold.As for applications to topology, the delocalized eta invariant can beapplied to estimate the size of the structure group of a given closedtopological manifold [33]. The delocalized eta invaraint is also closelyrelated to the Baum-Connes conjecture. The second and third authorsshowed that if the Baum-Connes conjecture holds for a given group Γ,then the delocalized eta invariant associated to any regular Γ-coveringspace is an algebraic number [37]. In particular, if a delocalized etainvariant is transcendental, then it would lead to a counterexample tothe Baum-Connes conjecture. We refer the reader to [38] for a moredetailed survey of the delocalized eta invariant and its higher analogues.The delocalized eta invariant, despite being defined in terms of anexplicit integral formula, is difficult to compute in general, due to itsnon-local nature. The main purpose of this article is to study whenthe delocalized eta invariant associated to the universal covering of aspace can be approximated by the delocalized eta invariants associatedto finite-sheeted coverings, where the latter are easier to compute.Let us first recall the definition of delocalized eta invariants. Let M be a closed manifold and D a self-adjoint elliptic differential operatoron M . Suppose Γ is a discrete group and f M is a Γ-regular coveringspace of M . Denote by e D the lift of D from M to f M . For any non-identity element α ∈ Γ, the delocalized eta invariant η h α i ( e D ) of e D atthe conjugacy class h α i is defined to be η h α i ( e D ) := 2 √ π Z ∞ X γ ∈h α i Z F tr( K t ( x, γx )) dxdt, (1.1)where K t ( x, y ) is the Schwartz kernel of the operator e De − t e D and F is a fundamental domain of M Γ under the action of Γ.We point out that it is still open question whether the convergenceof the integral in line (1.1) holds in general. A list of cases wherethe convergence is known to hold is given right after Definition 3.1. Inparticular, if Γ is finite, then the integral in line (1.1) always converges.Now suppose { Γ i } is a sequence of finite-index normal subgroups ofΓ. Let M Γ i = f M / Γ i be the associated finite-sheeted covering space of There is also an extra technical assumption that the conjugacy class h α i used inthe definition of the delocalized eta invariant is required to have polynomial growth. PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 3 M and D Γ i the lift of D from M to M Γ i . The delocalized eta invariant η h π Γ i ( α ) i ( D Γ i ) of D Γ i is defined similarly as in line (1.1), where π Γ i isthe canonical quotient map from Γ to Γ / Γ i . Suppose { Γ i } distinguishesthe conjugacy class h α i of a non-identity element α ∈ Γ (cf. Definition2 . i →∞ η h π Γ i ( α ) i ( D Γ i ) exist?(2) If lim i →∞ η h π Γ i ( α ) i ( D Γ i ) exists, is the limit equal to η h α i ( e D )?For simplicity, we assume that M is a closed spin manifold equippedwith a Riemannian metric of positive scalar curvature throughout thepaper. Positive scalar curvature implies e D has a spectral gap. In fact,the majority of results in this paper can be proved in the same wayunder the assumption that e D has a spectral gap or a sufficiently largespectral gap , or the operators D Γ i have a uniformly sufficiently largespectral gap .Here is one of the main results of our paper. Theorem 1.1.
With the above notation, assume that e D is invertibleand { Γ i } distinguishes the conjugacy class h α i of a non-identity element α ∈ Γ . If the maximal Baum-Connes assembly map for Γ is rationallyan isomorphism, then the limit lim i →∞ η h π Γ i ( α ) i ( D Γ i ) stabilizes, that is, ∃ k > such that η h π Γ i ( α ) i ( D Γ i ) = η h π Γ k ( α ) i ( D Γ k ) forall i > k. Here we say { Γ i } distinguishes the conjugacy class h α i if for anyfinite set F in Γ, there exists k ∈ N + such that ∀ β ∈ F, β / ∈ h α i = ⇒ π Γ i ( β ) / ∈ h π Γ i ( α ) i for all i > k .By a theorem of Higson and Kasparov [18, Theorem 1.1], the maxi-mal Baum-Connes assembly map is an isomorphism for all a-T-menablegroups. We have the following immediate corollary. Corollary 1.2.
With the above notation, assume that e D is invertibleand { Γ i } distinguishes the conjugacy class h α i of a non-identity element such as Theorem 1 . such as Theorem 1 . such as Theorem 5 . . JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU α ∈ Γ . If Γ is a-T-menable, then the limit lim i →∞ η h π Γ i ( α ) i ( D Γ i ) stabilizes. Note that Theorem 1 . i →∞ η h π Γ i ( α ) i ( D Γ i ). Onthe other hand, if in addition there exists a smooth dense subalgebra A of the reduced group C ∗ -algebra C ∗ r (Γ) of Γ such that C Γ ⊂ A andthe trace map tr h α i : C Γ → C extends continuously to a trace maptr h α i : A → C , then we havelim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . See the discussion at the end of Section 4 for more details. Just as anexample, we have the following theorem.
Theorem 1.3. If Γ is both a-T-menable and word hyperbolic , then lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Here is another main result of our paper.
Theorem 1.4.
With the above notation, suppose { Γ i } distinguishesthe conjugacy class h α i of a non-identity element α ∈ Γ . If the spectralgap of e D at zero is sufficiently large, then we have lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Here “sufficiently larger spectral gap” means that the spectral gapof e D at zero is greater than σ Γ , where σ Γ is the constant given inDefinition 5 .
1. In particular, if the group Γ has subexponential growth,then it follows from Definition 5 . σ Γ = 0. In this case, if e D hasa spectral gap, then it is automatically sufficiently large, hence thefollowing immediate corollary. Corollary 1.5.
With the above notation, suppose { Γ i } distinguishesthe conjugacy class h α i of a non-identity element α ∈ Γ . If Γ hassubexponential growth and e D has a spectral gap at zero , then we have lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . A smooth dense subalgebra of C ∗ r (Γ) is a dense subalgebra of C ∗ r (Γ) that isclosed under holomorphic functional calculus. The trace map tr h α i is given by the formula: P β ∈ Γ a β β P β ∈h α i a β . For example, if Γ is a virtually free group, then it is both a-T-menable and wordhyperbolic.
PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 5
There are other variants of Theorem 1 . . . C ∗ -algebras. In Section 3, we review some basics of delocalized etainvariants. In Section 4, we prove one of main results, Theorem 1 . . Preliminaries
In this section, we review some basic facts about conjugacy separablegroups and certain geometric C ∗ -algebras.2.1. Conjugacy separable groups.
We will prove our main approx-imation results for a particular class of groups, called conjugacy sep-arable groups. In this subsection, we review some basic properties ofconjugacy separable groups. In the following, all groups are assumedto be finitely generated, unless otherwise specified.
Definition 2.1.
Let Γ be a finitely generated discrete group. We saythat γ ∈ Γ is conjugacy distinguished if for any β ∈ Γ that is notconjugate to γ , there exists a finite-index normal subgroup Γ ′ of Γ suchthat the image of β in Γ / Γ ′ is not conjugate to γ .If every element in Γ is conjugacy distinguished, then we say that Γis conjugacy separable. In other words, we have the following definitionof conjugacy separability. Definition 2.2.
A finitely generated group Γ is conjugacy separableif for any γ , γ ∈ Γ that are not conjugate, there exists a finite-indexnormal subgroup Γ ′ of Γ such that the image of γ and γ in Γ / Γ ′ arenot conjugate.For any normal subgroup Γ ′ of Γ, we denote by π Γ ′ the quotient mapfrom Γ to Γ / Γ ′ . Definition 2.3.
Suppose that { Γ i } is a sequence of finite-index normalsubgroups of Γ. For any non-trivial conjugacy class h α i of Γ, we saythat { Γ i } distinguishes h α i , if for any finite set F in Γ there exists k ∈ N + such that ∀ β ∈ F, β / ∈ h α i = ⇒ π Γ i ( β ) / ∈ h π Γ i ( α ) i for all i > k . JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU If α ∈ Γ is conjugacy distinguished, then such sequence always exists.More generally, let N be the net of all normal subgroups of Γ with finiteindices. If α ∈ G is conjugacy distinguished in the sense of Definition2 .
1, then N distinguishes h α i , that is, for any finite set F ⊂ Γ, thereexists a finite index normal subgroup Γ F of Γ such that ∀ β ∈ F, β / ∈ h γ i = ⇒ π Γ ′ ( β ) / ∈ h π Γ ′ ( γ ) i for all Γ ′ ∈ N with Γ ′ ⊇ Γ F .Let C Γ be the group algebra of Γ and ℓ (Γ) be the ℓ -completionof C Γ. For any normal subgroup Γ ′ of Γ, the quotient map π Γ ′ : Γ → Γ / Γ ′ naturally induces an algebra homomorphism π Γ ′ : C Γ → C (Γ / Γ ′ ),which extends to a Banach algebra homomorphism π Γ ′ : ℓ (Γ) → ℓ (Γ / Γ ′ ).For any conjugacy class h γ i of Γ, let tr h γ i : C Γ → C be the trace mapdefined by the formula: X β ∈ Γ a β β X β ∈h γ i a β . The following lemma is obvious.
Lemma 2.4. If h α i is a non-trivial conjugacy class of Γ and { Γ i } is asequence of finite-index normal subgroups that distinguishes h α i , then lim i →∞ tr h π Γ i ( α ) i ( π Γ i ( f )) = tr h α i ( f ) for all f ∈ ℓ (Γ) . Moreover, if f ∈ C Γ , then the limit on the left handside stabilizes, that is, ∃ k > h π Γ i ( α ) i ( π Γ i ( f )) = tr h α i ( f ) , for all i > k. As we will mainly work with integral operators whose associatedSchwartz kernels are smooth, let us fix some notation further and re-state the above lemma in the context of integral operators. Let M bea closed manifold and f M be the universal covering space of M . Denotethe fundamental group π ( M ) of M by Γ. Suppose T is a Γ-equivariantbounded smooth function on f M × f M , that is, T ( γx, γy ) = T ( x, y )for all x, y ∈ f M and γ ∈ Γ. We say that T has finite propagation ifthere exists a constant d > x, y ) > d = ⇒ T ( x, y ) = 0 , where dist( x, y ) is the distance between x and y in f M . In this case, wedefine the propagation of T to be the infimum of such d . PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 7
Definition 2.5.
A Γ-equivariant bounded function T on f M × f M issaid to be ℓ -summable if k T k ℓ := sup x,y ∈F X γ ∈ Γ | T ( x, γy ) | < ∞ , where F is a fundamental domain of f M under the action of Γ. Weshall call k T k ℓ the ℓ -norm of T from now on.Clearly, every T with finite propagation is ℓ -summmable.If a Γ-equivariant bounded smooth function T ∈ C ∞ ( f M × f M ) is ℓ -summable, then it defines a bounded operator on L ( f M ) by theformula: f Z f M T ( x, y ) f ( y ) dy (2.1)for all f ∈ L ( f M ). For notational simplicity, we shall still denote thisoperator by T .Now suppose that Γ ′ is a finite-index normal subgroup of Γ. Let M Γ ′ = f M / Γ ′ be the quotient space of f M by the action of Γ ′ . Inparticular, M Γ ′ is a finite-sheeted covering space of M with the decktransformation group being Γ / Γ ′ . Let π Γ ′ be the quotient map from f M to M Γ ′ . Any Γ-equivariant bounded smooth function T ∈ C ∞ ( f M × f M )that is ℓ -summable naturally descends to a smooth function π Γ ′ ( T ) on M Γ ′ × M Γ ′ by the formula: π Γ ′ ( T )( π Γ ′ ( x ) , π Γ ′ ( y )) := X γ ∈ Γ ′ T ( x, γy )for all ( π Γ ′ ( x ) , π Γ ′ ( y )) ∈ M Γ ′ × M Γ ′ . Clearly, π Γ ′ ( T ) is a Γ / Γ ′ -equivariantsmooth function on M Γ ′ × M Γ ′ and, similar to the formula in (2.1), de-fines a bounded operator on L ( M Γ ′ ).For any non-trivial conjugacy class h α i of Γ, we define the followingtrace map: tr h α i ( T ) = X γ ∈h α i Z F T ( x, γx ) dx, for all Γ-equivariant ℓ -summable smooth function T ∈ C ∞ ( f M × f M ),where F is a fundamental domain of f M under the action of Γ. Moregenerally, for each finite-index normal subgroup Γ ′ of Γ, a similar tracemap is defined for Γ / Γ ′ -equivariant smooth functions on M Γ ′ × M Γ ′ .With the above notation, Lemma 2 . Lemma 2.6.
Suppose h α i is a non-trivial conjugacy class of Γ and { Γ i } is a sequence of finite-index normal subgroups that distinguishes JINMIN WANG, ZHIZHANG XIE, AND GUOLIANG YU h α i . Let T be a Γ -equivariant ℓ -summable bounded smooth functionon f M × f M . Then we have lim i →∞ tr h π Γ i ( α ) i ( π Γ i ( T )) = tr h α i ( T ) . Moreover, if T has finite propagation, the the limit on the left handside stabilizes, that is, ∃ k > such that tr h π Γ i ( α ) i ( π Γ i ( T )) = tr h α i ( T ) ,for all i > k. Geometric C ∗ -algebras. In this subsection, we review the defi-nitions of some geometric C ∗ -algebras, cf. [34, 40] for more details.Let X be a proper metric space, i.e. every closed ball in X is com-pact. An X -module is a separable Hilbert space equipped with a ∗ -representation of C ( X ). An X -module is called non-degenerated if the ∗ -representation of C ( X ) is non-degenerated. An X -module is calledstandard if no nonzero function in C ( X ) acts as a compact operator.In addition, we assume that a discrete group Γ acts on X properlyand cocompactly by isometries. Assume H X is an X -module equippedwith a covariant unitary representation of Γ. If we denote by ϕ and π the representations of C ( X ) and Γ respectively, this means π ( γ )( ϕ ( f ) v ) = ϕ ( γ ∗ f )( π ( γ ) v ) , where f ∈ C ( X ) , γ ∈ Γ , v ∈ H X and γ ∗ f ( x ) = f ( γ − x ). In this case,we call ( H X , Γ , ϕ ) a covariant system. Definition 2.7 ([41]) . A covariant system ( H X , Γ , ϕ ) is called admis-sible if(1) H X is a non-degenerate and standard X -module;(2) for each x ∈ X , the stabilizer group Γ x acts regularly in thesense that the action is isomorphic to the action of Γ x on l (Γ x ) ⊗ H for some infinite dimensional Hilbert space H . Here Γ x actson l (Γ x ) by translations and acts on H trivially.We remark that for each locally compact metric space X with aproper, cocompact and isometric action of Γ, an admissible covariantsystem ( H X , Γ , ϕ ) always exists. In particular, if Γ acts on X freely,then the condition (2) above holds automatically. Definition 2.8.
Let ( H X , Γ , ϕ ) be a covariant system and T a Γ-equivariant bounded linear operator acting on H X . • The propagation of T is defined to besup { d ( x, y ) : ( x, y ) ∈ supp ( T ) } , PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 9 where supp ( T ) is the complement (in X × X ) of points ( x, y ) ∈ X × X for which there exists f, g ∈ C ( X ) such that gT f = 0and f ( x ) = 0 , g ( y ) = 0; • T is said to be locally compact if f T and T f are compact forall f ∈ C ( X ). Definition 2.9.
Let X be a locally compact metric space with aproper and cocompact isometric action of Γ. Let ( H X , Γ , ϕ ) be anadmissible covariant system. We denote by C [ X ] Γ the ∗ -algebra of allΓ-equivariant locally compact bounded operators acting on H X withfinite propagations. We define the equivariant Roe algebra C ∗ ( X ) Γ tobe the completion of C [ X ] Γ under the operator norm.Indeed, C ∗ ( X ) Γ is isomorphic to C ∗ r (Γ) ⊗ K , the C ∗ -algebraic ten-sor product of the reduced group C ∗ -algebra of Γ and the algebra ofcompact operators. Definition 2.10.
We define the localization algebra C ∗ L ( X ) Γ to be the C ∗ -algebra generated by all uniformly bounded and uniformly norm-continuous function f : [0 , ∞ ) → C ∗ ( X ) Γ such that the propagation of f ( t ) goes to zero as t goes to infinity. Define C ∗ L, ( X ) Γ to be the kernelof the evaluation mapev : C ∗ L ( X ) Γ → C ∗ ( X ) , ev( f ) = f (0) . Now let us also review the construction of higher rho invariants forinvertible differential operators. For simplicity, let us focus on the odddimensional case. Suppose M is closed manifold of odd dimension.Let M Γ be the regular covering space of M whose deck transformationgroup is Γ. Suppose D is a self-adjoint elliptic differential operator on M and e D is the lift of D to M Γ . If e D is invertible, then its higher rhoinvariant is defined as follows. Definition 2.11.
With the same notation as above, the higher rhoinvariant ρ ( e D ) of an invertible operator e D is defined to be ρ ( e D ) := [ e πi χ ( e D/t )+12 ] ∈ K ( C ∗ L, ( M Γ ) Γ ) , where χ (called a normalizing function) is a continuous odd functionsuch that lim x →±∞ χ ( x ) = ± ρ ( e D ) is a uniformly norm-continuous function from [0 , ∞ ) to C ∗ ( X ) Γ . It is a secondary invariantthat serves as an obstruction for the higher index of e D to be bothtrivial and local (i.e. having small propagation) at the same time,cf. [11]. More precisely, for each fixed t , the unitary e πi χ ( e D/t )+12 is a representative of the higher index class of e D . On one hand, since e D isinvertible, e D has a spectral gap near zero. It follows that e πi χ ( e D/t )+12 converges in norm to the trivial unitary 1, as t goes to zero. On theother hand, the propagation of e πi χ ( e D/t )+12 goes to zero (up to operatorswith small norm) , as t goes to infinity. For an invertible e D , we canchoose a representative of the higher index of e D to be either trivialor local (i.e. having small propagation), but generally not both atthe same time. In other words, the higher rho invariant measures thetension between the triviality and locality of the higher index of aninvertible operator.The above discussion has an obvious maximal analogue (cf. [15,Lemma 3.4]). Definition 2.12.
For an operator T ∈ C [ X ] Γ , its maximal norm is k T k max := sup ϕ (cid:8) k ϕ ( T ) k : ϕ : C [ X ] Γ → B ( H ) is a ∗ -representation (cid:9) . The maximal equivariant Roe algebra C ∗ max ( X ) Γ is defined to be thecompletion of C [ X ] Γ with respect to k · k max . Similarly, we define(1) the maximal localization algebra C ∗ L, max ( X ) Γ to be the C ∗ -algebra generated by all uniformly bounded and uniformly norm-continuous function f : [0 , ∞ ) → C ∗ max ( X ) Γ such that the prop-agation of f ( t ) goes to zero as t goes to infinity.(2) and C ∗ L, , max ( X ) Γ to be the kernel of the evaluation mapev : C ∗ L, max ( X ) Γ → C ∗ max ( X ) , ev( f ) = f (0) . Now suppose M is a closed spin manifold. Assume that M is en-dowed with a Riemannian metric g of positive scalar curvature. Let M Γ be the regular covering space of M whose deck transformation groupis Γ. Suppose D is the associated Dirac operator on M and e D is thelift of D to M Γ . In this case, we can define the maximal higher rhoinvariant of e D as follows. Definition 2.13.
The maximal higher rho invariant ρ max ( e D ) of e D isdefined to be ρ max ( e D ) := [ e πi χ ( e D/t )+12 ] ∈ K ( C ∗ L, , max ( M Γ ) Γ ) , To be precise, one needs to use a normalizing function χ whose distributionalFourier transform has compact support, and furthermore approximate the function e πix by an appropriate polynomial. PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 11
Here χ is again a normalizing function, but the functional calculus fordefining χ ( t − e D ) is performed under the maximal norm instead. Seefor example [16, Section 3] for a discussion of such a functional calculus.3. Delocalized eta invariants and their approximations
In this section, we review the definition of delocalized eta invariantsand formulate the main question of this article.We assume that M is a closed spin manifold equipped with a Rie-mannian metric of positive scalar curvature throughout the paper. LetΓ be a finitely generated discrete group and f M a Γ-regular coveringspace of M . Suppose D is the associated Dirac operator on M and e D is the lift of D to f M .Positive scalar curvature implies e D has a spectral gap. In fact, themajority of results in this paper can be proved in the same way underthe assumption that e D has a spectral gap or a sufficiently large spectralgap , or the operators D Γ i have a uniformly sufficiently large spectralgap . For simplicity, we shall only discuss the case where M is aclosed spin manifold equipped with a Riemannian metric of positivescalar curvature. Definition 3.1 ([22]) . For any conjugacy class h α i of Γ, Lott’s delo-calized eta invariant η h α i ( e D ) of e D is defined to be η h α i ( e D ) := 2 √ π Z ∞ tr h α i ( e De − t e D ) dt (3.1)whenever the integral converges. Heretr h α i ( e De − t e D ) = X γ ∈h α i Z F tr( k t ( x, γx )) dx, where k t ( x, y ) is the corresponding Schwartz kernel of the operator e De − t e D and F is a fundamental domain of f M under the action of Γ.It is known that the integral formula (3.1) for η h α i ( e D ) converges if e D is invertible and any one of the following conditions is satisfied.(1) The scalar curvature of M is sufficiently large (see [12, Defini-tion 3.2] for the precise definition of “sufficiently large”). such as Theorem 1 . such as Theorem 1 . such as Theorem 5 . . (2) There exists a smooth dense subalgebra of C ∗ r (Γ) onto which thetrace map tr h α i extends continuously (cf. [22, Section 4]). Forexample, when Γ is a Gromov’s hyperbolic group, Puschnigg’ssmooth dense subalgebra [25] is such an subalgebra which ad-mits a continuous extension of the trace map tr h α i for all con-jugacy classes h h i .(3) h α i has subexponential growth (cf. [12, Corollary 3.4]).In general, it is still an open question when the integral in (3.1) con-verges for invertible operators.Now suppose that Γ ′ is a finite-index normal subgroup of Γ. As be-fore, let M Γ ′ = f M / Γ ′ be the associated finite-sheeted covering space of M . Similarly, let D Γ ′ be the lift of D to M Γ ′ , and define the delocalizedeta invariant η h π Γ ′ ( α ) i ( D Γ ′ ) of D Γ ′ to be η h π Γ ′ ( α ) i ( D Γ ′ ) := 2 √ π Z ∞ tr h π Γ ′ ( α ) i ( D Γ ′ e − t D ′ ) dt, (3.2)where α ∈ Γ and h π Γ ′ ( α ) i is conjugacy class of π Γ ′ ( α ) in Γ / Γ ′ . As M Γ ′ is compact, it is not difficult to verify that the integral in (3.2) alwaysconverges absolutely.The above discussion naturally leads to the following questions. Question . Given a non-identity element α ∈ Γ, suppose { Γ i } is asequence of finite-index normal subgroups that distinguishes the con-jugacy class h α i .(I) When does lim i →∞ η h π Γ i ( α ) i ( D Γ i ) exist?(II) If η h α i ( e D ) is well-defined and lim i →∞ η h π Γ i ( α ) i ( D Γ i ) exists, when dowe have lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D )? (3.3)4. Maximal higher rho invariants and their functoriality
In this section, we use the functoriality of higher rho invariants togive some sufficient conditions under which the answer to part (I) ofQuestion 3 . Proposition 4.1.
With the same notation as in Question . , if Γ isa-T-menable and { Γ i } is a sequence of finite-index normal subgroupsthat distinguishes the conjugacy class h α i for a non-identity element α ∈ Γ , then the limit lim i →∞ η h π Γ i ( α ) i ( D Γ i ) PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 13 stabilizes, that is, ∃ k > such that η h π Γ i ( α ) i ( D Γ i ) = η h π Γ k ( α ) i ( D Γ k ) , forall i > k. In particular, lim i →∞ η h π Γ i ( α ) i ( D Γ i ) exists.Proof. This is a consequence of Theorem 4 . (cid:3) Given a finitely presented discrete group Γ, let E Γ be the universalΓ-space for proper Γ-actions. The Baum-Connes conjecture [4] can bestated as follows.
Conjecture 4.2 (Baum-Connes conjecture) . The following map ev ∗ : K i ( C ∗ L ( E Γ) Γ ) → K i ( C ∗ ( E Γ) Γ ) is an isomorphism. Although this was not how the Baum-Connes conjecture was origi-nally stated, the above formulation is equivalent to the original Baum-Connes conjecture, after one makes the following natural identifica-tions: K i ( C ∗ L ( E Γ) Γ ) ∼ = K Γ i ( E Γ) and K i ( C ∗ ( E Γ) Γ ) ∼ = K i ( C ∗ r (Γ)) . Under this notation, we usually write the mapev ∗ : K i ( C ∗ L ( E Γ) Γ ) → K i ( C ∗ ( E Γ) Γ as follows: µ : K Γ i ( E Γ) → K i ( C ∗ r (Γ))and call it the Baum-Connes assembly map. Similarly, there is a max-imal version of the Baum-Connes assembly map: µ max : K Γ i ( E Γ) → K i ( C ∗ max (Γ)) . The maximal Baum-Connes assembly map µ max is not an isomorphismin general. For example, µ max fails to be surjective for non-finite prop-erty (T) groups.Before we discuss the functoriality of higher rho invariants, let usrecall the functoriality of higher indices. More precisely, let D be aDirac-type operator on a closed n -dimensional manifold X . Considerthe following commutative diagram B Γ Bϕ (cid:15) (cid:15) X f % % ▲▲▲▲▲▲ f rrrrrr B Γ where f , f are continuous maps and Bϕ is a continuous map from B Γ to B Γ induced by a group homomorphism ϕ : Γ → Γ . Let X Γ (resp. X Γ ) be the Γ (resp. Γ ) regular covering space of X inducedby the map f (resp. f ), and D X Γ1 (resp. D X Γ2 ) be the lift of D to X Γ (resp. X Γ ). We have the following functoriality of the higher indices: ϕ ∗ ( Ind max ( D X Γ1 )) = Ind max ( D X Γ2 ) in K n ( C ∗ max (Γ )) , where C ∗ max (Γ i ) is the maximal group C ∗ -algebra of Γ i , the notationInd max stands for higher index in the maximal group C ∗ -algebra, and ϕ ∗ : K n ( C ∗ max (Γ )) → K n ( C ∗ max (Γ )) is the morphism naturally inducedby ϕ .Now let us consider the functoriality of higher rho invariants. Follow-ing the same notation from above, in addition, assume X is a closedspin manifold endowed with a Riemannian metric of positive scalarcurvature. In this case, the maximal higher rho invariants ρ max ( D X Γ1 )of D X Γ2 and ρ max ( D X Γ1 ) of D X Γ2 are defined. Let E Γ (resp. E Γ )be universal Γ -space (resp. Γ -space ) for free Γ -actions (resp. Γ -actions). Denote by Φ the equivariant map X Γ → X Γ induced by ϕ : Γ → Γ , which in turn induces a morphismΦ ∗ : K n ( C ∗ L, , max ( X Γ ) Γ ) → K n ( C ∗ L, , max ( X Γ ) Γ ) . By [17], the maximal higher rho invariants are functorial:Φ ∗ ( ρ max ( D X Γ1 )) = ρ max ( D X Γ2 )in K n ( C ∗ L, , max ( X Γ ) Γ ).Now suppose M is an odd-dimensional closed spin manifold endowedwith a positive scalar curvature metric and Γ is a finitely generateddiscrete group. Let f M be a Γ-regular covering space of M and e D be theDirac operator lifted from M . For each finite-index normal subgroupΓ ′ of Γ, let M Γ ′ = f M / Γ ′ be the associated finite-sheeted covering spaceof M . Denote by D Γ ′ the Dirac opeartor on M Γ ′ lifted from M . Theorem 4.3.
With the above notation, given a non-identity element α ∈ Γ , suppose { Γ i } is a sequence of finite-index normal subgroups thatdistinguishes the conjugacy class h α i . If the maximal Baum-Connesassembly map for Γ is rationally an isomorphism, then lim i →∞ η h π Γ i ( α ) i ( D Γ i ) stabilizes, that is, ∃ k > such that η h π Γ i ( α ) i ( D Γ i ) = η h π Γ k ( α ) i ( D Γ k ) , forall i > k. Proof.
We have the short exact sequence of C ∗ -algebras:0 → C ∗ L, , max ( E Γ) Γ → C ∗ L, max ( E Γ) Γ → C ∗ max ( E Γ) Γ → PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 15 which induces the following long exact sequence in K -theory: K ( C ∗ L, , max ( E Γ) Γ ) ⊗ Q −−−→ K ( C ∗ L, max ( E Γ) Γ ) ⊗ Q µ −−−→ K ( C ∗ max (Γ)) ⊗ Q x y ∂ K ( C ∗ max (Γ)) ⊗ Q µ ←−−− K ( C ∗ L, max ( E Γ) Γ ) ⊗ Q ←−−− K ( C ∗ L, , max ( E Γ) Γ ) ⊗ Q (4.1)Note that K i ( C ∗ L, max ( E Γ) Γ ) is naturally isomorphic to K Γ i ( E Γ). Simi-larly, we have K i ( C ∗ L, max ( E Γ) Γ ) ∼ = K Γ i ( E Γ) . The morphism K Γ i ( E Γ) → K Γ i ( E Γ) induced by the inclusion from E Γ to E Γ is rationally injective(cf: [5, Section 7]). It follows that if the rational maximal Baum-Connes conjecture holds for Γ, that is, the maximal Baum-Connes as-sembly map µ max : K i ( C ∗ L, max ( E Γ) Γ ) ⊗ Q → K i ( C ∗ max (Γ)) ⊗ Q is anisomorphism, then the maps µ i in the above commutative diagram areinjective and the map ∂ is surjective. In particular, for the higher rhoinvariant ρ ( e D ) of e D , there exists[ p ] ∈ K ( C ∗ max ( E Γ) Γ ) ∼ = K ( C ∗ max (Γ))such that ∂ [ p ] = ρ ( e D ) rationally, that is, ∂ [ p ] = λ · ρ ( e D ) for some λ ∈ Q .By the surjectivity of the Baum-Connes assembly map µ max : K i ( C ∗ L, max ( E Γ) Γ ) ⊗ Q → K i ( C ∗ max (Γ)) ⊗ Q , we can assume p is an idempotent with finite propagation in K ⊗ C Γ.Indeed, let [ q ] be an element in K ( C ∗ L, max ( E Γ) Γ ) ⊗ Q ∼ = K Γ0 ( E Γ) ⊗ Q such that µ max ([ q ]) = [ p ]. It follows from the Baum–Douglas model ofK-homology [6] that [ q ] is the K-homology class of a twisted spin c Diracoperator and [ p ] is its higher index. More precisely, there is a spin c Γ-manifold X together with a Γ-equivariant vector bundle E such thatrationally [ p ] equals the Γ-index of the twisted Dirac operator /D E on X . For the convenience of the reader, we shall review the constructionof this index. Recall that a function χ on R is called a normalizingfunction if χ : R → [ − ,
1] is an odd continuous function such that χ ( x ) > x >
0, and χ ( x ) → ± x → ±∞ . Let f be asmooth normalizing function whose distributional Fourier transformhas compact support. Let us denote f ( /D E ) = (cid:18) F + F − (cid:19) . Using theformula f ( /D ) = 12 π Z R b f ( s ) e isD ds and the fact that e isD has propagation less than or equal to | s | , itfollows that f ( /D ) has finite propagation, since the Fourier transforms b f has compact support. Now we define w = (cid:18) F + (cid:19) (cid:18) − F − (cid:19) (cid:18) F + (cid:19) (cid:18) −
11 0 (cid:19) . Note that w − = (cid:18) − (cid:19) (cid:18) − F + (cid:19) (cid:18) F − (cid:19) (cid:18) − F + (cid:19) . The higher index of /D is given as the following formal difference ofidempotents: h w (cid:18) (cid:19) w − i − h (cid:18) (cid:19) i . By construction, we have w (cid:18) (cid:19) w − ∈ S ⊗ C Γwhere S is the algebra of trace class operators on a Hilbert space. Inparticular, we see that the element [ p ] ∈ K ( C ∗ max (Γ)) from above canbe (rationally) represented by a formal difference of idempotents withfinite propagation.Let Ψ i be the canonical quotient map from f M to M Γ i = f M / Γ i and(Ψ i ) ∗ : K ( C ∗ L, , max ( f M ) Γ ) → K ( C ∗ L, , max ( M Γ i ) Γ / Γ i )the corresponding morphism induced by Ψ i . By [17], we have(Ψ i ) ∗ ( ρ max ( e D )) = ρ ( D Γ i ) in K ( C ∗ L, , max ( M Γ i ) Γ / Γ i ) . By passing to the universal spaces, we have(Ψ i ) ∗ ( ρ max ( e D )) = ρ ( D Γ i ) in K ( C ∗ L, , max ( E (Γ / Γ i )) Γ / Γ i ) . Consider the following commutative diagram of long exact sequences : K ( C ∗ L, max ( E Γ) Γ ) ⊗ Q (cid:15) (cid:15) / / K ( C ∗ max (Γ)) ⊗ Q ∂ / / ( π Γ i ) ∗ (cid:15) (cid:15) K ( C ∗ L, , max ( E Γ) Γ ) ⊗ Q (Ψ i ) ∗ (cid:15) (cid:15) K ( C ∗ L ( E (Γ / Γ i )) Γ / Γ i ) ⊗ Q / / K ( C ∗ r (Γ / Γ i )) ⊗ Q ∂ / / K ( C ∗ L, ( E (Γ / Γ i )) Γ / Γ i ) ⊗ Q (4.2)where ( π Γ i ) ∗ : K ( C ∗ max (Γ)) → K ( C ∗ r (Γ / Γ i )) is the natural morphisminduced by the canonical quotient map π Γ i : Γ → Γ / Γ i . Let us denote( π Γ i ) ∗ ( p ) by p i . It follows from the commutative diagram above that ∂ ( p i ) = ρ ( D Γ i ) . (4.3) Since Γ / Γ i is finite, we have C ∗ L, , max ( E (Γ / Γ i )) Γ / Γ i ∼ = C ∗ L, ( E (Γ / Γ i )) Γ / Γ i . PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 17
By [37, Lemma 3.9 & Theorem 4.3], for each Γ i , there exists a deter-minant map τ i : K ( C ∗ L, ( E (Γ / Γ i )) Γ / Γ i ) → C such that 12 η h π Γ i ( α ) i ( D Γ i ) = − τ i ( ρ ( D Γ i )) = tr h π Γ i ( α ) i ( p i ) . Since the idempotent p has finite propagation, it follows from Lemma2.4 that lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = 2 lim i →∞ tr h π Γ i ( α ) i ( p i ) = 2tr h α i ( p ) , and the limit stabilizes. (cid:3) Remark . (1) In Theorem 4 . { Γ i } isa sequence of finite-index normal subgroups that distinguishesthe conjugacy class h α i , we assume that { Γ i } is a decreasingsequence of finite-index normal subgroups of Γ. The sameproof shows that if the maximal Baum-Connes assembly mapfor Γ is rationally an isomorphism, thenlim i →∞ η h π Γ i ( α ) i ( D Γ i )stabilizes. On the other hand, to eventually relate the limitlim i →∞ η h π Γ i ( α ) i ( D Γ i ) to η h α i ( e D ), if the latter exists, one willlikely need to assume the condition that { Γ i } distinguishes h α i .(2) Note that Theorem 4 . . . . ρ max ( e D ) is in the image of the composition of thefollowing maps: K Γ0 ( E Γ) ⊗ Q → K ( C ∗ max (Γ)) ⊗ Q ∂ −−→ K ( C ∗ L, , max ( E Γ) Γ ) ⊗ Q . By a theorem of Higson and Kasparov [18, Theorem 1.1], the max-imal Baum-Connes conjecture holds for all a-T-menable groups. To-gether with Theorem 4 . . We say { Γ i } is a decreasing sequence of finite-index normal subgroups of Γ ifΓ i ⊇ Γ i +1 for all i . As mentioned above, the maximal Baum-Connes assembly map µ max fails to be an isomorphism in general. For example, µ max fails to besurjective for non-finite property (T) groups. On the contrary, thereis no counterexample to the Baum-Connes conjecture, at the time ofwriting. In particular, the Baum-Connes conjecture is known to holdfor all hyperbolic groups [20, 23], many of which have property (T).For this reason, we shall now investigate Question 3 .
2, in particular,the convergence of lim i →∞ η h π Γ i ( α ) i ( D Γ i )when the group Γ satisfies the Baum-Connes conjecture.One of the first difficulties we face is that reduced group C ∗ -algebrasare not functorial with respect to group homomorphisms in general.As a result, the functoriality of higher rho invariants is, a priori, lostin the reduced C ∗ -algebra setting. Note that a key step (cf. Equation(4.3)) in the proof of Theorem 4 . p ∈ S ⊗ C Γ such that ∂ ( p i ) = ρ ( D Γ i ) , where p i = ( π Γ i ) ∗ ( p ). In the maximal setting, the existence of sucha universal idempotent follows if the rational maximal Baum-Connesconjecture holds for Γ. In the following, we shall discuss some geometricconditions that are sufficient for deriving an analogue of Theorem 4 . A .Recall that M is a closed spin manifold equipped with a Riemannianmetric h of positive scalar curvature. Let ϕ : M → B Γ be the classi-fying map for the covering f M → M , that is, the pullback of E Γ by ϕ is f M . In the following, we denote by B the Bott manifold, a simplyconnected spin manifold of dimension 8 with b A ( B ) = 1. This manifoldis not unique, but any choice will work for the following discussion. Definition 4.5.
We say a multiple of (
M, ϕ, h ) stably bounds withrespect to B Γ if there exists a compact spin manifold W and a mapΦ : W → B Γ such that ∂W = F ℓi =1 M ′ and Φ | ∂W = F ℓi =1 ϕ ′ , where( M ′ , ϕ ′ , h ′ ) is the direct product of ( M, ϕ, h ) with finitely many copiesof B and F ℓi =1 M ′ is the disjoint union of ℓ copies of M ′ . Definition 4.6.
Let ˜ h be the metric on f M lifted from h . We say amultiple of ( f M , ˜ h ) positively stably bounds with respect to E Γ if there
PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 19 exists a spin cocompact Γ-manifold e V equipped with a Γ-equivariantpositive scalar curvature metric g e V such that ∂ e V = F ℓi =1 f M ′ (as Γ-manifolds) and g e V has product structure near ∂ e V , where ( f M ′ , e h ′ ) isthe direct product of ( f M , e h ) with finitely many copies of B .The following proposition is an analogue of Theorem 4 . M, ϕ, h ) sta-bly bounds with respect to B Γ and a multiple of ( f M , ˜ h ) positivelystably bounds with respect to E Γ. For example, if M is a Lens spaceequipped with the metric inherited from the standard round metric on S n and Γ = π ( M ), then both of these assumptions are satisfied. Ingeneral, the validity of these two assumptions is closely related to thereduced Baum-Connes conjecture and the Stolz conjecture on positivescalar curvature metrics. We refer the reader to Appdendix A for moredetails. Proposition 4.7.
Let M be a closed spin manifold equipped with a Rie-mannian metric h of positive scalar curvature. Given a non-identityelement α ∈ Γ , suppose { Γ i } is a sequence of finite-index normal sub-groups of Γ that distinguishes the conjugacy class h α i . If a multipleof ( M, ϕ, h ) stably bounds with respect to B Γ and a multiple of ( f M , ˜ h ) positively stably bounds with respect to E Γ , then lim i →∞ η h π Γ i ( α ) i ( D Γ i ) stabilizes, that is, ∃ k > such that η h π Γ i ( α ) i ( D Γ i ) = η h π Γ k ( α ) i ( D Γ k ) , forall i > k. Proof.
For notational simplicity, let us assume (
M, ϕ, h ) itself boundswith respect to B Γ, that is, there exists a compact spin manifold W and a map Φ : W → B Γ such that ∂W = M and Φ | ∂W = ϕ . Similarly,let us assume ( f M , ˜ h ) itself positively stably bounds with respect to E Γ,that is, there exists a cocompact Γ-spin manifold e V with ∂ e V = F ℓi =1 f M ′ (as Γ-manifolds) and e V is equipped with a Γ-equivariant positive scalarcurvature metric that has product structure near ∂ e V . The general casecan be proved in exactly the same way.Endow W with a Riemannian metric g which has product structurenear ∂W = M and whose restriction on ∂W is the positive scalarcurvature metric h . Let f W be the covering space of W induced bythe map Φ : W → B Γ and ˜ g be the lift of g from W to f W . Due Here a Γ-manifold is a Riemannian manifold equipped with a proper isometricaction of Γ. to the positive scalar curvature of ˜ g near the boundary of f W , thecorresponding Dirac operator D f W on f W with respect to the metric ˜ g has a well-defined higher index Ind( D f W , ˜ g ) in KO n +1 ( C ∗ r (Γ; R )) . Now for each normal subgroup Γ i of Γ, let M Γ i = f M / Γ i , W Γ i = f W / Γ i and g i be the lift of g to W Γ i . Similarly, the corresponding Diracoperator D W Γ i on W Γ i with respect to the metric g i has a well-definedhigher index Ind( D W Γ i , g i ) in KO n +1 ( C ∗ r (Γ / Γ i ; R )) . Moreover, we have ∂ ( Ind( D W Γ i , g i )) = ρ ( D M Γ i ) in KO n ( C ∗ L, ( E (Γ / Γ i ); R ) Γ / Γ i ) , cf. [24, Theorem 1.14][34, Theorem A].By [37, Lemma 3.9 & Theorem 4.3], for each Γ i , there exists a de-terminant map τ i : K ( C ∗ L, , max ( E (Γ / Γ i )) Γ / Γ i ) → C such that12 η h π Γ i ( α ) i ( D M Γ i ) = − τ i ( ρ ( D M Γ i )) = tr h π Γ i ( α ) i ( Ind( D W Γ i , g i )) . Therefore, to prove the proposition, it suffices to show that there ex-ists [ p ] ∈ KO n +1 ( C ∗ max (Γ; R )) such that [ p ] is represented by a formaldifference of idempotents in S ⊗ C Γ and( π Γ i ) ∗ ([ p ]) = Ind( D W Γ i , g i )for all k , where ( π Γ i ) ∗ : C ∗ max (Γ; R ) → C ∗ r (Γ / Γ i ; R ) is the morphisminduced by the quotient homomorphism π Γ i : Γ → Γ / Γ i . The existenceof such a “universal” K -theory element with finite propagation can beseen as follows.Let Y be the spin Γ-manifold obtained by gluing e V and f W along theircommon boundary f M . Since the scalar curvature on e V is uniformlybounded below by a positive number, it follows from the relative indextheorem [10, 35] thatInd max ( D Y ) = Ind max ( D f W , ˜ g ) in KO n +1 ( C ∗ max (Γ; R )) . Let p = Ind max ( D Y ). By the discussion in the proof of Theorem 4 . max ( D Y ) can be represented by a formal differenceof idempotents in S ⊗ C Γ. On the other hand, we have( π Γ i ) ∗ ( Ind max ( D f W , ˜ g )) = Ind( D W Γ i , g i )for all i . To summarize, we have( π Γ i ) ∗ ([ p ]) = ( π Γ i ) ∗ ( Ind max ( D f W , ˜ g )) = Ind( D W Γ i , g i ) . This finishes the proof. (cid:3)
PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 21
In Theorem 4 . .
7, we have mainly focused on thepart (I) of Question 3 .
2. In the following, we shall try to answer part(II) of Question 3 . . . K -theory element [ p max ] ∈ K n +1 ( C ∗ max (Γ) ⊗ K ) that is represented bya formal difference of idempotents in S ⊗ C Γ such that ∂ ( p max ) = ρ max ( e D ) , where ∂ : K n +1 ( C ∗ max (Γ)) → KO n ( C ∗ L, , max ( E Γ) Γ )is the usual boundary map in the corresponding K -theory long exactsequence. We shall assume the existence of such a K -theory element p max throughout the rest of the section.In addition, suppose there exists a smooth dense subalgebra A of C ∗ r (Γ) such that A ⊃ C Γ and the trace map tr h α i : C Γ → C extends toa trace map A → C . In this case, tr h α i : A → C induces a trace maptr h α i : K ( C ∗ r (Γ)) ∼ = K ( A ) → C and a determinant map (cf. [37]) τ α : K ( C ∗ L, ( E Γ) Γ ) → C such that the following diagram commutes: K ( C ∗ r (Γ)) ∂ / / − tr h α i (cid:15) (cid:15) K ( C ∗ L, ( E Γ) Γ ) τ α (cid:15) (cid:15) C = / / C Such a smooth dense subalgebra indeed exists if h α i has polynomialgrowth (cf. [13][37]) or Γ is word hyperbolic (cf. [25][12]).Note that the canonical morphism K ( C ∗ L, , max ( E Γ) Γ ) → K ( C ∗ L, ( E Γ) Γ )maps ρ max ( e D ) to ρ ( e D ). Let p r be the image of p max under the canonicalmorphism K ( C ∗ max (Γ)) → K ( C ∗ r (Γ)). The same argument from theproof of Theorem 4 . ∂ ( p r ) = ρ ( e D ) and12 η h α i ( e D ) = − τ α ( ρ ( e D )) = tr h α i ( p r ) . Similarly, for each finite-index normal subgroup Γ i ⊂ Γ, let( π Γ i ) ∗ : K ( C ∗ max (Γ)) → K ( C ∗ r (Γ / Γ i )) In the case of Proposition 4 .
7, we map KO -theory to K -theory. be the natural morphism induced by the quotient map π Γ i : Γ → Γ / Γ i .Let us denote p i := ( π Γ i ) ∗ ( p ). We have ∂ ( p i ) = ρ ( D Γ i ) and12 η h π Γ i ( α ) i ( D Γ i ) = − τ i ( ρ ( D Γ i )) = tr h π Γ i ( α ) i ( p i ) . where τ i : K ( C ∗ L, ( E (Γ / Γ i )) Γ / Γ i ) → C is a determinant map induced by the trace map tr h π Γ i ( α ) i , cf. [37,Lemma 3.9 & Theorem 4.3]. Since p max is a formal difference of idem-potents in S ⊗ C Γ, it follows that the limit lim i →∞ tr h π Γ i ( α ) i ( p i ) stabilizesand is equal to tr h α i ( p r ). Thus the limitlim i →∞ η h π Γ i ( α ) i ( D Γ i )stabilizes and is equal to η h α i ( e D ).In particular, as a consequence of the above discussion and Theorem4 .
3, we have the following theorem
Theorem 4.8. If Γ is both a-T-menable and word hyperbolic , then lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Scalar curvature and ℓ -summability In this section, we show that the answers to both part (I) and part(II) of Question 3 . M is bounded below by a sufficiently large positive num-ber.Throughout this section, assume M is an odd-dimensional closedspin manifold endowed with a positive scalar curvature metric and Γ isa finitely generated discrete group. Let f M be a regular Γ-covering spaceof M and e D be the Dirac operator lifted from M . For each finite-indexnormal subgroup Γ ′ of Γ, let M Γ ′ = f M / Γ ′ be the associated finite-sheeted covering space of M . Denote by D Γ ′ the Dirac opeartor on M Γ ′ lifted from M .Let S be a symmetric finite generating set of Γ and ℓ be the associatedword length function on Γ. There exist C >
B > { γ ∈ Γ : ℓ ( g ) n } Ce B · n . (5.1)for all n ≥
0. Let K Γ be the infimum of all such numbers B . For example, if Γ is a virtually free group, then it is both a-T-menable andword hyperbolic.
PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 23
Furthermore, there exist θ , θ , c , c > θ · ℓ ( β ) − c dist( x, βx ) θ · ℓ ( β ) + c (5.2)for all x ∈ F and β ∈ Γ, where F is a fundamental domain of f M underthe action of Γ. In particular, we may define θ as follows: θ = lim inf ℓ ( β ) →∞ (cid:18) inf x ∈F dist( x, βx ) ℓ ( β ) (cid:19) . (5.3) Definition 5.1.
With the above notation, let us define σ Γ := 2 K Γ θ . The following theorem answers both part (I) and part (II) of Ques-tion 3 . e D atzero is sufficiently large. Theorem 5.2.
With the same notation as above, suppose { Γ i } is asequence of finite-index normal subgroups that distinguishes the conju-gacy class h α i of a non-identity element α ∈ Γ . If the spectral gap of e D at zero is greater than σ Γ , then lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Proof.
It suffices to find a function of t that is a dominating functionfor all the following functions:tr h α i ( e De − t e D ) = X γ ∈h α i Z F tr( K t ( x, γx )) dx and tr h π Γ i ( α ) i ( D Γ i e − t D i ) = X ω ∈h π Γ i ( α ) i Z F tr(( K i ) t ( x, ωx )) dx. and show that tr h π Γ i ( α ) i ( D Γ i e − t D i ) converges to tr h α i ( e De − t e D ), as i →∞ , for each t . Indeed, the theorem then follows by the dominatedconvergence theorem.Recall that K t ( x, y ) (resp. ( K i ) t ( x, y )) is the Schwartz kernel of e De − t e D (resp. D Γ i e − t D i ). We have the following estimates (cf. [12,Section 3]).(1) By [12, Lemma 3.8], for any µ > r >
0, there exists aconstant c µ,r > k K t ( x, y ) k c µ,r · F t (cid:18) dist( x, y ) µ (cid:19) , (5.4) for ∀ x, y ∈ f M with dist( x, y ) > r . Here k K t ( x, y ) k is the opera-tor norm of the matrix K t ( x, y ), and the function F t is definedby F t ( s ) := sup n dim M +3 Z | ξ | >s (cid:12)(cid:12)(cid:12)(cid:12) d n dξ n b f t ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) dξ, where b f t is the Fourier transform of f t ( x ) = xe − t x . It followsthat for µ > r >
0, there exist c µ,r > , n > m > k K t ( x, y ) k c µ,r (1 + dist( x, y )) n t m exp (cid:18) − dist( x, y ) µt (cid:19) , (5.5)for all t > x, y ∈ f M with dist( x, y ) > r .(2) By [12, Lemma 3.5], there exists c > x,y ∈ M k K t ( x, y ) k c · sup k + j dim M +3 k e D k ( e De − t e D ) e D j k op , (5.6)for all x, y ∈ f M , where k · k op stands for the operator norm. Itfollows that there exist positive numbers c , m and δ such that k K t ( x, y ) k c t m exp( − ( σ Γ + δ ) · t ) , (5.7)for all t > x, y ∈ f M .In fact, since the manifolds f M and M Γ i have uniformly boundedgeometry, the constants c µ,r , n , m , c , m and δ from above can bechosen so that for all i ≥
1, we have k ( K i ) t ( x, y ) k c µ,r (1 + dist( x, y )) n t m exp (cid:18) − dist( x, y ) µt (cid:19) , (5.8)for all t > x, y ∈ M Γ i with dist( x, y ) > r ; and k ( K i ) t ( x, y ) k c t m exp( − ( σ Γ + δ ) · t ) , (5.9)for all t > x, y ∈ M Γ i .For the rest of the proof, let us fix r >
0. Note that we havedist( x, γy ) ≤ dist( x, γx ) + dist( γx, γy )= dist( x, γx ) + dist( x, y )for all x, y ∈ f M and γ ∈ Γ. Similarly, we havedist( x, γx ) − dist( x, y ) ≤ dist( x, γy ) . By line (5.2), we have θ · ℓ ( γ ) − c − dist( x, y ) dist( x, γy ) θ · ℓ ( γ ) + c + dist( x, y ) PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 25 for all x, y ∈ f M and γ ∈ Γ. In particular, there exist c ′ > c ′ > θ · ℓ ( γ ) − c ′ dist( x, γy ) θ · ℓ ( γ ) + c ′ (5.10)for all x, y ∈ F and γ ∈ Γ, where F is a precompact fundamentaldomain of f M under the action of Γ. Let us define F := { β ∈ Γ | dist( x, βy ) ≤ r for some x, y ∈ F } . Clearly, F is a finite subset of Γ.For any given t >
0, it follows from line (5.1) , (5.5) and (5.10) thatthe Schwartz kernel K t is ℓ -summable (cf. Definition 2 . f t ( x ) = xe − t x by smooth functions { ϕ j } whoseFourier transforms are compactly supported. By applying the estimatesin line (5.4) and (5.6) to the Schwartz kernel K ϕ j ( e D ) (resp. K ϕ j ( D Γ i ) ) ofthe operator ϕ j ( e D ) (resp. ϕ j ( D Γ i )), it is not difficult to see that K ϕ j ( e D ) (resp. K ϕ j ( D Γ i ) ) converges to K t (resp. ( K i ) t ) in ℓ -norm (defined inDefinition 2 . ϕ j ( e D ) has finite propagation. Since e D locallycoincides with D Γ i , it follows from finite propagation estimates of waveoperators that (cf. [17]): K ϕ j ( D Γ i ) ( π Γ i ( x ) , π Γ i ( y )) = X β ∈ Γ i K ϕ j ( e D ) ( x, βy )for all x, y ∈ f M . As a consequence of the above discussion, we have( K i ) t ( π Γ i ( x ) , π Γ i ( y )) = X β ∈ Γ i K t ( x, βy ) . (5.11)for all x, y ∈ f M and for all t >
0. Furthermore, by Lemma 2 .
6, wehave the following convergence:tr h π Γ i ( α ) i ( D Γ i e − t D i ) → tr h α i ( e De − t e D ) , as j → ∞ , for each t >
0, since { Γ i } distinguishes the conjugacy class h α i .By line (5.5) and (5.7), there exists a positive number c such that k K t ( x, y ) k ≤ c (1 + dist( x, y )) n t m + m exp (cid:18) − dist( x, y ) µt (cid:19) e − ( σ Γ + ε ) · t e − ε t ≤ c (1 + dist( x, y )) n t m + m exp (cid:18) − dist( x, y ) · ( σ Γ + ε ) µ (cid:19) e − ε t , for all x, y ∈ f M with dist( x, y ) > r , where ε = δ/
2. By choosing µ > c > λ > that k K t ( x, y ) k ≤ c e − ε t t ( m + m ) / exp( − λ · σ Γ · dist( x, y ))for all x, y ∈ f M with dist( x, y ) > r . It follows that there exist c > m > X γ ∈ Γ k K t ( x, γy ) k X γ ∈ F c e − ( σ Γ + δ ) · t t m + c e − ε t t ( m + m ) / X γ / ∈ F e − λ · K Γ · ( ℓ ( γ ) − c ′ τ − ) c · | F | · e − ε t t m + c e − ε t t ( m + m ) / ∞ X n =0 e K Γ · n e − λ · K Γ · ( n − c ′ τ − ) < c t m e − ε t for all t >
0. In particular, we have X γ ∈h α i (cid:12)(cid:12) tr K t ( x, γy ) (cid:12)(cid:12) < c t m e − ε t for all x, y ∈ f M and all t >
0. By the same argument, we also have X ω ∈h π Γ i ( α ) i (cid:12)(cid:12) tr( K i ) t ( x ′ , ωy ′ ) (cid:12)(cid:12) < c t m e − ε t for all x ′ , y ′ ∈ M Γ i and all t >
0. Therefore, the functions | tr h α i ( e De − t e D ) | and | tr h π Γ i ( α ) i ( D Γ i e − t D i ) | are all bounded by the function c · t − m e − ε t . The latter is clearly absolutely integrable on [1 , ∞ ).We have found above an appropriate dominating function on theinterval [1 , ∞ ). Now let us find the dominating function on (0 , f M freely and cocompactly, it follows that there exists ε > x, γx ) > ε for all x ∈ f M and all γ = e ∈ Γ. By applying line (5.5), a similarcalculation as above shows that there exist ε > c > X γ ∈ Γ k K t ( x, γx ) k < c t m e − ε · t − PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 27 for all x ∈ F and all t ≤
1. The same estimate also holds for ( K i ) t .Therefore, on the interval (0 , | tr h α i ( e De − t e D ) | and | tr h π Γ i ( α ) i ( D Γ i e − t D i ) | are all bounded by the function c · t − m e − ε · t − . The latter is absolutely integrable on (0 , (cid:3) If the group Γ has subexponential growth, then it follows from Def-inition 5 . σ Γ = 0. In this case, if e D has a spectral gap, then it isautomatically sufficiently large, hence the following immediate corol-lary. Corollary 5.3.
With the above notation, suppose { Γ i } distinguishesthe conjugacy class h α i of a non-identity element α ∈ Γ . If Γ hassubexponential growth and e D has a spectral gap at zero , then we have lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Remark . Recall that if Γ has subexponential growth, then Γ isamenable and thus a-T-menable [8]. So the convergence of the limitlim i →∞ η h π Γ i ( α ) i ( D Γ i )also follows from Proposition 4 .
1. In fact, Proposition 4 . . . Remark . In this remark, we shall briefly comment on the conditionthat the spectral gap of e D at zero is greater than σ Γ in Theorem 5 . e D atzero is greater than σ Γ and the higher rho invariant ρ ( e D ) is nonzero.Suppose that N is a closed spin manifold equipped with a positivescalar curvature metric g N , whose fundamental group F = π ( N ) isfinite and its higher rho invariant ρ ( e D N ) is nontrivial. Here e D N is theDirac operator on the universal covering e N of N . For instance, let N tobe a lens space, that is, the quotient of the 3-dimensional sphere by afree action of a finite cyclic group. In this case, the classical equivariantAtiyah-Patodi-Singer index theorem implies that the delocalized higherrho invariant of N is nontrivial, cf. [14]. As we have seen in the proofs of Theorem 4 . .
8, the delocalizedeta invariant η h α i ( e D ) is essentially the pairing between the higher rho invariant ρ ( e D )and the delocalized trace tr h α i , cf. [37, Theorem 4.3]. Now let X be an even dimensional closed spin manifold, whose Diracoperator D X has nontrivial higher index in K ( C ∗ r (Γ)), where Γ = π ( X ). In particular, it follows that D X defines a nonzero element inthe equivariant K -homology K ( C ∗ L ( E Γ) Γ ) of the universal space E Γfor free Γ actions. Consider the product space M = V × N equippedwith a metric g M = g X + ε · g N , where g X is an arbitrary Riemannianmetric on X and the metric g N on N is scaled by a positive number ε . Denote the Dirac operator on the universal covering f M of M by e D M . The spectral gap of e D M at zero can always be made sufficientlylarge, as long as we choose ε to be sufficiently small. To see that ρ ( e D M ) is nonzero in K ( C ∗ L, ( E (Γ × F )) Γ × F ), we apply the productformula for secondary invariants (cf. [34, Claim 2.19], [42, Corollary4.15]), which states that the higher rho invariant ρ ( e D M ) is the productof the K -homology class of D X and the higher rho invariant ρ ( e D N ).It follows from the above construction that the higher rho invariant ρ ( e D M ) is nonzero in K ( C ∗ L, ( E (Γ × F )) Γ × F ). In fact, if the Baum-Connes conjecture holds for Γ, then the K -theory group K ( C ∗ L, ( E (Γ × F )) Γ × F ) is (at least rationally) generated by the higher rho invariantsof the above examples, cf. [39, Theorem 3.7 & Corollary 3.16].On the other hand, we also would like to point out that for theoperator e D M from the above examples, a straightfoward calculationshows the delocalized eta invariant η h α i ( e D M ) is nonzero only in thecase when α = (1 , a ) ∈ Γ × F with a being a non-identity element of F = π ( N ). This essentially reduces the computation of η h α i ( e D M ) tothe case of finite fundamental groups. Consequently, Question 3 . . | tr h α i ( e De − t e D ) | , it suffices to assume the spectral gap of e D at zero to be greater than σ h α i := 2 · K h α i θ , where θ is the constant from line (5.3) and K h α i is nonnegative constantsuch that there exists some constant C > { γ ∈ h α i : ℓ ( γ ) n } Ce K h α i · n (5.12)for all n . In fact, if we have a uniform control of the spectral gap of D Γ i at zero and the growth rate of the conjugacy class {h π Γ i ( α ) i} forall i ≥
1, a notion to be made precise in the following, then the same
PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 29 proof above also implies thatlim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D )in this case.Recall that S a symmetric finite generating set of Γ. For each normalsubgroup Γ i of Γ, the map π Γ i : Γ → Γ / Γ i is the canonical quotient map.The set π Γ i ( S ) is a symmetric generating set for Γ / Γ i , hence induces aword length function ℓ Γ i on Γ / Γ i . More explicitly, we have ℓ Γ i ( ω ) := inf { ℓ ( β ) : β ∈ π − i ( ω ) } . (5.13)for all ω ∈ Γ / Γ i . Definition 5.6.
For a given conjugacy class h α i of Γ, we say that h α i has uniform exponential growth with respect to a family of normalsubgroups { Γ i } , if there exist C > A ≥ (cid:8) ω ∈ h π Γ i ( α ) i : ℓ Γ i ( ω ) n (cid:9) Ce A · n . (5.14)for all i ≥ n ≥
0. In this case, we define K u to be the infimumof all such numbers A . Definition 5.7.
With the above notation, we define σ u := 2 · K u θ , where θ is the constant from line (5.3).The same argument from the proof of Theorem 5 . Theorem 5.8.
With the same notation as in Theorem . , suppose { Γ i } is a sequence of finite-index normal subgroups that distinguishesthe conjugacy class h α i of a non-identity element α ∈ Γ . Assume h α i has uniform exponential growth with respect to { Γ i } . If there exists ε > such that the spectral gap of D Γ i at zero is greater than σ u + ε for sufficiently large i ≫ , then lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Remark . Here is a geometric condition on M that guarantees thespectral gap of D Γ i at zero to be greater than σ u + ε for all i ≥
1. Ifthe scalar curvature of M is strictly bounded below by 4 · σ u , thenit follows from the Lichnerowicz formula that there exists ε > D Γ i at zero is greater than σ u + ε for all i ≥ The scalar curvature function κ ( x ) of M satisfies that κ ( x ) > · σ u for all x ∈ M. Separation rates of conjugacy classes
In this section, we introduce a notion of separation rate for how fasta sequence of normal subgroups { Γ i } of Γ distinguishes a conjugacyclass h α i of Γ and use it to answer Question 3 . Definition 6.1.
For each normal subgroup Γ ′ of Γ, let π Γ ′ : Γ → Γ / Γ ′ be the quotient map from Γ to Γ / Γ ′ . Given a conjugacy class h α i of Γ,we define the injective radius of π Γ ′ with respect to h α i to be r (Γ ′ ) := max { n | if γ / ∈ h α i and ℓ ( γ ) ≤ n, then π Γ ′ ( γ ) / ∈ h π Γ ′ ( α ) i} . (6.1) Definition 6.2.
Suppose that { Γ i } is a sequence of finite-index normalsubgroups of Γ that distinguishes h α i . We say that { Γ i } distinguishes h α i sufficiently fast if there exist C >
R > |h π Γ i ( α ) i| Ce R · r (Γ i ) . (6.2)In this case, we define the separation rate R h α i , { Γ i } of h α i with respectto { Γ i } to be the infimum of all such numbers R .We have the following proposition. Proposition 6.3.
Let h α i be the conjugacy class of a non-identityelement α ∈ Γ . Suppose { Γ i } is a sequence of finite-index normalsubgroups that distinguishes h α i sufficiently fast with separation rate R = R h α i , { Γ i } . If η h α i ( e D ) is finite and there exists ε > such thatthe spectral gap of D Γ i at zero is greater than σ R + ε for all sufficientlylarge i ≫ , where σ R := 2( K Γ · R ) / θ , then we have lim i →∞ η h π Γ i ( α ) i ( D Γ i ) = η h α i ( e D ) . Proof.
By assumption, the integral η h α i ( e D ) := 2 √ π Z ∞ tr h α i ( e De − t e D ) dt converges. To prove the proposition, it suffices to show that there existsa sequence of positive real numbers { s i } such that s i → ∞ , as i → ∞ ,and To be precise, η h α i ( e D ) is finite if the integral in line (3.1) converges. In partic-ular, the integral in line (3.1) does not necessarily absolutely converge. PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 31 (1)lim i →∞ Z s i (cid:16) tr h π Γ i ( α ) i ( D Γ i e − t D i ) − tr h α i ( e De − t e D ) (cid:17) dt = 0 (6.3)(2) and lim i →∞ Z ∞ s i tr h π Γ i ( α ) i ( D Γ i e − t D i ) dt = 0 . (6.4)Let K t ( x, y ) (resp. ( K i ) t ( x, y )) be the Schwartz kernel of e De − t e D (resp. D Γ i e − t D i ). Recall that we have (cf. line (5.11))( K i ) t ( π Γ i ( x ) , π Γ i ( y )) = X β ∈ Γ i K t ( x, βy )for all x, y ∈ f M . It follows that (cid:12)(cid:12) tr h π Γ i ( α ) i ( D Γ i e − t D i ) − tr h α i ( e De − t e D ) (cid:12)(cid:12) X γ ∈ π − i h π Γ i ( α ) i but γ / ∈h α i Z x ∈F k K t ( x, γx ) k dx. By the definition of r (Γ i ) in line (6.1), we see that { γ ∈ Γ | γ ∈ π − i h π Γ i ( α ) i but γ / ∈ h α i} ⊆ { γ ∈ Γ | ℓ ( γ ) > r (Γ i ) } . From line (5.5), we have X ℓ ( γ ) > r (Γ i ) Z x ∈F k K t ( x, γx ) k dx c µ,r ∞ X m = r (Γ i ) e − ( θ · m − c µt e K Γ · m . Note that Z s i ∞ X m = r (Γ i ) e − ( θ · m − c µt e K Γ · m dt s i ∞ X m = r (Γ i ) e − ( θ · m − c µs i + K Γ · m . The right hand side goes to zero, as s i → ∞ , as long as there exists λ > θ · r (Γ i ) − c ) µs i > λ · K Γ · r (Γ i ) (6.5)for all sufficiently large i ≫
1. Since { Γ i } distinguishes h α i , we havethat r (Γ i ) → ∞ , as i → ∞ . So the condition in line (6.5) is equivalentto s i < θ · r (Γ i )4 µ · λ · K Γ for sufficiently large i ≫ On the other hand, by the inequality from line (5.9), there exist c > ε > (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ s i tr h π Γ i ( α ) i ( D Γ i e − t D i ) dt (cid:12)(cid:12)(cid:12)(cid:12) c · e − ( σ R + ε ) · s i · |h π Γ i ( α ) i| for all sufficiently large i ≫
1. Note that the right hand side goes tozero, as s i → ∞ , as long as there exists λ > σ R + ε ) · s i > λ · R · r (Γ i ) . (6.6)for all sufficiently large j ≫
1. Combining the two inequalities in line(6.5) and (6.6) together, we can choose a sequence of real numbers { s i } that satisfies the limits in both line (6.3) and (6.4), as long as thereexists λ > θ · r (Γ i )4 µ · λ · K Γ > λ R · r (Γ i )( σ R + ε ) (6.7)for all sufficiently large i ≫
1. By choosing µ sufficiently close to 1, theinequality in line (6.7) follows from the definition of σ R . This finishesthe proof. (cid:3) We finish this section with the following calculation of the separationrates of conjugacy classes of SL ( Z ). The group SL ( Z ) SL ( Z ) is aconjugacy separable group [29]. It has a presentation: h x, y | x = 1 , x = y i , where x = ( −
11 0 ) and y = ( −
11 1 ). In particular, it follows that SL ( Z )is an amalgamated free product of a cyclic group of order 4 and a cyclicgroup of order 6. We now show that for any finite order element α ∈ SL ( Z ), there exists a sequence of finite-index normal subgroups { Γ i } of SL ( Z ) that distinguishes h α i such that the corresponding separationrate R h α i , { Γ i } = 0.Since SL ( Z ) is an amalgamated free product of a cyclic group oforder 4 and a cyclic group of order 6, every finite order element inSL ( Z ) is conjugate to an element in one of the factors. It follows thatevery finite order element of SL ( Z ) is conjugate to a power of x and y . Let ψ : SL ( Z ) → Z / Z be the group homomorphism defined by ψ ( x ) = 3 and ψ ( y ) = 2. In particular, we have ψ ( e ) = 0 , ψ ( x ) = 3 , ψ ( x ) = ψ ( y ) = 6 , ψ ( x ) = 9 ,ψ ( y ) = 2 , ψ ( y ) = 4 , ψ ( y ) = 8 , and ψ ( y ) = 10 . It follows that any finite order elements γ and γ of SL ( Z ) are con-jugate in SL ( Z ) if and only if ψ ( γ ) = ψ ( γ ).Now given any finite-index normal subgroup N of SL ( Z ), the group N = N ∩ ker( ψ ) is a finite-index normal subgroup of SL ( Z ). By the PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 33 discussion above, we see that any finite order elements γ and γ ofSL ( Z ) are conjugate in SL ( Z ) if and only if they are conjugate inSL ( Z ) /N . In other words, the set { N } consisting of a single finite-index normal subgroup distinguishes the conjugacy class h α i of anyfinite order element α ∈ SL ( Z ). Moreover, the injective radius r ( N )of π N : SL ( Z ) → SL ( Z ) /N with respect to h α i is infinity. It followsthat the separation rate R h α i , { N } = 0 in this case. Remark . Since SL ( Z ) is hyperbolic, Puschnigg’s smooth dense sub-algebra A of C ∗ r (SL ( Z )) admits a continuous extension of the tracemap tr h α i for any conjugacy class h α i of SL ( Z ) (cf. [25]). In this case,for any element α = e ∈ Γ, the delocalized eta invariant η h α i ( e D ) isfinite (cf. [22, Section 4][12, Section 6]). Hence we can apply Propo-sition 6 . . ( Z ), when α is a finite order element.On other hand, since SL ( Z ) is also a-T-menable, we can equallyapply Proposition 4 . . ( Z ) (cf. the discussion at the end ofSection 4). Appendix A. Positive scalar curvature and the Stolzconjecture
In this appendix, we shall explain how the geometric conditions givenin Definition 4.5 & 4.6 are related to the reduced Baum-Connes con-jecture and the Stolz conjecture on positive scalar curvature metrics.
Definition A.1.
Given a topological space Y , let R spin n ( Y ) be the fol-lowing bordism group of triples ( L, f, h ), where L is an n -dimensionalcompact spin manifold (possibly with boundary), f : L → Y is acontinuous map, and h is a positive scalar curvature metric on theboundary ∂M . Two triples ( L , f , h ) and ( L , f , h ) are bordant if(a) there is a bordism ( V, F, H ) between ( ∂L , f , h ) and ( ∂L , f , h )such that H is a positive scalar curvature metric on V withproduct structure near ∂L i and H | ∂L i = h i , and the restrictionof the map F : V → Y on ∂L i is f i ;(b) and the closed spin manifold L ∪ ∂L V ∪ ∂L L (obtained bygluing L , V and L along their common boundaries) is theboundary of a spin manifold W with a map E : W → Y suchthat E | L i = f i and E | V = F . For any hyperbolic group and the conjugacy class of any nonidentity element,the integral in line (3.1) absolutely converges.
The above definition has the following obvious analogue for the caseof proper actions.
Definition A.2.
Let X be a proper metric space equipped with aproper and cocompact isometric action of a discrete group Γ. We de-note by R spin n ( X ) Γ the set of bordism classes of pairs ( L, f, h ), where L is an n -dimensional complete spin manifold equipped with a proper andcocompact isometric action of Γ, the map f : L → X is a Γ-equivariantcontinuous map and h is a Γ-invariant positive scalar curvature metricon ∂L . Here the bordism equivalence relation is defined similarly asthe non-equivariant case above.If the action of Γ on X is free and proper, then it follows by definitionthat R spin n ( X ) Γ ∼ = R spin n ( X/ Γ) . Suppose (
L, f, h ) is an element in R spin n ( E Γ) Γ ∼ = R spin n ( B Γ), where B Γ = E Γ / Γ is the classifying space for free Γ-actions. Let L Γ be theΓ-covering space of M induced by the map f : L → B Γ and D L Γ be theassociated Dirac operator. Due to the positive scalar curvature metric h on ∂L , the Γ-equivariant operator D L Γ has a well-defined higherindex class Ind( D L Γ ) in K n ( C ∗ r (Γ)), cf. [26, Proposition 3.11] [27].By the relative higher index theorem [10, 35], we have the followingwell-defined index mapInd : R spin n ( B Γ) → KO n ( C ∗ r (Γ; R )) , ( L, f, h ) Ind( D L Γ ) , where C ∗ r (Γ; R ) is the reduced group C ∗ -algebra of Γ with real coef-ficients. Now let B be the Bott manifold, a simply connected spinmanifold of dimension 8 with b A ( B ) = 1. This manifold is not unique,but any choice will work for the following discussion. To make thediscussion below more transparent, let us choose a B that is equippedwith a scalar flat curvature metric. The fact that such a choice existsfollows for example from the work of Joyce [19, Section 6].Let ( L, f, h ) be an element R spin n ( B Γ), that is, L is an n -dimensionalspin manifold whose boundary ∂L carries a positive scalar curvaturemetric h , together with a map f : L → B Γ. Taking direct productwith k copies of B produces an element ( L ′ , f ′ , h ′ ) in R spin n +8 k ( B Γ), where L ′ = L × B × · · · × B , f ′ = f ◦ p with the map p being the projectionfrom L ′ to L , and h ′ is the product metric of h with the Riemannianmetric on B . By our choice of B above, the Riemannian metric h ′ alsohas positive scalar curvature since h does. Define R spin n ( B Γ)[ B − ] tobe the direct limit of the following directed system: R spin n ( B Γ) × B −−→ R spin n +8 ( B Γ) × B −−→ R spin n +16 ( B Γ) → · · · . PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 35
Since the higher index class Ind( D L Γ ) associated to ( L, f, h ) is invariantunder taking direct product with B , it follows that the above index mapinduces the following well-defined index map: θ : R spin n ( B Γ)[ B − ] → KO n ( C ∗ r (Γ; R )) , ( L, f, h ) Ind( D L Γ ) . Conjecture A.3 (Stolz conjecture [31, 30]) . The index map θ : R spin n ( B Γ)[ B − ] → KO n ( C ∗ r (Γ; R )) is an isomorphism. Similarly, if one works with the universal space E Γ for proper Γ-actions instead, then the same argument from above also produces asimilar index mapΘ : R spin n ( E Γ) Γ [ B − ] → KO n ( C ∗ r (Γ; R ))where R spin n ( E Γ) Γ [ B − ] is the direct limit of the following directed sys-tem: R spin n ( E Γ) Γ × B −−→ R spin n +8 ( E Γ) Γ × B −−→ R spin n +16 ( E Γ) Γ → · · · . One has the following analogue of the Stolz conjecture above, whichwill be called the generalized Stolz conjecture from now on.
Conjecture A.4 (Generalized Stolz conjecture) . The index map
Θ : R spin n ( E Γ) Γ [ B − ] → KO n ( C ∗ r (Γ; R )) is an isomorphism. If Γ is torsion-free, then clearly the generalized Stolz conjecture co-incides with the original Stolz conjecture. By definition, the surjec-tivity of the Stolz map θ in Conjecture A. A.
4. On the other hand,the surjectivity of the Stolz map θ follows from the surjectivity of theBaum-Connes assembly map µ R : KO Γ • ( E Γ) → KO • ( C ∗ r (Γ; R ))cf. [39, Corollary 3.15]. As the writing of this paper, the injectivity ofthe Stolz map θ or generalized Stolz map Θ is wild open, and it is noteven known in the case where Γ is the trivial group.Similarly, one could also formulate the maximal version of the (gen-eralized) Stolz conjecture by considering the index maps θ max : R spin n ( B Γ)[ B − ] → KO n ( C ∗ max (Γ; R ))and Θ max : R spin n ( E Γ) Γ [ B − ] → KO n ( C ∗ max (Γ; R )) respectively. Again, the surjectivity of θ max , hence that of Θ max , followsfrom the surjectivity of the maximal Baum-Connes assembly map µ R : KO Γ • ( E Γ) → KO • ( C ∗ max (Γ; R )) . A.1.
Stable bounding with respect to B Γ . In this subsection, weshall discuss how the assumption that a multiple of (
M, ϕ, h ) stablybounds with respect to B Γ (cf. Definition 4 .
5) is related to the Baum-Connes conjecture and the Stolz conjecture. Here again M is a closedspin manifold equipped with a Riemannian metric h of positive scalarcurvature. Let ϕ : M → B Γ be the classifying map for the covering f M → M , that is, the pullback of E Γ by ϕ is f M .To be more precise, in this subsection, let us assume the Baum-Connes assembly map µ R : KO Γ • ( E Γ) → KO • ( C ∗ r (Γ; R ))is rationally isomorphic . There is a long exact sequence for KO -theory of reduced C ∗ -algebras analogous to commutative diagram (4.1).Now a similar argument as in the proof of Theorem 4 . ρ ( e D ) = ∂ [ p ] (up to rational multiples) for someelement [ p ] ∈ KO n +1 ( C ∗ r (Γ; R )), where n = dim M . Now the rationalsurjectivity of the Baum-Connes assembly map µ R : KO Γ • ( E Γ) → KO • ( C ∗ r (Γ; R ))implies the rational surjectivity of the Stolz map θ : R spin • ( B Γ)[ B − ] → KO • ( C ∗ r (Γ; R ) , cf. [39, Corollary 3.15]. And the rational surjectivity of θ impliesthat there exists an element ( L, f, h ) ∈ R spin n +1 ( B Γ)[ B − ] such that θ ( L, f, h ) = [ p ] (up to a rational mutiple). Recall that (cf. [24, Theo-rem 1.14][34, Theorem A]) ∂ ( θ ( L, f, g )) = ρ ( D f ∂L ) in KO n ( C ∗ L, ( E Γ; R ) Γ ) , where ρ ( D f ∂L ) is the higher rho invariant of D f ∂L with respect to thepositive scalar curvature metric h . In particular, this implies that ρ ( e D ) = ρ ( D f ∂L ). Hence, as far as ρ ( e D ) is concerned, we could workwith ( ∂L, f, g ), which clearly bounds, instead of ( M, ϕ, h ). On theother hand, it is an open question whether the higher rho invariantsfor M and ∂L remain equal to each other, for corresponding finite-sheeted covering spaces of M and ∂L . The rational bijectivity of µ R follows from the rational bijectivity of the com-plex version µ : K Γ • ( E Γ) → K • ( C ∗ r (Γ)), cf. [7]. PPROXIMATIONS OF DELOCALIZED ETA INVARIANTS 37
A.2.
Positively stable bounding with respect to E Γ . In this sub-section, we shall discuss how the assumption that a multiple of ( f M , e h )positively stably bounds with respect to E Γ (cf. Definition 4 .
6) isrelated to the Baum-Connes conjecture and the generalized Stolz con-jecture.Observe that the injectivity of the generalized Stolz mapΘ : R spin n ( E Γ) Γ [ B − ] → KO n ( C ∗ r (Γ; R ))has the following immediate geometric consequence. Let ( L, f, h ) bean element in R spin n ( E Γ) Γ , that is, L is a n -dimensional spin Γ-manifoldwhose boundary ∂L carries a Γ-invariant positive scalar curvature met-ric h , together with a Γ-equivariant map f : L → E Γ. Suppose thehigher index Ind( D L ) associated to ( L, f, h ) vanishes, then the injec-tivity of the generalized Stolz map implies that (
L, f, h ) is stably Γ-equivariantly cobordant to the empty set. More precisely, if ( L ′ , f ′ , h ′ )is the direct product of ( L, f, h ) with sufficiently many copies of B ,then ( L ′ , f ′ , h ′ ) is Γ-equivariantly cobordant to the empty set. In par-ticular, this implies that, if the higher index of ( L, f, h ) vanishes, then ∂L positively stably bounds with respect to E Γ, cf. Definition 4 . ∂L ′ bounds a spin Γ-manifold V such that V admitsa Γ-invariant positive scalar curvature metric g , which has productstructure near the boundary ∂V = ∂L ′ , and the restriction of g to theboundary is equal to h ′ .Now let M be a closed spin manifold equipped with a Riemannianmetric h of positive scalar curvature. Let ϕ : M → B Γ be the classify-ing map for the covering f M → M and ˜ h the metric on f M lifted from h . The above discussion has the following consequence. Lemma A.5.
With the above notation, suppose a multiple of ( M, ϕ, h ) stably bounds with respect to B Γ . If the Baum-Connes assembly map µ R : KO Γ • ( E Γ) → KO • ( C ∗ r (Γ; R )) is rationally surjective and the generalized Stolz map Θ : R spin n ( E Γ) Γ [ B − ] → KO n ( C ∗ r (Γ; R )) is rationally injective, then ( f M , e h ) positively stably bounds with respectto E Γ .Proof. For notational simplicity, let us assume (
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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
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