Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume
aa r X i v : . [ m a t h . G T ] J a n DEGREE THEOREMS AND LIPSCHITZ SIMPLICIAL VOLUMEFOR NON-POSITIVELY CURVED MANIFOLDSOF FINITE VOLUME
CLARA L ¨OH AND ROMAN SAUER
Abstract.
We study a metric version of the simplicial volume on Riemann-ian manifolds, the Lipschitz simplicial volume, with applications to degreetheorems in mind. We establish a proportionality principle and a productinequality from which we derive an extension of Gromov’s volume compari-son theorem to products of negatively curved manifolds or locally symmetricspaces of non-compact type. In contrast, we provide vanishing results for theordinary simplicial volume; for instance, we show that the ordinary simplicialvolume of non-compact locally symmetric spaces with finite volume of Q -rankat least 3 is zero. Introduction and statement of results
The prototypical degree theorem bounds the degree deg f of a proper, continuousmap f : N → M between n -dimensional Riemannian manifolds of finite volume bydeg( f ) ≤ const n · vol( N )vol( M ) . For example, Gromov’s volume comparison theorem [16, p. 13] is a degree theo-rem where the target M has negative sectional curvature and the domain N satisfiesa lower Ricci curvature bound. In loc. cit. Gromov also pioneered the use of the simplicial volume to prove theorems of this kind. Recall that the simplicial vol-ume k M k of a manifold M without boundary is defined by k M k = inf (cid:8) | c | ; c fundamental cycle of M with R -coefficients (cid:9) . Here | c | denotes the ℓ -norm with respect to the basis given by the singular sim-plices. If M is non-compact then one takes locally finite fundamental cycles in theabove definition. Under the given curvature assumptions, Gromov’s comparisontheorem is proved by the following three steps (of which the third one is elemen-tary):(1) Upper volume estimate for target: k M k ≥ const n vol( M ).(2) Lower volume estimate for domain: k N k ≤ const n vol( N ).(3) Degree estimate: deg( f ) ≤ const n k N k / k M k . Unless stated otherwise, all manifolds in this text are assumed to be connectedand without boundary. As Riemannian metrics on locally symmetric spaces of non-compact type we always choose the standard metric, i.e., the one given by the Killingform [12, Section 2.3.11] . Date : October 31, 2018.2000
Mathematics Subject Classification.
Primary 53C23; Secondary 53C35.
Main results.
In this article, we prove degree theorems where the target isnon-positively curved and has finite volume. More specifically, we consider the casewhere the target is a product of negatively curved manifolds of finite volume orlocally symmetric spaces of finite volume. To this end, we study a variant of thesimplicial volume, the
Lipschitz simplicial volume , and pursue a Lipschitz versionof the three step strategy above. The properties of the Lipschitz simplicial volumewe show en route are also of independent interest.Before introducing the Lipschitz simplicial volume, we give a brief overview ofthe properties of the ordinary simplicial volume of non-compact locally symmetricspaces of finite volume: On the one hand by a classic result of Thurston [29, Chap-ter 6] the simplicial volume of finite volume hyperbolic manifolds is proportionalto the Riemannian volume. According to Gromov and Thurston the simplicial vol-ume of complete Riemannian manifolds with pinched negative curvature and finitevolume is positive [16, Section 0.3]. In addition, we proved by different means thatthe simplicial volume of Hilbert modular varieties is positive [23] (see also Theo-rem 1.14 below). In accordance with these examples we expect positivity for alllocally symmetric spaces of Q -rank 1.On the other hand, in Section 5 we show that the simplicial volume of locallysymmetric spaces of Q -rank at least 3 vanishes – in particular, the ordinary simpli-cial volume does not give rise to the desired degree theorems. Theorem 1.1.
Let Γ be a torsion-free, arithmetic lattice of a semi-simple, center-free Q -group G with no compact factors. Let X = G ( R ) /K be the associatedsymmetric space where K is a maximal compact subgroup of G ( R ) . If Γ has Q -rankat least , then k Γ \ X k = 0 . This result is based on a more general vanishing theorem (Corollary 5.4) derivedfrom Gromov’s vanishing-finiteness theorem [16, Corollary (A) on p. 58] by con-structing suitable amenable coverings for manifolds with nice boundary and whosefundamental groups admit small classifying spaces.Gromov’s original applications of the vanishing-finiteness theorem contain thesurprising fact that the simplicial volume of any product of three open manifolds iszero [16, p. 59]. However, there are products of two open manifolds whose simplicialvolume is non-zero (see Example 5.5), and Gromov’s argument fails for products oftwo open surfaces. In particular, the Q -rank 2 case is still open.In contrast to Theorem 1.1, Lafont and Schmidt showed the following positivityresult in the closed case [18]; the proof is based on work of Connell-Farb [9], as wellas – for the exceptional cases – Thurston, Savage, and Bucher-Karlsson: Theorem 1.2 (Lafont, Schmidt) . Let M be a closed locally symmetric space ofnon-compact type. Then k M k > . In view of the fact that the simplicial volume of non-compact manifolds is zeroin a large number of cases, Gromov studied geometric variants of the simplicialvolume [16, Section 4.4f], i.e., simplicial volumes where the simplices allowed infundamental cycles respect a geometric condition. In this article, we consider thefollowing Lipschitz version of simplicial volume:
Definition 1.3.
Let M be an n -dimensional, oriented Riemannian manifold. Fora locally finite chain c ∈ C lf n ( M ) we denote the supremum of the Lipschitz con-stants of the simplices occurring in c by Lip( c ) ∈ [0 , ∞ ]. The Lipschitz simplicial
IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 3 volume k M k Lip ∈ [0 , ∞ ] of M is defined by k M k Lip = inf (cid:8) | c | ; c ∈ C lf n ( M ) fundamental cycle of M with Lip( c ) < ∞ (cid:9) . By definition, we have the obvious inequality k M k ≤ k M k Lip . It is easy to seethat if f : N → M is a proper Lipschitz map between Riemannian manifolds, thendeg( f ) · k M k Lip ≤ k N k Lip . Remark 1.4. If M is a closed Riemannian manifold, then k M k = k M k Lip ; eachfundamental cycle involves only finitely many simplices, and hence this equalityis implied by the fact that singular homology and smooth singular homology areisometrically isomorphic [21, Proposition 5.3].In Section 4 we prove the following theorem, which leads to a degree theoremfor locally symmetric spaces of finite volume.
Theorem 1.5 (Proportionality principle) . Let M and N be complete, non-positivelycurved Riemannian manifolds of finite volume. Assume that their universal coversare isometric. Then k M k Lip vol( M ) = k N k Lip vol( N ) . The proportionality principle for closed Riemannian manifolds is a classical the-orem of Gromov [16, Section 0.4; 28, Chapter 5; 29, pp. 6.6–6.10]. The proportion-ality principle in the closed case does not require a curvature condition, and ourproof in the non-closed case uses non-positive curvature in a light way. It might bepossible to weaken the curvature condition in the non-compact case.By Theorem 1.2 the proportionality principle for the ordinary simplicial volumecannot hold in general since for every locally symmetric space of finite volume thereis always a compact one such that their universal covers are isometric [4]. For thesame reason, Theorems 1.5 and 1.2 and Remark 1.4 imply the following corollary.
Corollary 1.6.
The Lipschitz simplicial volume of locally symmetric spaces of finitevolume and non-compact type is non-zero.
Gromov [16, Section 4.5] states also a proportionality principle for non-compactmanifolds for geometric invariants related to the Lipschitz simplicial volume. Un-raveling his definitions, one sees that it implies a proportionality principle for finitevolume manifolds without a curvature assumption (which we need) provided one ofthe manifolds is compact (which we do not need). This would be sufficient for theprevious corollary. Gromov’s proof, which is unfortunately not very detailed, andours seem to be independent.The simplicial volume of a product of oriented, closed, connected manifolds canbe estimated from above as well as from below in terms of the simplicial volume ofboth factors [1, Theorem F.2.5; 16, p. 17f]. While the upper bound continues to holdfor the locally finite simplicial volume in the case of non-compact manifolds [22,Theorem C.7], the lower bound in general does not.The Lipschitz simplicial volume on the other hand is better behaved with respectto products. In addition to the estimate k M × N k Lip ≤ c (dim M +dim N ) ·k M k Lip ·k N k Lip , the presence of non-positive curvature enables us to derive also the non-trivial lower bound:
CLARA L ¨OH AND ROMAN SAUER
Theorem 1.7 (Product inequality for non-positively curved manifolds) . Let M and N be two complete, non-positively curved Riemannian manifolds. Then k M k Lip · k N k Lip ≤ k M × N k Lip . On a technical level, we mention two issues that often prevent one from extendingproperties of the simplicial volume for compact manifolds to non-compact ones, andthus force one to work with the Lipschitz simplicial volume instead. Firstly, thereis no straightening (see Section 2.2) for locally finite chains: The straightening ofa locally finite chain c is not necessarily locally finite. However, it is locally finiteprovided Lip( c ) < ∞ , which motivates a Lipschitz condition. Secondly, there is nowell-defined cup product for compactly supported cochains. This is an issue arisingin the proof of the product inequality. We circumvent this difficulty by introducingthe complex of cochains with Lipschitz compact support (see Definition 3.6), whichcarries a natural cup-product.1.2. Degree theorems.
To apply the theorems of the previous section to degreetheorems, we need upper and lower estimates of the volume by the Lipschitz sim-plicial volume.For the (locally finite) simplicial volume and all complete n -dimensional Rie-mannian manifolds, Gromov gives the bound k M k ≤ ( n − n n ! vol( M ) providedRicci( M ) ≥ − ( n −
1) [16]. The latter stands for Ricci( M )( v, v ) ≥ − ( n − k v k for all v ∈ T M . One can extract from loc. cit. a similar estimate for the Lipschitzsimplicial volume:
Theorem 1.8 (Gromov) . For every n ≥ there is a constant C n > suchthat every complete n -dimensional Riemannian manifold M with sectional curva-ture sec( M ) ≤ and Ricci curvature Ricci( M ) ≥ − ( n − satisfies k M k Lip ≤ C n · vol( M ) . Proof.
For sec( M ) ≤ U = M , R = 1, a fundamental cycle c , and ε →
0: One obtains afundamental cycle c ′ made out of straight simplices whose diameter is less than R + ε .In particular, Lip( c ′ ) < ∞ by Proposition 2.4. Further, the estimate k M k Lip ≤k c ′ k ≤ C n vol( M ) follows from (4) in loc. cit. and the Bishop-Gromov inequality,which provides a bound of l ′ v ( R ) in terms of n [15, Theorem 4.19 on p. 214; 16, (C)in Section 4.3].Gromov also explains why these arguments carry over to the general case thatsec( M ) ≤ c ′ is made out ofstraight simplices of diameter less than π/ c ′ ) < ∞ followsfrom Proposition 2.6. (cid:3) Corollary 1.9.
Any complete Riemannian manifold of finite volume that has anupper sectional curvature and lower Ricci curvature bound has finite Lipschitz sim-plicial volume.
Connell and Farb [9] prove, building upon techniques of Besson-Courtois-Gallot,a degree theorem where the target M is a locally symmetric space (closed or finitevolume) with no local R , H , or SL(3 , R ) / SO(3 , R )-factor. For non-compact M theyhave to assume that f : N → M is (coarse) Lipschitz. Using the simplicial volume(and the work by Connell-Farb, Thurston, Savage, and Bucher-Karlsson), Lafont IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 5 and Schmidt [18] prove degree theorems for closed locally symmetric spaces includ-ing the exceptional cases. The following theorem includes also the non-compactexceptional cases.
Theorem 1.10 (Degree theorem, complementing [9, 18]) . For every n ∈ N there isa constant C n > with the following property: Let M be an n -dimensional locallysymmetric space of non-compact type with finite volume. Let N be an n -dimensionalcomplete Riemannian manifold of finite volume with Ricci( N ) ≥ − ( n − and sec( N ) ≤ , and let f : N → M be a proper Lipschitz map. Then deg( f ) ≤ C n · vol( N )vol( M ) . Proof.
By Theorem 1.5 and Corollary 1.6 we know that k M k Lip = const n vol( M )where const n > f M . Because there are onlyfinitely many symmetric spaces (with the standard metric) in each dimension, thereis D n > n such that k M k Lip ≥ D n vol( M ). So Theorem 1.8applied to N and k N k Lip ≥ deg( f ) k M k Lip yield the assertion. (cid:3)
Unfortunately, the Lipschitz simplicial volume cannot be used to prove positivityof Gromov’s minimal volume minvol( M ) of a smooth manifold M ; the minimalvolume is defined as the infimum of volumes vol( M, g ) over all complete Riemannianmetrics g on M whose sectional curvature is pinched between − M ) in our setting: The Lipschitz class [ g ] of a complete Riemannian metric g on M is defined as the set ofall complete Riemannian metrics g ′ such that the identity id : ( M, g ′ ) → ( M, g ) isLipschitz. Then we define the minimal volume of [ g ] asminvol Lip ( M, [ g ]) = (cid:8) vol( M, g ′ ); − ≤ sec( g ′ ) ≤ g ′ ∈ [ g ] (cid:9) . Of course, we have minvol
Lip ( M, [ g ]) = minvol( M ) whenever M is compact. The-orem 1.10, applied to the identity map and varying metrics, implies: Theorem 1.11.
The minimal volume of the Lipschitz class of the standard metricof a locally symmetric space of non-compact type and finite volume is positive.
Excluding certain local factors, Connell and Farb have the following strongerstatement for the minimal volume instead of the Lipschitz minimal volume.
Theorem 1.12 (Connell-Farb) . The minimal volume of a locally symmetric spaceof non-compact type and finite volume that has no local H - or SL(3 , R ) / SO(3 , R ) -factors is positive. A little caveat: Connell and Farb state this theorem erroneously as a corollary ofa degree theorem for which they have to assume a Lipschitz condition. This wouldonly give the positivity of the Lipschitz minimal volume. However, Chris Connellexplained to us how to modify their proof to get the positivity of the minimalvolume.As an application of the product inequality we obtain a new degree theoremfor products of manifolds with (variable) negative curvature or locally symmetricspaces.
Theorem 1.13 (Degree theorem for products) . For every n ∈ N there is a con-stant C n > with the following property: Let M be a Riemannian n -manifold CLARA L ¨OH AND ROMAN SAUER of finite volume that decomposes as a product M = M × · · · × M m of Riemann-ian manifolds, where for every i ∈ { , . . . , m } the manifold M i is either negativelycurved with −∞ < − k < sec( M i ) ≤ − or a locally symmetric space of non-compact type. Let N be an n -dimensional, complete Riemannian manifold of finitevolume with sec( N ) ≤ and Ricci( N ) ≥ − ( n − . Then for every proper Lipschitzmap f : N → M we have deg( f ) ≤ C n · vol( N )vol( M ) . Proof.
In the sequel, D i , D ′ i , E n , and C n stand for constants depending only on n .If M i is negatively curved then Thurston’s theorem [16, Section 0.3; 29] yieldsvol( M i ) ≤ D n k M i k ≤ D n k M i k Lip . If M i is locally symmetric of non-compact type then, as in the proof of Theo-rem 1.10, we also obtain vol( M i ) ≤ D ′ n k M i k Lip . By the product inequality (Theo-rem 1.7), vol( M ) ≤ max i ∈{ ,...,m } ( D i , D ′ i ) m k M k Lip . On the other hand, by Theorem 1.8, we have k N k Lip ≤ E n vol( N ). Combin-ing everything with k N k Lip ≥ deg( f ) k M k Lip , proves the theorem with the con-stant C n = E n / max i ∈{ ,...,m } ( D i , D ′ i ) m . (cid:3) As a concluding remark, we mention a computational application of the pro-portionality principle. We proved that k M k = k M k Lip for Hilbert modular vari-eties [23]. This fact combined with the proportionality principle 1.5 and work ofBucher-Karlsson [8] leads then to the following computation [23]:
Theorem 1.14.
Let Σ be a non-singular Hilbert modular surface. Then k Σ k = 32 π vol(Σ) . Conversely, the proportionality principle 1.5 together with Thurston’s computa-tion of the simplicial volume of hyperbolic manifolds shows that the simplicial vol-ume of hyperbolic manifolds of finite volume equals the Lipschitz simplicial volume.More generally, this holds true for locally symmetric spaces of R -rank 1 [23, Theo-rem 1.5; see also beginning of Section 1.5]. However, in the general Q -rank 1 case,the relation between the simplicial volume and the Lipschitz simplicial volume re-mains open. Organization of this work.
Section 2 reviews the basic properties of geodesicsimplices and Thurston’s straightening. The product inequality (Theorem 1.7) isproved in Section 3. Section 4 contains the proof of the proportionality principle(Theorem 1.5). Finally, Section 5 is devoted to the proof of the vanishing result(Theorem 1.1).
Acknowledgements.
The first author would like to thank the Graduiertenkolleg“Analytische Topologie und Metageometrie” at the WWU M¨unster for its financialsupport. The second author acknowledges support of the German Science Founda-tion (DFG), made through grant SA 1661/1-1.Both authors thank the University of Chicago for an enjoyable working atmo-sphere (C.L. visited UC in March/April 2007). We are very grateful to Chris
IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 7
Connell and Benson Farb for discussions about their work [9]. The second authoralso thanks Juan Souto and Shmuel Weinberger for several helpful discussions.2.
Straightening and Lipschitz estimates of straight simplices
In Section 2.1, we collect some basic properties of geodesic simplices. We recallthe technique of straightening singular chains for non-positively curved manifoldsin Section 2.2. Variations of this straightening play an important role in the proofsof the proportionality principle (Theorem 1.5) and the product inequality (Theo-rem 1.7).2.1.
Geodesic simplices.
Let M be a simply connected, complete Riemann-ian manifold. Firstly assume that M has non-positive sectional curvature. Forpoints x , x ′ in M , we denote by [ x, x ′ ] : [0 , → M the unique geodesic joining x and x ′ . The geodesic join of two maps f and g : X → M from a space X to M isthe map defined by[ f, g ] : X × [0 , → M, ( x, t ) (cid:2) f ( x ) , g ( x ) (cid:3) ( t ) . We recall the notion of geodesic simplex: The standard simplex ∆ n is given by∆ n = { ( z , . . . , z n ) ∈ R n +1 ≥ ; P i z i = 1 } , and we identify ∆ n − with the sub-set { ( z , . . . , z n ) ∈ ∆ n ; z n = 0 } . Moreover, the standard simplex is alwaysequipped with the induced Euclidean metric. Let x , . . . , x n ∈ M . The geodesicsimplex [ x , . . . , x n ] : ∆ n → M with vertices x , . . . , x n is defined inductively as[ x , . . . , x n ] (cid:0) (1 − t ) s + t (0 , . . . , , (cid:1) = (cid:2) [ x , . . . , x n − ]( s ) , x n (cid:3) ( t )for s ∈ ∆ n − and t ∈ [0 , M admits an upper bound K ∈ (0 , ∞ ) of the sectional curva-ture, then every pair of points with distance less than K − / π/ M is joined bya unique geodesic. Thus we can define the geodesic simplex with vertices x , . . . , x n as before whenever { x , . . . , x n } has diameter less than K − / π/ Lipschitz estimates for geodesic joins.
Proposition 2.1.
Let M be a simply connected, complete Riemannian manifold ofnon-positive sectional curvature, and let n ∈ N . Let f , g ∈ map(∆ n , M ) be smoothmaps. Then [ f, g ] is smooth and has a Lipschitz constant that depends only on theLipschitz constants for f and g .Proof. Using the exponential map we can rewrite [ f, g ] as(2.2) [ f, g ]( x, t ) = exp f ( x ) (cid:0) t · exp − f ( x ) ( g ( x )) (cid:1) . Since the exponential map viewed as a map
T M → M × M is a diffeomorphism,[ f, g ] is smooth. The assertion about the Lipschitz constant is a consequence of thefollowing lemma. (cid:3) Lemma 2.3.
Let X be a compact metric space and M as above. If f and g : X → M are two Lipschitz maps, then the geodesic join [ f, g ] : X × [0 , → M is also aLipschitz map, and we have Lip[ f, g ] ≤ · (cid:0) Lip f + Lip g + diam(im f ∪ im g ) (cid:1) . CLARA L ¨OH AND ROMAN SAUER
Proof.
Let ( x, t ), ( x ′ , t ′ ) ∈ X × [0 , d M (cid:0) [ f, g ]( x, t ) , [ f, g ]( x ′ , t ′ ) (cid:1) ≤ d M (cid:0) [ f ( x ) , g ( x )]( t ) , [ f ( x ) , g ( x )]( t ′ ) (cid:1) + d M (cid:0) [ f ( x ) , g ( x )]( t ′ ) , [ f ( x ′ ) , g ( x ′ )]( t ′ ) (cid:1) . Because Lip[ f ( x ) , g ( x )] = d M ( f ( x ) , g ( x )), the first term satisfies d M (cid:0) [ f ( x ) , g ( x )]( t ) , [ f ( x ) , g ( x )]( t ′ ) (cid:1) ≤ | t − t ′ | · d M (cid:0) f ( x ) , g ( x ) (cid:1) ≤ | t − t ′ | · diam (cid:0) im f ∪ im g (cid:1) ;notice that diam(im f ∪ im g ) is finite because X is compact. The CAT (0)-inequalityallows us to simplify the second term as follows d M (cid:0) [ f ( x ) , g ( x )]( t ′ ) , [ f ( x ′ ) , g ( x ′ )]( t ′ ) (cid:1) ≤ d M (cid:0) [ f ( x ) , g ( x )]( t ′ ) , [ f ( x ) , g ( x ′ )]( t ′ ) (cid:1) + d M (cid:0) [ f ( x ) , g ( x ′ )]( t ′ ) , [ f ( x ′ ) , g ( x ′ )]( t ′ ) (cid:1) ≤ (1 − t ′ ) · d M (cid:0) g ( x ) , g ( x ′ ) (cid:1) + t ′ · d M (cid:0) f ( x ) , f ( x ′ ) (cid:1) ≤ d M (cid:0) g ( x ) , g ( x ′ ) (cid:1) + d M (cid:0) f ( x ) , f ( x ′ ) (cid:1) ≤ Lip f · d X ( x, x ′ ) + Lip g · d X ( x, x ′ ) . Therefore, we obtain d M (cid:0) [ f, g ]( x, t ) , [ f, g ]( x ′ , t ′ ) (cid:1) ≤ (cid:0) Lip f + Lip g + diam(im f ∪ im g ) (cid:1) · · d X × [0 , (cid:0) ( x, t ) , ( x ′ , t ′ ) (cid:1) . (cid:3) Lipschitz estimates for geodesic simplices.
Similarly to geodesic joins alsogeodesic simplices admit a uniform Lipschitz estimate and analogous smoothnessproperties.
Proposition 2.4.
Let M be a complete, simply connected, non-positively curvedRiemannian manifold. Then every geodesic simplex in M is smooth. Moreover, forevery D > and k ∈ N there is L > such that every geodesic k -simplex σ ofdiameter less than D satisfies k T x σ k < L for every x ∈ ∆ k . Remark 2.5.
Let M be a simply connected, complete Riemannian manifold ofnon-positive sectional curvature. If x , . . . , x k ∈ M , then applying the triangleinequality inductively shows that ∀ y ∈ ∆ k d M (cid:0) [ x , . . . , x k ]( y ) , x k (cid:1) ≤ k · max i, j ∈{ ,...,k } d M ( x i , x j )and hence that diam (cid:0) im[ x , . . . , x k ] (cid:1) ≤ · k · max i, j ∈{ ,...,k } d M ( x i , x j ) . In the proof of Theorem 1.8, it is necessary to have a more general version ofProposition 2.4 dealing with a positive upper sectional curvature bound. In thiscase, locally, the same arguments apply:
Proposition 2.6.
Let M be a complete, simply connected Riemannian manifoldwhose sectional curvature is bounded from above by K ∈ (0 , ∞ ) . Then everygeodesic simplex σ of diameter less than K − / π/ is smooth. Further, there is aconstant L > such that every geodesic k -simplex σ of diameter less than K − / π/ satisfies k T x σ k < L for every x ∈ ∆ k . IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 9
The proofs of the following two lemmas used to prove Propositions 2.4 and 2.6are elementary and thus omitted. The proof of the first one is very similar to Lee’sproof of the Sturm comparison theorem [20, Proof of Theorem 11.1].
Lemma 2.7.
Let u : [0 , → R ≥ be a smooth function such that u (0) = 0 , and u ( t ) > for t ∈ (0 , , as well as ∀ t ∈ [0 , d dt u ( t ) + π · u ( t ) ≥ . Then for all t ∈ [0 , we have u ( t ) ≤ u (1) · sin (cid:0) t · π/ (cid:1) . Lemma 2.8.
Let f : V → W be a linear map between finite-dimensional vectorspaces with inner products. Let H ⊂ V be a subspace of co-dimension , and let z ∈ V be a vector such that z and H span V . Let { y , . . . , y k − } be an orthonormalbasis of H . Assume that for some C > ∀ w ∈{ z,y ,...,y k − } k f ( w ) k ≤ C · k w k . Further, assume that the angle α between z and H lies in [ ε, π/ with < ε ≤ π/ .Then there is a constant L > that depends only on dim( V ) , C , and ε such that k f k < L. Proof of Proposition 2.4 and 2.6.
That geodesic simplices are smooth is easily seenusing the fact that the exponential map is a diffeomorphism. Let K ≥ M . By normalizing the metric we mayassume that either K = 0 or K = 1. In the case K = 1, it is understoodthat D = π/
2. Led by the inductive definition of geodesic simplices, we prove theproposition by induction over k : For k = 0 or k = 1 there is nothing to show.We now assume that there is an L ′ > k − D is smooth and that the norm of its differential is less than L ′ .Let σ := [ x , . . . , x k ] : ∆ k → M be a geodesic k -simplex of diameter less than D .By the induction hypothesis,(2.9) ∀ p ∈ ∆ k − (cid:13)(cid:13) T p [ x , . . . , x k − ] (cid:13)(cid:13) < L ′ . In the following, we write v , . . . , v k for the vertices of ∆ k . Let p ∈ ∆ k − ,and let γ : [0 , → M denote the geodesic from x k to [ x , . . . , x k − ]( p ). Choosean orthonormal basis { X , . . . , X k − } of the hyperplane in R k spanned by ∆ k − ;then we can view { X , . . . , X k − } as an orthonormal frame of T ∆ k − . For i ∈{ , . . . , k − } we consider the following variation of γ : H i : ( − ε i , ε i ) × [0 , → M, ( s, t ) (cid:2) x k , σ ( s · X i + p ) (cid:3) ( t ) . Let X i ( t ) := dds H i ( s, t ) | s =0 ∈ T M . By definition, X i is a Jacobi field along γ .Moreover, we have at each point p ( t ) := [ v k , p ]( t ) of ∆ k the relation(2.10) T p ( t ) σ ( X i ) = t · X i ( t ) . In order to obtain the desired bounds for k T p ( t ) σ k we first give estimates for k X i ( t ) k and then apply Lemma 2.8 to conclude the proof.For the following computation, let D t denote the covariant derivative along γ at γ ( t ), and let K and R denote the sectional curvature and the curvature tensor, respectively. Straightforward differentiation and the Jacobi equation yield d dt (cid:13)(cid:13) X i ( t ) (cid:13)(cid:13) = 2 · k D t X i ( t ) k − D R (cid:16) X i ( t ) , ddt γ (cid:17) ddt γ, X i ( t ) E ≥ − K · k X i ( t ) k · (cid:13)(cid:13)(cid:13) ddt γ (cid:13)(cid:13)(cid:13) ≥ − K · k X i ( t ) k · D . By definition, X i (0) = 0, and by (2.10) and (2.9), k X i (1) k = k T p σ ( X i ) k = k T p [ x , . . . , x k − ]( X i ) k < L ′ . First assume that K = 0. Then the smooth function t
7→ k X i ( t ) k starts with thevalue 0, is non-negative, and convex. So it is non-decreasing. This implies that(2.11) ∀ t ∈ [0 , k X i ( t ) k ≤ k X i (1) k < L ′ . Next assume that K = 1, thus D = π/
2. Lemma 2.7 yields (2.11). Thus, in bothcases K = 0 or K = 1 we see that k X i ( t ) k ≤ L ′ for all t ∈ [0 , (cid:13)(cid:13)(cid:13) T p ( t ) σ (cid:16) ddt p (cid:17)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ddt γ (cid:13)(cid:13)(cid:13) ≤ D. So Lemma 2.8 implies that there is a constant
L > L ′ , D ,and k such that k T γ ( t ) σ k < L because the angle between the line p ( t ) and ∆ k − is at least ε > ε dependingonly on ∆ k . (cid:3) Geodesic straightening.
In the following, we recall the definition of thegeodesic straightening map on the level of chain complexes, as introduced byThurston [29, p. 6.2f].Let M be a connected, complete Riemannian manifold of non-positive sectionalcurvature. A singular simplex on M is straight if it is of the form p M ◦ σ for somegeodesic simplex σ on f M , where p M : f M → M is the universal covering map. Thesubcomplex of the singular complex C ∗ ( M ) generated by the straight simplices isdenoted by Str ∗ ( M ); the elements of Str ∗ ( M ) are called straight chains . Everystraight simplex is uniquely determined by the (ordered set of) vertices of its liftto the universal cover.The straightening s M : C ∗ ( M ) → Str ∗ ( M ) is defined by s M ( σ ) := p M ◦ (cid:2)e σ ( v ) , . . . , e σ ( v ∗ ) (cid:3) for σ ∈ map(∆ ∗ , M ) , where p M : f M → M is the universal covering map, v , . . . , v ∗ are the vertices of ∆ ∗ ,and e σ is some p M -lift of σ .Notice that the definition of s M ( σ ) is independent of the chosen lift e σ becausethe fundamental group π ( M ) acts isometrically on f M . Proposition 2.12 (Thurston) . Let M be a connected, complete Riemannian man-ifold of non-positive sectional curvature. Then the straightening s M : C ∗ ( M ) → Str ∗ ( M ) and the inclusion Str ∗ ( M ) → C ∗ ( M ) are mutually inverse chain homo-topy equivalences. The easy proof is based on Lemma 2.13 below, which is a standard device forconstructing chain homotopies. Because we need this lemma later, we reproducethe short argument for Proposition 2.12 here.
IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 11
Proof of Proposition 2.12.
For each singular simplex σ : ∆ n → M on M , we define H σ := p M ◦ (cid:2)e σ, [ e σ ( v ) , . . . , e σ ( v n )] (cid:3) : ∆ n × [0 , → M, where v , . . . , v n are the vertices of ∆ n , and e σ is a lift of σ with respect to theuniversal covering map p M . It is not difficult to see that H σ is independent of thechosen lift e σ and that H σ satisfies the hypotheses of Lemma 2.13 below.Therefore, Lemma 2.13 provides us with a chain homotopy between id C ∗ ( M ) andthe straightening map s M . (cid:3) Lemma 2.13.
Let X be a topological space. For each i ∈ N and each singular i -simplex σ : ∆ i → X let H σ : ∆ i × I → X be a homotopy such that for each facemap ∂ k : ∆ i − → ∆ i we have H σ ◦ ∂ k = H σ ◦ ( ∂ k × id I ) . Then f (0) and f (1) : C ∗ ( X ) → C ∗ ( X ) , defined by f ( m ) ( σ ) = H σ ◦ i m for m ∈ { , } ,are chain maps. For every i ∈ N there are i +1 affine simplices G k,i : ∆ i +1 → ∆ i × I such that H : C i ( X ) → C i +1 ( X ) , h ( σ ) = i X k =0 H σ ◦ G k,i defines a chain homotopy f (0) ≃ f (1) .Proof. This is literally proved in Lee’s book [19, Proof of Theorem 16.6, p. 422-424]although the lemma above is not stated as such. (cid:3)
Remark 2.14.
The simplices G k,i in the previous lemma arise from decomposingthe prism ∆ i × I into ( i + 1)-simplices.3. Product inequality for the Lipschitz simplicial volume
This section is devoted to the proof of the product inequality (Theorem 1.7).The corresponding statement in the compact case is proved by first showingthat the simplicial volume can be computed in terms of bounded cohomology andthen exploiting the fact that the cohomological cross-product is compatible withthe semi-norm on bounded cohomology [1, Theorem F.2.5; 16, p. 17f]. In a similarfashion, the product inequality for the locally finite simplicial volume can be shownif one of the factors is compact [16, p. 17f; 22, Appendix C].To prove the Lipschitz version, we proceed in the following steps:(1) We show that the Lipschitz simplicial volume can be computed in termsof a suitable semi-norm on cohomology with Lipschitz compact supports;this semi-norm is a variant of the supremum norm parametrized by locallyfinite supports (Sections 3.1, 3.2, and 3.3).(2) The failure of the product inequality for the locally finite simplicial volumeis linked to the fact that there is no well-defined cross product on compactlysupported cochains. In contrast, we show in Lemma 3.15 in Section 3.4that there is a cross-product for cochains with Lipschitz compact support(Definition 3.6), and we analyze the interaction between this semi-normand the cross-product on cohomology with compact supports (Section 3.4).(3) Finally, we prove that the presence of non-positive curvature allows us torestrict attention to locally finite fundamental cycles of the product thathave nice supports (Section 3.5). This enables us to use the information on cohomology with Lipschitz compact supports to derive the product in-equality (Section 3.6).3.1.
Locally finite homology with a Lipschitz constraint.
The locally finitesimplicial volume is defined in terms of the locally finite chain complex. In thesame way, the Lipschitz simplicial volume is related to the chain complex of chainswith
Lipschitz locally finite support . Definition 3.1.
For a topological space X , we define K ( X ) to be the set of allcompact, connected, non-empty subsets of X .For simplicity, we consider only connected compact subsets. This is essentialwhen considering relative fundamental classes of pairs of type ( M, M − K ). Definition 3.2.
Let X be a metric space, and let k ∈ N . Then we write S lf k ( X ) := (cid:8) A ⊂ map(∆ k , X ); ∀ K ∈ K ( X ) |{ σ ∈ A | im( σ ) ∩ K = ∅}| < ∞ (cid:9) S lf , Lip k ( X ) := (cid:8) A ∈ S lf k ( X ); ∃ L ∈ R > ∀ σ ∈ A Lip( σ ) < L (cid:9) . The elements of S lf , Lip k ( X ) are said to be Lipschitz locally finite . The subcomplexof C lf ∗ ( X ) of all chains with Lipschitz locally finite support is denoted by C lf , Lip ∗ ( M ),and the corresponding homology – so-called homology with Lipschitz locally finitesupport – is denoted by H lf , Lip ∗ ( X ). Theorem 3.3.
Let M be a connected Riemannian manifold. Then the homomor-phism H lf , Lip ∗ ( M ) → H lf ∗ ( M ) induced by the inclusion C lf , Lip ∗ ( M ) → C lf ∗ ( M ) is an isomorphism. During the course of the proof of this theorem, we rely on the following notation:
Definition 3.4.
Let X be a proper metric space, and let A ⊂ X be a subspace.Let L ∈ R > .(1) We write C lf , Proof of Theorem 3.3. We divide the proof into three steps:(1) For all L ∈ R > and all K ∈ K ( M ), the inclusion C Let M be a Riemannian manifold and let U ⊂ M be an open subset.Then the inclusion C Lip ∗ ( U ) → C ∗ ( U ) induces an isomorphism on homology.Proof. The proof consists of an induction as, for example, in Bredon’s proof of thede Rham theorem [6, Section V.9]: If U ⊂ R n is a bounded convex subset, then one can easily construct a chaincontraction for C Lip ∗ ( U ); therefore, the lemma holds for bounded convex subsets inEuclidean spaces.If U , V ⊂ M are open subsets such that the lemma holds for both of them aswell as for the intersection U ∩ V , then the lemma also holds for U ∪ V : The classicalconstruction of barycentric subdivision (and the corresponding chain homotopy tothe identity) [6, Section IV.17] restricts to the Lipschitz chain complex and thusLipschitz homology admits a Mayer-Vietoris sequence.Proceeding by induction we see that the lemma holds for finite unions of boundedconvex subsets of Euclidean space. Then a standard colimit argument shows thatthe lemma holds for arbitrary open subsets of Euclidean space.We call an open subset V of M admissible if there is a smooth chart V ′ → R n and a compact set K ⊂ M such that V ⊂ K ⊂ V ′ . In particular, any admissiblesubset of M is bi-Lipschitz homeomorphic to an open subset of R n , and hence thelemma holds for admissible subsets of M .Noting that the intersection of two admissible sets is admissible, the Mayer-Vietoris argument shows that the lemma holds for finite unions of admissible sets.Any open subset of M can be written as a union of admissible sets; hence, astandard colimit argument yields that the lemma holds for arbitrary open subsetsof M . (cid:3) Cohomology with compact supports with a Lipschitz constraint. Thenatural cohomological counterpart of locally finite homology is cohomology withcompact supports. Similarly, the cohomology theory corresponding to Lipschitzlocally finite homology is cohomology with Lipschitz compact supports; here, “cor-responding” means in particular that there is an evaluation map linking homologyand cohomology (Remark 3.7). Definition 3.6. Let X be a metric space. A cochain f ∈ hom R ( C Lip ∗ ( X ) , R ) issaid to have Lipschitz compact support if for all L ∈ R > there exists a compactsubset K ⊂ X such that ∀ σ ∈ map(∆ k ,X ) (cid:0) Lip( σ ) < L ∧ im( σ ) ⊂ X − K (cid:1) = ⇒ f ( σ ) = 0 . The cochains with Lipschitz compact support form a subcomplex of the cochaincomplex hom R ( C Lip ∗ ( X ) , R ); this subcomplex is denoted by C ∗ cs , Lip ( X ).The cohomology of C ∗ cs , Lip ( X ), denoted by H ∗ cs , Lip ( X ), is called cohomology withLipschitz compact supports . Remark 3.7. Let X be a metric space. By construction of the chain com-plexes C lf , Lip ∗ ( X ) and C ∗ cs , Lip ( X ), the evaluation map h · , · i : C ∗ cs , Lip ( X ) ⊗ C lf , Lip ∗ ( X ) −→ R f ⊗ X i ∈ I a i · σ i X i ∈ I a i · f ( σ i )is well-defined. Moreover, the same computations as in the case of locally finitehomology/cohomology with compact supports show that this evaluation descendsto a map h · , · i : H ∗ cs , Lip ( X ) ⊗ H lf , Lip ∗ ( X ) −→ R on the level of (co)homology.Dually to Theorem 3.3, we obtain: IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 15 Theorem 3.8. For all connected Riemannian manifolds, the natural homomor-phism C ∗ cs ( M ) → C ∗ cs , Lip ( M ) given by restriction induces an isomorphism on coho-mology.Proof. We start by disassembling the cochain complex C ∗ cs , Lip ( M ) into pieces thatare accessible by the universal coefficient theorem: C ∗ cs , Lip ( M ) = lim ←− L →∞ C ∗ cs , Any ori-ented, connected manifold possesses a (integral) fundamental class , which is adistinguished generator of the locally finite homology H lf n ( M ; Z ) ∼ = Z with inte-gral coefficients in the top dimension n = dim( M ). The fundamental class in H lf n ( M ) = H lf n ( M ; R ) is, by definition, the image of the integral fundamental classunder the coefficient change H lf n ( M ; Z ) → H lf n ( M ; R ). Correspondingly, one definesthe cohomological or dual fundamental class as a distinguished generator of the topcohomology with compact supports. Definition 3.9. Let M be an oriented, connected Riemannian n -manifold (withoutboundary). The Lipschitz fundamental class of M is the homology class [ M ] Lip ∈ H lf , Lip n ( M ) that corresponds to the fundamental class [ M ] ∈ H lf n ( M ) via the isomor-phism H lf , Lip ∗ ( M ) → H lf ∗ ( M ) (Theorem 3.3). Analogously, one defines the Lipschitzdual fundamental class [ M ] ∗ Lip ∈ H n cs , Lip ( M ) of M . Remark 3.10. The proofs of Theorem 3.3 and 3.6 work for any coefficient module.Thus one can equivalently define the Lipschitz fundamental class as the image ofthe generator of H lf , Lip n ( M ; Z ) that corresponds to the integral fundamental classin H lf n ( M ; Z ) ∼ = H lf , Lip n ( M ; Z ) under the change of coefficients Z → R . Similarconsiderations apply to the Lipschitz dual fundamental class.In the compact case, the simplicial volume can be expressed as the inverse ofthe semi-norm of the dual fundamental class [16, p. 17]. In the non-compact case,however, one has to be a bit more careful [16, p. 17; 22, Theorem C.2]. Similarly,also the Lipschitz simplicial volume can be computed in terms of certain semi-normson cohomology (Proposition 3.12). Definition 3.11. Let M be a topological space, k ∈ N , and let A ⊂ map(∆ k , M ).(1) For a locally finite chain c = P i ∈ I a i · σ i ∈ C lf k ( M ), let | c | A := ( | c | if supp( c ) ⊂ A , ∞ otherwise. , Here, supp( c ) := { i ∈ I ; a i = 0 } .(2) The semi-norms on (Lipschitz) locally finite/relative homology inducedby | · | A are denoted by k · k A .(3) If M is an oriented, connected n -manifold, then k M k A := k [ M ] k A . If moreover, K ∈ K ( M ), then k M, M − K k A := k [ M, M − K ] k A , where [ M, M − K ] ∈ H n ( M, M − K ) is the relative fundamental class.(4) For f ∈ C k ( M ) we write k f k A ∞ := sup σ ∈ A | f ( σ ) | ∈ [0 , ∞ ] . (5) The semi-norms on (relative) cohomology with (Lipschitz) compact sup-ports induced by k · k A ∞ are also denoted by k · k A ∞ . Proposition 3.12 (Duality principle for the Lipschitz simplicial volume) . Let M be an oriented, connected Riemannian n -manifold.(1) Then k M k Lip = inf (cid:8) k M k A ; A ∈ S lf , Lip n ( M ) (cid:9) . (2) Moreover, for all A ∈ S lf , Lip n ( M ) , we have k M k A = 1 k [ M ] ∗ Lip k A ∞ . Proof. The first part follows directly from the definitions. For the second part let A ∈ S lf , Lip n ( M ). Then k M k A = sup K ∈ K ( M ) k M, M − K k A = sup K ∈ K ( M ) k [ M, M − K ] ∗ Lip k A ∞ = 1 k [ M ] ∗ Lip k A ∞ . IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 17 We now explain these steps in more detail: • The first equality is shown by constructing an appropriate diagonal se-quence out of “small” relative fundamental cycles of the ( M, M − K ) sup-ported on A [22, Proposition C.3]. • The class [ M, M − K ] ∗ Lip ∈ H n (hom R ( C Lip ∗ ( M, M − K ) , R )) is the dual ofthe relative fundamental class in H Lip n ( M, M − K ) ∼ = H n ( M, M − K ) ∼ = R .Therefore, the second equality is a consequence of the Hahn-Banachtheorem – this is exactly the same argument as in the non-Lipschitz case [22,Proposition C.6], but applied to functionals on C Lip ∗ ( M ) instead of C ∗ ( M );this is possible because A is Lipschitz. • The last equality is equivalent toinf K ∈ K ( M ) k [ M, M − K ] ∗ Lip k A ∞ = k [ M ] ∗ Lip k A ∞ . Here the ≥ -inequality is clear. For the ≤ -inequality, let ε > f ∈ C n cs , Lip ( M ) with k f k A ∞ ≤ k [ M ] ∗ Lip k A ∞ + ε . By Theorem 3.8, thereis a compactly supported cochain g and a ( n − h with Lipschitzcompact support such that f = g + δh . Since A ∈ S lf , Lip n ( M ), the chain h ′ defined by h ′ ( σ ) = ( h ( τ ) if σ ∈ S nj =0 { ∂ j σ ; σ ∈ A } ,0 otherwise,is compactly supported. Further, f ′ := g + δh ′ is compactly supported,cohomologous in C ∗ cs , Lip ( M ) to f , and k f ′ k A ∞ = k f k A ∞ . In particular, thereis K ∈ K ( M ) with f ′ ∈ C n Lip ( M, M − K ) and k [ M, M − K ] ∗ Lip k A ∞ ≤ k f ′ k A ∞ = k f k A ∞ ≤ k [ M ] ∗ Lip k A ∞ + ε. This finishes the proof of the duality principle. (cid:3) Product structures in the Lipschitz setting. The definition of productstructures in singular (co-)homology is based on the following maps: Let X and Y betopological spaces. Then there exist chain maps EZ : C ∗ ( X ) ⊗ C ∗ ( Y ) → C ∗ ( X × Y )and AW : C ∗ ( X × Y ) → C ∗ ( X ) ⊗ C ∗ ( Y ), called the Eilenberg-Zilber map and the Alexander-Whitney map , respectively, such that EZ ◦ AW and AW ◦ EZ both arenaturally homotopic to the identity; explicit formulas are, for example, given inDold’s book [10, 12.26 on p. 184].The map EZ and the composition C ∗ ( X ) ⊗ C ∗ ( Y ) → C ∗ ( X × Y ), f ⊗ g ( f ⊗ g ) ◦ AW, induce the so-called cross-products × : H m ( X ) ⊗ H n ( Y ) → H m + n ( X × Y ) , (3.13) × : H m ( X ) ⊗ H n ( Y ) → H m + n ( X × Y )in homology and cohomology, respectively.Next we describe these cross-products more explicitly on the (co)chain level: Let f ∈ C m ( X ) and g ∈ C n ( Y ). Let π X and π Y be the projections from X × Y to X and Y , respectively. For a k -simplex σ , let σ ⌋ l and ( k − l ) ⌊ σ the l -front face and the( k − l ) -back face of σ , respectively. Then the explicit formula for AW in loc. cit. yields(3.14) ( f × g )( σ ) = f ( π X ◦ σ ⌋ m ) · g ( π Y ◦ n ⌊ σ ) . For simplices σ : ∆ m → X and ̺ : ∆ n → Y , the chain EZ( σ ⊗ ̺ ) can be describedas follows: The product ∆ n × ∆ m → X × Y of σ and ̺ is not a simplex butcan be chopped into a union of ( m + n )-simplices (like a square can be choppedinto triangles, or a prism into tetrahedra). Then EZ( σ ⊗ ̺ ) is the sum of these( m + n )-simplices.From this description we see that if c = P i a i σ i and d = P j b j ̺ j are (Lip-schitz) locally finite chains in (metric) spaces X and Y , then P i,j a i b j ( σ i × ̺ j ) isa (Lipschitz) locally finite chain in X × Y . Thus, (3.13) extends to maps × : H lf m ( X ) ⊗ H lf n ( Y ) → H lf m + n ( X × Y ) , × : H lf , Lip m ( X ) ⊗ H lf , Lip n ( Y ) → H lf , Lip m + n ( X × Y ) . In general, the cross-product of two cocycles with compact supports has notnecessarily compact support. However, the cross-product of two cochains withLipschitz compact supports again has Lipschitz compact support: Lemma 3.15. Let M and N be two complete metric spaces, and let m , n ∈ N .Then the cross-product on C ∗ ( M ) ⊗ C ∗ ( N ) → C ∗ ( M × N ) restricts to a cross-product × : C m cs , Lip ( M ) ⊗ C n cs , Lip ( N ) → C m + n cs , Lip ( M × N ) , which induces a cross-product H m cs , Lip ( M ) ⊗ H n cs , Lip ( N ) → H m + n cs , Lip ( M × N ) .Proof. Let f ∈ C m cs , Lip ( M ) and g ∈ C n cs , Lip ( N ). Let L ∈ R > . Because f and g arecochains with Lipschitz compact supports, there are compact sets K M ⊂ M and K N ⊂ N with ∀ σ ∈ map(∆ m ,M ) (cid:0) Lip( σ ) ≤ L ∧ im( σ ) ⊂ M − K M (cid:1) = ⇒ f ( σ ) = 0 , and analogously for g and K N .We now consider the compact set K := U L ( K M ) × U L ( K N ) ⊂ M × N , where U L ( X ) denotes the set of all points with distance at most L from X . Becausethe diameter of the image of a Lipschitz map on a standard simplex is at mostas large as √ σ ∈ map(∆ m + n , M × N ) with Lip( σ ) ≤ L and im( σ ) ⊂ M × N − K , thenim( π M ◦ σ ) ⊂ M − K M or im( π N ◦ σ ) ⊂ N − K N . In particular, f ( π M ◦ σ ⌋ m ) = 0 or g ( π N ◦ n ⌊ σ ) = 0. By (3.14), ( f × g )( σ ) = 0. Inother words, the cross-product f × g lies in C m + n cs , Lip ( M × N ). (cid:3) Definition 3.16. Let M and N be two topological spaces, let m , n ∈ N , and let A ⊂ map(∆ m + n , M × N ). Then we write A M := (cid:8) π M ◦ σ ⌋ m ; σ ∈ A (cid:9) ,A N := (cid:8) π N ◦ n ⌊ σ ; σ ∈ A (cid:9) , where π M : M × N → M and π N : M × N → N are the projections. Notice that A M and A N depend on m and n , but the context will always make clear whichindices are involved; therefore, we suppress m and n in the notation.The cross-product of cochains with Lipschitz compact support is continuous inthe following sense: IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 19 Remark 3.17. Let M and N be two topological spaces, and let m , n ∈ N . Then– by the explicit description (3.14) – the cross-product satisfies k f × g k A ∞ ≤ k f k A M ∞ · k g k A N ∞ for all A ⊂ map(∆ m + n , M × N ) and all f ∈ C m cs , Lip ( M ), g ∈ C n cs , Lip ( N ).Notice however, that in general the sets A M and A N are not locally finite evenif A is locally finite. This issue is addressed in Section 3.5. Lemma 3.18. Let M and N be oriented, connected, complete Riemannian mani-folds. Then [ M × N ] ∗ Lip = [ M ] ∗ Lip × [ N ] ∗ Lip ∈ H ∗ cs , Lip ( M × N ) . Proof. In view of Remark 3.10, it is enough to show (cid:10) [ M ] ∗ Lip × [ N ] ∗ Lip , [ M ] Lip × [ N ] Lip (cid:11) = 1 . Let f ∈ C m cs , Lip ( M ) and g ∈ C n cs , Lip ( N ) be fundamental cocycles that vanish ondegenerate simplices. Such fundamental cocycles always exist; for example, let f bethe cocycle σ R ∆ m σ ∗ ω where ω ∈ Ω m ( M ) is a compactly supported differential m -form with R M ω = 1. Note that the integral exists by Rademacher’s theorem [14].Let w = P i a i σ i ∈ C lf , Lip m ( M ) and z = P j b j ̺ j ∈ C lf , Lip n ( N ) be fundamentalcycles of M and N respectively. The Eilenberg-Zilber and Alexander-Whitneymaps have the property that AW ◦ EZ differs from the identity by degeneratechains [13, Theorem 2.1a (2.3)]. Thus, we obtain h f × g, w × z i = X i,j a i b j ( f × g )( σ i × ̺ j )= X i,j a i b j ( f ⊗ g ) (cid:0) AW ◦ EZ( σ i ⊗ ̺ j ) (cid:1) = X i,j a i b j ( f ⊗ g ) (cid:0) σ i ⊗ ̺ j + degenerate simplices (cid:1) = X i,j a i b j f ( σ i ) g ( ̺ j ) = f ( w ) g ( z ) = 1 . (cid:3) Representing the fundamental class of the product by sparse cycles. The functor C lf ∗ is only functorial with respect to proper maps. For example, ingeneral, the projection of a locally finite chain on a product of non-compact spacesto one of its factors is not locally finite. Definition 3.19. Let M and N be two topological spaces, and let k ∈ N . A locallyfinite set A ∈ S lf k ( M × N ) is called sparse if (cid:8) π M ◦ σ ; σ ∈ A (cid:9) ∈ S lf k ( M ) and (cid:8) π N ◦ σ ; σ ∈ A (cid:9) ∈ S lf k ( N ) , where π M : M × N → M and π N : M × N → N are the projections.A locally finite chain c ∈ C lf ∗ ( M × N ) is called sparse if its support is sparse.The following proposition is crucial in proving the product inequality for theLipschitz simplicial volume. Proposition 3.20. Let M and N be two oriented, connected, complete Riemannianmanifolds (without boundary) with non-positive sectional curvature. (1) For any cycle c ∈ C lf , Lip ∗ ( M × N ) there is a sparse cycle c ′ ∈ C lf , Lip ∗ ( M × N ) satisfying | c ′ | ≤ | c | and c ∼ c ′ in C lf , Lip ∗ ( M × N ) . (2) In particular, the Lipschitz simplicial volume can be computed via sparsefundamental cycles, i.e., k M × N k Lip = inf (cid:8) k M × N k A ; A ∈ S lf , Lipdim M +dim N ( M × N ) , A sparse (cid:9) . Proof. The second part is a direct consequence of the first part. For the first part,we take advantage of a straightening procedure:Let F M ⊂ M and F N ⊂ N be locally finite subsets with U ( F M ) = M and U ( F N ) = N . Then the corresponding preimages e F M := p − M ( F M ) ⊂ f M and e F N := p − N ( F N ) ⊂ e N satisfy U ( e F M ) = f M and U ( e F N ) = e N , where p M : f M → M and p N : e N → N are the Riemannian universal covering maps.Furthermore, also the product F := F M × F N ⊂ M × N is locally finite, andthere is a π ( M ) × π ( N )-equivariant map f : f M × e N → e F M × e F N =: e F such that d f M × e N (cid:0) z, f ( z ) (cid:1) ≤ √ z ∈ f M × e N .For σ ∈ map(∆ k , M × N ), we define h σ := ( p M × p N ) ◦ (cid:2)e σ, [ f ( e σ ( v )) , . . . , f ( e σ ( v k ))] (cid:3) : ∆ k × [0 , → M × N, where v , . . . , v k are the vertices of the standard simplex ∆ k , and e σ is a lift of σ .By Lemma 2.3 and Proposition 2.4 (and Remark 2.5), the map h σ is Lipschitz,and the Lipschitz constant can be estimated from above in terms of the Lipschitzconstant of σ . Moreover, the fact that f is equivariant and covering theory showthat h σ ◦ ∂ j = h σ ◦ ( ∂ j × id [0 , )(3.21)for all σ ∈ map(∆ k , M × N ) and all j ∈ { , . . . , k − } , where ∂ j : ∆ k − → ∆ k isthe inclusion of the j -th face.We now consider the map H : C lf , Lip ∗ ( M × N ) −→ C lf , Lip ∗ +1 ( M × N ) X i ∈ I a i · σ i X i ∈ I a i · h σ i , where h σ is the singular chain constructed out of h σ by subdividing the prism ∆ k × [0 , 1] in the canonical way into a sum of k + 1 simplices of dimension k + 1 (compareLemma 2.13).The map H is indeed well-defined: As discussed above, for all c ∈ C lf , Lip k ( M × N ), all simplices occurring in the (formal) sum H ( c ) satisfy a uniform Lipschitzcondition depending on Lip( c ). Further, it follows from im( h σ ) ⊂ U √ (im( σ )) that H maps locally finite chains to locally finite chains. As next step, we define ϕ : C lf , Lip ∗ ( M × N ) −→ C lf , Lip ∗ ( M × N ) X i ∈ I a i · σ i X i ∈ I a i · h σ i ( · , . IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 21 In other words, ϕ is given by replacing each simplex by a straight simplex whosevertices lie in F M × F N and whose vertices are close to the ones of the originalsimplex. Property (3.21) implies that ϕ is a chain map and that H is a chainhomotopy between the identity and ϕ (see Lemma 2.13).By construction, k ϕ k ≤ 1. Therefore, it remains to show that the image of ϕ contains only sparse chains:Let c ∈ C lf , Lip k ( M × N ). Let A := supp( ϕ ( c )). Because the geodesics in f M × e N are just products of geodesics in f M and e N , it follows that the projection π M : M × N → M preserves straight simplices. Thus, the set (cid:8) π M ◦ σ ; σ ∈ A (cid:9) consistsof straight simplices whose Lipschitz constant is bounded by Lip( c ) and whosevertices lie in F M . The fact that F M is locally finite and that there are only finitelymany straight simplices with a bounded Lipschitz constant and the same verticesimply that { π M ◦ σ ; σ ∈ A } is locally finite. Similarly for the projection to N . Sothe chain ϕ ( c ) is sparse. (cid:3) Conclusion of the proof of the product inequality. Finally, we can putall the pieces collected in the previous sections together to give a proof of theproduct inequality: Proof of Theorem 1.7. In the following, we write m := dim M and n := dim N . Inorder to prove the product inequality, it suffices to find for each ε ∈ R > locallyfinite sets A M ∈ S lf , Lip m ( M ) and A N ∈ S lf , Lip n ( N ) with k M k A M · k N k A N ≤ k M × N k Lip + ε. For every ε ∈ R > , Proposition 3.20 provides us with a sparse fundamentalcycle c ∈ C lf , Lip m + n ( M × N ) with support A satisfying | c | ≤ k M × N k Lip + ε . Inparticular, k M × N k A Lip ≤ k M × N k Lip + ε. By sparseness, the sets A M and A N associated to A (see Definition 3.16) lie in S lf , Lip m ( M ) and S lf , Lip n ( N ), respectively. The duality principle (Proposition 3.12)yields k M k A M = 1 k [ M ] ∗ Lip k A M ∞ , k N k A N = 1 k [ N ] ∗ Lip k A N ∞ , k M × N k A = 1 k [ M × N ] ∗ Lip k A ∞ ;the cohomological terms are related as follows k [ M × N ] ∗ Lip k A ∞ ≤ k [ M ] ∗ Lip k A M ∞ · k [ N ] ∗ Lip k A N ∞ because [ M × N ] ∗ Lip = [ M ] ∗ Lip × [ N ] ∗ Lip (Lemma 3.18) and the cohomological cross-product is compatible with the semi-norms (Remark 3.17).Therefore, we obtain k M k Lip · k N k Lip ≤ k M k A M · k N k A N ≤ k M × N k A ≤ k M × N k Lip + ε. (cid:3) Proportionality principle for non-compact manifolds Thurston’s proof of the proportionality principle in the compact case is based on“smearing” singular chains to so-called measure chains [28, Chapter 5; 29, p. 6.6–6.10]. We prove the proportionality principle in the non-compact case by combiningthe smearing technique with a discrete approximation of it; to this end, we replacemeasure homology by Lipschitz measure homology, a variant that incorporates aLipschitz constraint (Section 4.2). Throughout Section 4, we often refer to the following setup: Setup 4.1. Let M and N be oriented, connected, complete, non-positively curvedRiemannian manifolds of finite volume without boundary whose universal coversare isometric. We denote the common universal cover by U . Let G = Isom + ( U ) beits group of orientation-preserving isometries. Then Γ = π ( M ) and Λ = π ( N ) arelattices in G by Lemma 4.2 below. Let µ Λ \ G denote the normalized Haar measureon Λ \ G . The universal covering maps of M and N are denoted by p M and p N ,respectively.The following lemma is well known for locally symmetric spaces and compactmanifolds but we were unable to find a reference in the general case. Lemma 4.2. Let M be a complete Riemannian manifold of finite volume. Then Γ = π ( M ) is a lattice in G = Isom( f M ) .Proof. The isometry group G acts smoothly and properly on f M . It is easy to seethat Γ is a discrete subgroup. Let x ∈ f M , and let K ⊂ G be the stabilizer of x .Let ν → Gx be the normal bundle of Gx , and let ν ( r ) denote the sub-bundle ofvectors of length at most r . By the slice theorem [11, Chapter 2; 24, Section 2.2],there exists r > ν ( r ) → V is a diffeomorphismonto a tubular neighborhood V of Gx . The map f : G × K ν x → ν , ( g, z ) T g ( z )is a diffeomorphism. Define g = f − ◦ exp − . We equip G/K with the Riemannianmetric that turns the diffeomorphism G/K → Gx into an isometry. Since ν x can be equipped with a K -invariant metric ( K is compact), it is easy to see that G × K ν x ( r ) carries a G -invariant Riemannian metric such that the projection G × K ν x ( r ) → G/K is a Riemannian submersion. By compactness, there is λ > T z g has norm at most λ for all z ∈ exp( ν x ( r )). By G -invariance of themetrics, T z g has norm at most λ for all z ∈ V , thus, g is λ -Lipschitz, and so is theinduced map between the Γ-quotients. We obtain thatvol (cid:0) Γ \ ( G × K ν x ( r )) (cid:1) ≤ λ dim( M ) vol(Γ \ V ) < ∞ . Fubini’s theorem for Riemannian submersions [26, Theorem 5.6 on p. 66] yieldsvol(Γ \ G/K ) vol( ν x ( r )) = vol (cid:0) Γ \ ( G × K ν x ( r )) (cid:1) < ∞ . Thus vol(Γ \ G/K ) < ∞ . Now equip G with a G -equivariant metric such that G → G/K is a Riemannian submersion. By uniqueness, the corresponding Riemannianmeasure on G is a Haar measure. Fubini’s theorem and vol(Γ \ G/K ) < ∞ showthat vol(Γ \ G ) < ∞ . (cid:3) Integrating Lipschitz chains. Before introducing the smearing operation inSection 4.2, we first discuss integration of Lipschitz chains, which provides a meansto detect which class in locally finite homology a given Lipschitz cycle represents.Let M be an n -dimensional Riemannian manifold, and let K ⊂ M be a com-pact, connected subset with non-empty interior. Let Ω ∗ ( M, M − K ) be the kernelof the restriction homomorphism Ω ∗ ( M ) → Ω ∗ ( M − K ) on differential forms. Thecorresponding cohomology groups are denoted by H ∗ dR ( M, M − K ). The de Rhammap Ω ∗ ( M ) → C ∗ ( M ) restricts to the respective kernels and thus induces a homo-morphism, called relative de Rham map ,Ψ ∗ : H ∗ dR ( M, M − K ) → H ∗ ( M, M − K ) . IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 23 The relative de Rham map is an isomorphism, which follows from the bijectivityof the absolute de Rham map and an application of the five lemma. Note thatintegration gives a homomorphism R : H n dR ( M, M − K ) → R . Moreover, it is wellknown that(4.3) (cid:10) Ψ n [ ω ] , [ M, M − K ] (cid:11) = Z M ω holds for all n -forms ω . Proposition 4.4. Let M be a Riemannian n -manifold, and let c = P k ∈ N a k σ k ∈ C lf n ( M ) be a cycle with | c | < ∞ and Lip( c ) < ∞ .(1) Then h dvol M , σ k i ≤ Lip( c ) n vol(∆ n ) for every k ∈ N .(2) Furthermore, we have the following equivalence: X k ∈ N a k · h dvol M , σ k i = vol( M ) ⇐⇒ c is a fundamental cycle.Proof. For the first part, it suffices to observe that all Lipschitz simplices σ arealmost everywhere differentiable, that σ ∗ dvol M is measurable (by Rademacher’stheorem [14]), and that (cid:12)(cid:12) h dvol M , σ i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∆ n σ ∗ dvol M (cid:12)(cid:12)(cid:12)(cid:12) ≤ ess-sup x ∈ ∆ n k T x σ k n vol (cid:0) ∆ n (cid:1) ≤ Lip( σ ) n vol (cid:0) ∆ n (cid:1) holds. In particular, we see that P k ∈ N a k h dvol M , σ k i converges absolutely.For the second part, let s ∈ R be the number defined by[ c ] = s · [ M ] ∈ H lf n ( M ) . In the following, we show that P k ∈ N a k h dvol M , σ k i = s · vol( M ): To this end, wefirst relate s · vol( K ) for compact K to a finite sum derived from the series on theleft hand side, and then use a limit process to compute the value of the whole series.Let K ⊂ M be a connected, compact subset with non-empty interior. For δ ∈ R > let g δ : M → [0 , 1] be a smooth function supported on the closed δ -neigh-borhood K δ of K with g | K = 1. Then g δ · dvol M ∈ Ω n ( M, M − K δ ) is a cocycle,and s · vol( K ) = lim δ → s · Z M g δ dvol M . On the other hand, the map H n ( j δ ) : H lf n ( M ) → H n ( M, M − K δ ) induced bythe inclusion j δ : ( M, ∅ ) → ( M, M − K δ ) maps the fundamental class of M to the relative fundamental class of ( M, M − K δ ) and H n ( j δ )[ c ] is representedby P im σ k ∩ K δ = ∅ a k σ k . Therefore, we obtain by (4.3)lim δ → X im σ k ∩ K δ = ∅ a k · h g δ · dvol M , σ k i = lim δ → (cid:10) Ψ n [ g δ · dvol M ] , s · [ M, M − K δ ] (cid:11) = lim δ → s · Z M g δ dvol M = s · vol( K ) . For each k ∈ N and δ ∈ R > we have |h g δ · dvol M , σ k i| ≤ Lip( c ) n vol(∆ n ), and hence (cid:12)(cid:12)(cid:12)(cid:12)X k ∈ N a k h dvol M , σ k i − X im σ k ∩ K δ =0 a k h g δ · dvol M , σ k i (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ) n vol(∆ n ) · X im σ k ⊂ M − K | a k | . Because P k ∈ N | a k | < ∞ , there is an exhausting sequence ( K m ) m ∈ N of compact,connected subsets of M with non-empty interior satisfyinglim m →∞ vol( K m ) = vol( M ) and lim m →∞ X im σ k ⊂ M − K m | a k | = 0 . Thus, the estimates of the previous paragraphs yield X k ∈ N a k · h dvol M , σ k i = lim m →∞ lim δ → X im σ k ∩ K mδ = ∅ a k · h g mδ · dvol M , σ k i = lim m →∞ s · vol( K m )= s · vol( M ) . If c is a fundamental cycle, then s = 1 and hence the series has value vol( M ).Conversely, if the series evaluates to vol( M ), then vol( M ) must be finite by thefirst part. Therefore, we can deduce from the computation above that s = 1, i.e., c is a fundamental cycle. (cid:3) One should be aware that the (locally finite) simplicial volume of a non-compactmanifold M might be finite even if vol( M ) = ∞ , e.g., (cid:13)(cid:13) R (cid:13)(cid:13) = 0 – unlike the Lip-schitz simplicial volume as the following direct corollary of Proposition 4.4 shows. Corollary 4.5. Let M be a Riemannian manifold. If k M k Lip is finite, then sois vol( M ) . The smearing homomorphism. Let M and N be smooth manifolds (withor without boundary). The set of smooth maps M → N equipped with the topologythat turns the differential map from this set to map( T M, T N ) into a homeomor-phism onto its image is denoted by C ( M, N ). This topology is called C -topology .The following defines a variant of Thurston’s measure homology [29, p. 6.6f]. Definition 4.6 (Lipschitz measure homology) . Let M be a Riemannian manifold.(a) A signed Borel measure µ on C (∆ n , M ) is said to have Lipschitz deter-mination if there is L > µ is determined on the subset of C -simplices whose Lipschitz constant is smaller than L .(b) Let C Lip ∗ ( M ) denote the set of signed Borel measures on C (∆ n , M ) thathave finite total variation and Lipschitz determination. Then ( C Lip n ( M )) n ≥ forms a chain complex whose elements are called Lipschitz measure chains .The differential is given by the alternating sum of push-forwards inducedby face maps [25, p. 539; 31, Corollary 2.9]. The total variation defines anorm on each of these chain groups.(c) The homology groups of C Lip ∗ ( M ) are denoted by H Lip ∗ ( M ). They areequipped with the quotient semi-norm. IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 25 The Lipschitz determination condition ensures that the function σ R σ ∗ dvol M is bounded on the supports of the measure chains in question. Therefore, Lipschitzmeasure chains can be evaluated against the volume form: Remark 4.7. Let M be a Riemannian n -manifold and let µ ∈ C Lip n ( M ). Then thefunction I : C (∆ n , M ) → R , σ 7→ h dvol M , σ i = Z ∆ n σ ∗ dvol M is well defined, measurable, and µ -almost everywhere bounded, thus µ -integrable.We denote the integral R Idµ by h dvol M , µ i . Definition 4.8. For a Riemannian manifold M , we define the following subcomplexof C lf , Lip ∗ ( M ) (see Definition 3.2) C ℓ , Lip ∗ ( M ) = (cid:26)X i ∈ N a i σ i ∈ C lf , Lip ∗ ( M ); σ i smooth for all i ∈ N , and X i ∈ N | a i | < ∞ (cid:27) . A cycle in C ℓ , Lipdim M ( M ) is called a fundamental cycle if it is a locally finite funda-mental cycle in C lfdim M ( M ).From now on, we refer to the setting in Setup 4.1. Thurston’s smearing techniqueis a cunning way of averaging the simplices over the isometry group of the universalcover: Proposition 4.9. Let σ : ∆ i → M be a smooth simplex, and let e σ : ∆ i → U be alift of σ to U . The push-forward of µ Λ \ G under the map smear e σ : Λ \ G → C (∆ i , N ) , Λ g p N ◦ g e σ does not depend on the choice of the lift of σ and is denoted by µ σ . Further thereis a well-defined chain map smear ∗ : C ℓ , Lip ∗ ( M ) −→ C Lip ∗ ( N ) , X σ a σ σ X σ a σ µ σ . Proof. One uses the right G -invariance of µ Λ \ G for showing that smear ∗ is indepen-dent of the choice of the lifts and compatible with the boundary. The computationsare similar to the ones in the classical case [28, Section 5.4]. (cid:3) In the proof of the proportionality principle (Theorem 1.5), it is essential to beable to determine the map induced by smearing in the top homology. We achievethis by evaluating with respect to the volume form. Lemma 4.10. For every fundamental cycle c ∈ C ℓ , Lip n ( M ) we have (cid:10) dvol N , smear n ( c ) (cid:11) = Z C (∆ n ,N ) Z ∆ n σ ∗ dvol N d smear n ( c )( σ ) = vol( M ) . Remark 4.11. There exists a fundamental cycle in C ℓ , Lip n ( M ) if and only if k M k Lip < ∞ . Equivalently, the Lipschitz simplicial volume can be computed bysmooth cycles:(4.12) k M k Lip = inf (cid:8) | c | ; c ∈ C lf n ( M ) smooth fundamental cycle, Lip( c ) < ∞ (cid:9) . This can be shown without curvature conditions using relative approximation the-orems for Lipschitz maps by smooth ones but in the case of non-positively curvedmanifolds the straightening technique gives a quick proof of (4.12): If c = P i ∈ I a i σ i ∈ C lf ∗ ( M ) satisfies Lip( c ) < ∞ , then Proposition 2.4 andRemark 2.5 show that also the straightened chain c ′ = P i ∈ I a i · s M ( σ i ) is bothLipschitz and locally finite. Moreover, it is smooth by 2.4. Thus, straighteningchains gives rise to a chain map C lf , Lip ∗ ( M ) → C lf , Lip ∗ ( M ). The same argumentsas in Proposition 2.12 and Lemma 2.13 also apply in the locally finite case, whichimplies that this chain map is homotopic to the identity. Hence [ c ′ ] = [ c ], which,combined with | c ′ | ≤ | c | , shows (4.12). Proof of Lemma 4.10. In view of Remark 4.7, the double integral in the lemma iswell-defined. Because the universal covering maps p M and p N are locally isometric,we obtain (where we write c = P σ a σ σ ) h dvol N , smear n ( c ) i = X σ a σ h dvol N , µ e σ i = X σ a σ Z C (∆ n ,N ) h dvol N , ̺ i dµ e σ ( ̺ )= X σ a σ Z Λ \ G h dvol N , p N ◦ g e σ i dµ Λ \ G ( g )= X σ a σ Z Λ \ G h dvol U , g e σ i dµ Λ \ G ( g )= X σ a σ Z Λ \ G h dvol U , e σ i dµ Λ \ G ( g )= X σ a σ Z Λ \ G h dvol M , σ i dµ Λ \ G ( g ) . By Proposition 4.4, the last expression equals vol( M ). (cid:3) Proof of Theorem 1.5. In order to prove the proportionality principle (The-orem 1.5), we proceed in the following steps:(1) First we construct a Λ-equivariant partition of U into Borel sets of smalldiameter and a corresponding Λ-equivariant 1-net.(2) Using the 1-net and a straightening procedure, we develop a discrete ver-sion of the smearing map – i.e., a mechanism turning fundamental cycleson M into cycles on N . This has some similarity with the construction byBenedetti and Petronio [1, p. 114f].(3) By comparing the discrete smearing with the original smearing, integrationenables us to identify which class the smeared cycle represents.(4) In the final step, we compute the ℓ -norm of the smeared cycle, therebyproving the theorem. Proof of Theorem 1.5. Like in the previous paragraphs, we refer to the notationestablished in Setup 4.1.4.3.1. Construction of a suitable Λ -equivariant partition of U into Borel sets. Bylocally subdividing a triangulation of N , it is possible to construct a locally finite(and hence countable) set T ⊂ N and a partition ( F x ) x ∈ T of N into Borel sets withthe following properties: For each x ∈ T we have x ∈ F x , the diameter of F x is atmost 1 / T is a 1-net in N ), and the universal cover p N is trivial over F x . IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 27 Let e T ⊂ U be a lift of T to U = e N . In view of the triviality condition, wefind a corresponding Λ-equivariant partition e F := ( e F x ) x ∈ Λ · e T of U into Borel sets ofdiameter at most 1 / 2. Note that Λ · e T is locally finite since Λ acts properly on e N .4.3.2. Discrete version of the smearing map. In order to construct the discrete ver-sion of the smearing map, we first define a version str of the geodesic straighteningthat turns simplices in U into geodesic simplices with vertices in Λ · e T : For an i -simplex ̺ : ∆ i → U we define the geodesic simplexstr i ( ̺ ) := [ x , . . . , x i ] , where x , . . . , x i ∈ Λ · e T are the elements uniquely determined by the requirementthat for all j ∈ { , . . . , i } the j -th vertex of ̺ lies in e F x j . By Proposition 2.4,the simplex str i ( ̺ ) is smooth. Because the partition e F is Λ-equivariant, so is str i .Using the fact that all elements of e F are Borel and that Λ · e T is countable, it is notdifficult to see that the map str i : C (∆ i , U ) → C (∆ i , N ) is Borel with respect tothe C -topology. Moreover, for all k ∈ { , . . . , i } (4.13) str i − ( ∂ k ̺ ) = ∂ k str i ( ̺ ) . For i ∈ N we write S i := (cid:8) p N ◦ σ ; σ : ∆ i → U geodesic simplex with vertices in Λ · e T (cid:9) ⊂ C (∆ i , N ) , and for every simplex σ : ∆ i → U we define a map f σ : G → S i , g p N ◦ str i ( gσ );The map f σ is Borel because str i is Borel and the action of G is C -continuous (thecompact-open topology on G coincides with the C -topology [28, Theorem 5.12]).Furthermore, f σ induces a well-defined Borel map f σ : Λ \ G → S i , which we denoteby the same symbol.We now consider the following discrete approximation of the smearing map de-fined in Proposition 4.9 ϕ ∗ : C ℓ , Lip ∗ ( M ) → C ℓ , Lip ∗ ( N ) ϕ i (cid:18)X k ∈ N a k σ k (cid:19) := X ̺ ∈ S i (cid:18)X k ∈ N a k · µ Λ \ G (cid:0) f − e σ k ( ̺ ) (cid:1)(cid:19) · ̺ (4.14)where each e σ k is a lift of σ k to U . First we show that ϕ ∗ is well-defined: Thenumber µ Λ \ G ( f − e σ ( ̺ )) does not depend on the choice of the lift e σ of the simplex σ because µ Λ \ G is invariant under right multiplication of G . If L = Lip( σ ), any lift e σ has diameter at most √ L . Hence, each pair of vertices of str i ( g e σ ) has distance atmost 1 + √ L . In view of Proposition 2.4 and Remark 2.5, str i ( g e σ ), and thus f e σ ( g ),are smooth and have a Lipschitz constant depending only on L . Hence there is auniform bound on the Lipschitz constants of simplices appearing in the right handsum of (4.14). This also implies that (4.14) defines a locally finite chain becauseboth Λ · e T and T are locally finite. Therefore, ϕ i is a well-defined homomorphismfor every i ∈ N .Next we prove that ϕ ∗ is a chain homomorphism: From (4.13) we obtain [ ̺ with ∂ k ̺ = ξ (cid:8) Λ g ∈ Λ \ G ; p N ◦ str i ( g e σ ) = ̺ (cid:9) = (cid:8) Λ g ∈ Λ \ G ; p N ◦ str i − ( g∂ k e σ ) = ξ (cid:9) for all σ ∈ map(∆ i , N ), k ∈ { , . . . , i } , and all ξ ∈ map(∆ i − , N ). Because the lefthand side is a disjoint, at most countable, union this implies that X ̺ with ∂ k ̺ = ξ µ Λ \ G (cid:0) f − e σ ( ̺ ) (cid:1) = µ Λ \ G (cid:0) f − ∂ k e σ ( ξ ) (cid:1) . Therefore, we deduce ∂ k ϕ i ( σ ) = X ̺ ∈ S i µ Λ \ G (cid:0) f − e σ ( ̺ ) (cid:1) · ∂ k ̺ = X ξ ∈ S i − X ̺ with ∂ k ̺ = ξ µ Λ \ G (cid:0) f − e σ ( ̺ ) (cid:1) · ξ = X ξ ∈ S i − µ Λ \ G (cid:0) f − ∂ k e σ ( ξ ) (cid:1) · ξ = ϕ i − ( ∂ k σ ) , which shows that ϕ ∗ is a chain map.4.3.3. Comparison with the original smearing map. Let j ∗ : C ℓ , Lip ∗ ( N ) → C Lip ∗ ( N )be the chain map that is the obvious extension of the map given by mapping asimplex σ to the atomic measure concentrated in { σ } . Next we show that there is achain homotopy between the smearing map smear ∗ given in Proposition 4.9 and thecomposition j ∗ ◦ ϕ ∗ : For any smooth simplex σ : ∆ i → U and g ∈ G the geodesichomotopy from str i ( gσ ) to gσ followed by p N defines a map h σ ( g ) : ∆ i × I → N . ByProposition 2.1, Proposition 2.4, and Remark 2.5, h σ ( g ) is smooth and its Lipschitzconstant is bounded from above in terms of the Lipschitz constant of σ . Moreover,Proposition 2.1 shows that the map h σ : G → C (∆ i , N ) is Borel with respect tothe C -topology. Because str ∗ is Λ-equivariant, we obtain a well-defined Borel map h σ : Λ \ G → C (∆ i × I, N )satisfying h σ (Λ g ) | ∆ i ×{ } = f σ ( g ) , (4.15) h σ (Λ g ) | ∆ i ×{ } = p N ◦ gσ. It is also clear that for each face map ∂ k : ∆ i − → ∆ i and every simplex σ : ∆ i → U we have h σ ◦ ∂ k (Λ g ) = h σ (Λ g ) ◦ (cid:0) ∂ k × id I (cid:1) . Retaining the notation of Lemma 2.13 and Remark 2.14, for every σ : ∆ i → U andevery k ∈ { , . . . , i } let ν σ,k be the push-forward of µ Λ \ G under the mapΛ \ G → C (∆ i +1 , N ) , Λ g h σ ( g ) ◦ G i,k . If σ is a simplex in M and e σ a lift to U , then ν e σ,k does not depend on the choiceof the lift and will be also denoted by ν σ,k . We now define the homomorphism H ∗ : C ℓ , Lip ∗ ( M ) → C Lip ∗ +1 ( N ) , H i ( σ ) := X k =0 ν σ,k . Lemma 2.13 and (4.15) yield [31, Theorem 2.1 (1)] ∂H i ( σ ) + H i − ∂σ = j i ( ϕ i ( σ )) − smear i ( σ ) IPSCHITZ SIMPLICIAL VOLUME OF NON-POSITIVELY CURVED MANIFOLDS 29 for every i -simplex σ in M . Thus H ∗ is the desired chain homotopy j ∗ ◦ ϕ ∗ ≃ smear ∗ .The evaluation with dvol N (cf. Remark 4.7) is compatible with j ∗ , that is, (cid:10) dvol N , j ∗ ( c ) (cid:11) = h dvol N , c i for every c ∈ C ℓ , Lip ∗ ( N ).Let c ∈ C ℓ , Lip n ( M ) be a fundamental cycle. Because evaluation with dvol N iswell-defined on homology classes and by Lemma 4.10, we obtain that (cid:10) dvol N , ϕ n ( c ) (cid:11) = (cid:10) dvol N , j n ( ϕ n ( c )) (cid:11) = (cid:10) dvol N , smear n ( c ) (cid:11) = vol( M ) . Now Proposition 4.4 lets us determine the homology class of ϕ n ( c ) as(4.16) [ ϕ n ( c )] = vol( M )vol( N ) · [ N ] . The norm estimate and conclusion of proof. By symmetry we only have toshow that k M k Lip vol( M ) ≥ k N k Lip vol( N ) , and in addition we can assume k M k Lip < ∞ . By Remark 4.11, we can computethe Lipschitz simplicial volume k M k Lip by fundamental cycles lying in the chaincomplex C ℓ , Lip ∗ ( M ). Let c = P k ∈ N a k σ k ∈ C ℓ , Lip n ( M ) be a fundamental cycleof M . Because of (4.16) it suffices to show that | ϕ n ( c ) | ≤ | c | , which is a consequence the following computation: | ϕ n ( c ) | ≤ X ̺ ∈ S n X k ∈ N | a k | · µ (cid:0) f − f σ k ( ̺ ) (cid:1) = X k ∈ N X ̺ ∈ S n | a k | · µ (cid:0) f − f σ k ( ̺ ) (cid:1) = X k ∈ N | a k | = | c | . This finishes the proof of the proportionality principle. (cid:3) Vanishing results for the locally finite simplicial volume In this section, we give a proof of the vanishing theorem (Theorem 1.1); the proof isbased on the fact that locally symmetric spaces of higher Q -rank admit “amenable”coverings of sufficiently small multiplicity and Gromov’s vanishing finiteness theo-rem.As a first step, we recall Gromov’s definition of amenable subsets and sequencesof subsets that are amenable at infinity [16, p. 58] and his vanishing-finitenesstheorem: Definition 5.1. Let X be a topological space.(1) A subset U ⊂ X is called amenable in X if for every basepoint x ∈ U thesubgroup im (cid:0) π ( U, x ) → π ( X, x ) (cid:1) is amenable. (2) A sequence ( U i ) i ∈ N of subsets of X is called amenable at infinity if there isan increasing sequence of compact subsets ( K i ) i ∈ N of X with U i ⊂ X − K i , X = S i ∈ N K i , and such that U i is amenable in X − K i for sufficientlylarge i ∈ N . Theorem 5.2 (Vanishing-finiteness theorem for simplicial volume [16, Corollary (A)on p. 58]) . Let M be a manifold without boundary of dimension n . Let ( U i ) i ∈ N be alocally finite covering of M by open, relatively compact subsets such that each pointof M is contained in at most n such subsets. If every U i is amenable in M and ( U i ) i ∈ N is amenable at infinity, then k M k = 0 . As a next step, we provide a construction of locally finite coverings with smallmultiplicity by relatively compact, open, amenable subsets; notice however thatsuch a covering is not necessarily amenable at infinity. Theorem 5.3. Let M be a manifold and Γ = π ( M ) . Assume that Γ admits afinite model for its classifying space B Γ of dimension k . Then there is a locallyfinite covering of M by relatively compact, amenable, open subsets such that everypoint of M is contained in at most k + 2 such subsets.Proof. Since B Γ is k -dimensional and compact, every open covering of B Γ has afinite refinement with multiplicity at most k +1 [17, Theorem V 1 on p. 54]. Startingwith a covering of B Γ by open, contractible sets, let ( V j ) j ∈ J be a finite refinementof multiplicity at most k + 1.We pull this covering back to M via the classifying map ϕ : M → B Γ: For j ∈ J let U j := ϕ − ( V j ) . By construction, ( U j ) j ∈ J is an open covering of M with multiplicity at most k + 1.However, the sets U j may not be relatively compact.To achieve a nice covering of M by relatively compact sets, we combine thecovering ( U j ) j ∈ J with another covering of M of small multiplicity consisting ofrelatively compact sets, which is constructed as follows: For every j ∈ J we choosea covering R j of R by bounded, open intervals such that each R j has multiplicity 2and for i = j the cover R i ⊔ R j (disjoint union) has multiplicity at most 3. This ispossible because J is finite.Let f : M → R be a proper function. We show now that the combined covering U := (cid:0) U j ∩ f − ( W ) (cid:1) j ∈ J, W ∈R j of M has the desired properties: In the following, by definition, we say that the J-index of U j ∩ f − ( W ) is j .Because f is proper and the elements of the R j are bounded, each set in U isrelatively compact.Since ϕ : π ( M ) → π ( B Γ) is an isomorphism, the inclusion U j ∩ f − ( W ) ֒ → M is trivial on the level of π if and only if its composition with ϕ is so. But thecomposition with ϕ factors over the inclusion V j ֒ → B Γ, which is trivial in π . Inparticular, each element of U is an amenable subset of M .It remains to verify that U has multiplicity at most k + 2: Suppose there is asubset U ⊂ U of k + 3 sets whose intersection is non-empty. Because the elementsof U have at most k + 1 different J -indices, and the multiplicity of each of the R j is at most 2, there must be i = j ∈ J such that there are at least two elements in U having J -index i , and at least two with J -index j . But this contradicts the fact that R i ⊔ R j has multiplicity at most 3. So the multiplicity of U is at most k + 2. (cid:3) In order to obtain a suitable amenable covering that is amenable at infinity,we impose additional constraints on the fundamental group of the boundary; oneshould compare this also with Gromov’s remark on subpolyhedra [16, p. 59]. Corollary 5.4. Let M be the interior of a compact, n -dimensional manifold W with boundary ∂W . Assume that Bπ ( M ) admits a finite model of dimension atmost n − and that at least one of the following conditions is satisfied(1) The fundamental group π ( ∂W ; x ) is amenable for all x ∈ ∂W .(2) For all x ∈ ∂W the inclusion induces an injection π ( ∂W ; x ) → π ( W ; x ) .Then k M k = 0 .Proof. By Theorem 5.3 we obtain a covering ( U i ) i ∈ N of M by open, relatively com-pact, amenable subsets in M which has multiplicity ≤ n . Let ( V i ) i ∈ N be a decreasingsequence of open neighborhoods in W of the boundary ∂W with T i ∈ N V i = ∂W and S i ∈ N V i = W . By choosing collar neighborhoods of ∂W we can assume that ∂W is a deformation retract of V i for all large i ∈ N . Because ( U i ) i ∈ N is locallyfinite, we additionally can assume that U i ⊂ V i for all i ∈ N .If π ( ∂W ; x ) is amenable for every basepoint x then U i is obviously an amenablesubset of V i for all large i ∈ N . If the inclusion maps ∂W → W are π -injectivethen so are the inclusion maps V i ∩ M → M for all large i ∈ N , and the amenabilityof the subset U i ⊂ V i ∩ M follows from the one of U i ⊂ M .In either case we can now apply Gromov’s vanishing-finiteness theorem 5.2. (cid:3) Example 5.5 (Products of open manifolds with non-zero simplicial volume) . Weconsider the open manifold M := W ◦ × R , where ( W, ∂W ) is the surface withboundary obtained by removing a finite number of pairwise disjoint, open discsfrom an oriented, closed, connected surface of genus at least 1.Then the fundamental group π ( M ) ∼ = π ( W ) is a finitely generated free groupand thus admits a finite model of dimension 1 = dim M − M as the interior of the compact manifold W × [0 , M to be infinite [16, p. 17; 22, Corollary 6.2].In fact, tracking down the construction of an open covering in the proof ofTheorem 5.3 shows that this particular covering is amenable but not amenable atinfinity.In particular, the finiteness hypothesis in the corollary is not sufficient for thevanishing of the simplicial volume. The following cohomological criterion helps tocheck whether the finiteness hypothesis in the corollary is satisfied. Lemma 5.6. Let Γ be a group that has a finite model for its classifying space B Γ .(1) If cd Γ = 2 , then there is a finite model of B Γ whose dimension equals theintegral cohomological dimension cd Γ of Γ .(2) If cd Γ = 2 , then there is a finite model of B Γ of dimension at most .Proof. Because there is a finite model for B Γ, the group Γ is finitely presented andof type FL. Therefore, a classic result of Eilenberg and Ganea shows that there isa finite model of B Γ of dimension max { cd Γ , } [7, Theorem VIII.7.1]. If cd Γ = 0, then Γ is the trivial group and hence the one-point space is a modelfor B Γ. If cd Γ = 1, then Γ is free by a theorem of Stallings and Swan [27]; becauseΓ is finitely presented, Γ is a finitely generated free group. In particular, we cantake a finite wedge of circles as a finite, one-dimensional model for B Γ. 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Graduiertenkolleg “Analytische Topologie und Metageometrie,” Westf¨alische Wil-helms-Universit¨at M¨unster, M¨unster, Germany E-mail address : [email protected] URL : University of Chicago, Chicago, USA Current address : Westf¨alische Wilhelms-Universit¨at M¨unster, M¨unster, Germany E-mail address : [email protected] URL ::