Cohomological induction and uniform measure equivalence
aa r X i v : . [ m a t h . G R ] A ug COHOMOLOGICAL INDUCTION AND UNIFORM MEASUREEQUIVALENCE
THOMAS GOTFREDSEN AND DAVID KYED
Abstract.
We construct a general cohomological induction isomorphism from a uniformmeasure equivalence of locally compact, second countable, unimodular groups which, as aspecial case, yields that the graded cohomology rings of quasi-isometric, connected, simplyconnected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer andalso provides new insight into the quasi-isometry classification problem for low dimensionalnilpotent Lie groups. Introduction
Geometric and measurable group theory originated in the the works of Gromov and isby now a well developed and important tool in the study of countable discrete groups; seee.g. [Gro84, Fur99a, Fur99b, OW80] and references therein. One of the many insights from thebeginning of the present century, due to Shalom, Shalom-Monod [MS06] and Sauer [Sau06],is that group cohomology interacts well with the notion of quasi-isometry, which is among themost fundamental concepts in geometric group theory [Roe03, CdlH16]. However, in recentyears, increasing emphasis has been put on the the more general setting of locally compactgroups [BHI18, BFS13, CLM18, BR18a, KKR17, KKR18], and results concerning the interplaybetween group cohomology and quasi-isometry are now beginning to emerge in the setting aswell [BR18b, SS18]. The present article provides a contribution to this line of research byproving that the central results due Shalom, Shalom-Monod and Sauer mentioned above,admit natural generalisations to the class of unimodular, locally compact, second countablegroups. Actually, our primary focus will not be on quasi-isometry but rather on the relatednotion known as (uniform) measure equivalence, which is a measurable analogue of quasi-isometry, introduced for discrete groups by Gromov in [Gro93]. Uniform measure equivalencecan be defined also for unimodular, locally compact second countable groups [BFS13, KKR17],and for compactly generated, unimodular groups it was shown in [KKR17] that, just as in thediscrete case, uniform measure equivalence implies quasi-isometry and that the two notionscoincide when the groups in question are amenable. Our primary focus will therefore be onuniformly measure equivalent topological groups and our first main result, drawing inspirationfrom [MS06, Proposition 4.6], is the following reciprocity principle in group cohomology; forthe basic definitions concerning cohomology and uniform measure equivalence, see Section 2.
Theorem A. If G and H are uniformly measure equivalent, locally compact, second countable,unimodular groups, then for any uniform measure equivalence coupling (Ω , η, X, µ, Y, ν, i, j ) Mathematics Subject Classification.
Key words and phrases.
Locally compact groups, measure equivalence, quasi-isometry, cohomology, nilpo-tent Lie groups. and any Fréchet G × H -module E there exists an isomorphism of topological vector spaces H n (cid:0) G, L (Ω , E ) H (cid:1) ∼ −→ H n (cid:0) H, L (Ω , E ) G (cid:1) , for all n > . In particular, when E = R with trivial G × H -action, this induces an isomorphism H n ( G, L ( X )) ≃ H n ( H, L ( Y )) . As a consequence of Theorem A we also obtain a generalisation of [Sau06, Theorem 5.1]. Forthe statement, recall that a group G is said to have (Shalom’s) property H T if H ∗ ( G, H ) = { } for every unitary G -module H with no non-trivial fixed-points. Recall also that the total coho-mology H ∗ ( G, R ) := ⊕ n > H n ( G, R ) and its reduced counterpart H ∗ ( G, R ) := ⊕ n > H n ( G, R ) become graded, unital rings with respect to the cup product; see Sections 2.1.1 and 2.1.2 forfurther details on this. Theorem B. If G and H are uniformly measure equivalent, locally compact, second countablegroups satisfying property H T then the associated reduced cohomology rings H ∗ ( G, R ) and H ∗ ( H, R ) are isomorphic as graded, unital rings. As already mentioned, for compactly generated amenable groups, quasi-isometry and uni-form measure equivalence coincide, and when the n -th Betti number β n ( G ) := dim R H n ( G, R ) is finite, then we have that H n ( G, R ) is automatically Hausdorff [Gui80, III, Prop. 3.1], so thatwe indeed recover Sauer’s result [Sau06, Theorem 5.1].We remark that both the class of connected, simply connected (csc) nilpotent Lie groupsand the class of finitely generated, nilpotent groups both satisfy the assumptions in TheoremB and that these furthermore have finite Betti numbers in all degrees; see Section 2.1.1 fordetails on this.For csc nilpotent Lie groups containing lattices, i.e. those admitting a rational structure[Mal51], the isomorphism in Theorem B was already known to experts in the field, as it canbe be deduced from [Sau06, Theorem 5.1] via [Nom54, Theorem 1]. In this way Theorem Bprovides a more natural approach, covering also csc nilpotent groups without lattices, and, asa special case, it also gives a ring-isomorphism H ∗ (Γ , R ) ≃ H ∗ ( G, R ) for any locally compactsecond countable, unimodular group G with a uniform lattice Γ G ; see [KKR18, Proposition6.11].As an application of our results, we show in Section 5 how Theorem A can be used to im-prove on the quasi-isometry classification programme for csc nilpotent Lie groups of dimension7. Moreover, we show that within each of the 1-parameter families of 7-dimensional nilpotentLie algebras (cf. [Gon98]), for which the corresponding csc nilpotent Lie groups are not dis-tinguished by Pansu’s theorem [Pan89, Théorème 3], all (but at most finitely many) membershave pairwise isomorphic cohomology rings, and thus showing that Theorem B cannot be usedto distinguish these. Standing assumptions.
Unless otherwise specified, all generic vector spaces will be over thereals, and this in particular applies to function spaces. Thus, if ( X, µ ) is a measure space L ( X ) will denote the Hilbert space of real valued square integrable functions, and so on andso forth. We remark that this convention is primarily chosen to streamline notation, and thatTheorem A and Theorem B hold verbatim over the complex numbers as well. OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 3
Acknowledgments.
The authors gratefully acknowledge the financial support from the Vil-lum Foundation (grant no. 7423) and from the Independent Research Fund Denmark (grantno. 7014-00145B and 9040-00107B). They also thank Henrik D. Petersen whose earlier jointwork with D.K. formed the basis for part of the present paper, Yves de Cornulier for suggest-ing the applications regarding quasi-isometry classification of 7-dimensional csc nilpotent Liegroups, and Nicky Cordua Mattsson for his assistance with programming issues in Maple.2.
Preliminaries
Group cohomology.
In this section we recall the basics on cohomology theory forlocally compact groups, from the point of view of relative homological algebra; the reader isreferred to [Gui80] for more details and proofs of the statements below. In what follows, G denotes a locally compact second countable group. Definition 2.1.
A (continuous) G -module is a Hausdorff topological vector space E endowedwith an action of G by linear maps such that the action map G × E → E is continuous. If E and F are G -modules, then a linear, continuous, G -equivariant map ϕ : E → F is called a morphism of G -modules and ϕ is said to be strengthened if there exists a linear, continuousmap η : F → E such that ϕ ◦ η ◦ ϕ = ϕ .Note that the map η in the definition of a strengthened morphism is not required to be G -equivariant. The definition of a strengthened morphism given above is easily seen to beequivalent to the more standard formulation given, for instance, in [Gui80, Definition D.1]. Definition 2.2. A G -module I is said to be relatively injective if for any strengthened, one-to-one homomorphism ι : E → F of G -modules E and F and any morphism ϕ : E → I thereexists a morphism ˜ ϕ : F → I making the following diagram commutative / / F ι / / ϕ (cid:15) (cid:15) E ˜ ϕ (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) F Definition 2.3.
Let E be a G -module. A strengthened, relatively injective resolution of E isan exact complex −→ E ι −→ I d −→ I d −→ I d −→ · · · such that each I n is a relatively injective G -module and ι and d n are strengthened morphisms.If ( I n , d n ) is a relatively injective strengthened resolution of a G -module E , then the coho-mology , H n ( G, E ) , and reduced cohomology of G , H n ( G, E ) , with coefficients in E is definedas that of the complex −→ ( I ) G d ↾ −→ ( I ) G d ↾ −→ ( I ) G d ↾ −→ · · · obtained by taking G -invariants; i.e. H n ( G, E ) = ker( d n ↾ )im( d n − ↾ ) and H n ( G, E ) = ker( d n ↾ )cl(im( d n − ↾ )) , where cl( − ) denotes closure in the topological vector space I n . We will denote by H ∗ ( G, E ) and H ∗ ( G, E ) the direct sums ⊕ n > H n ( G, E ) and ⊕ n > H n ( G, E ) , respectively. A standard ar-gument [Gui80, Corollaire 3.1.1] shows that H n ( G, E ) is independent of the choice of strength-ened, relative injective resolution up to isomorphism of (not necessarily Hausdorff!) topological THOMAS GOTFREDSEN AND DAVID KYED vector spaces and hence also H n ( G, E ) ≃ H n ( G, E ) / cl( { } ) is independent of the choice ofresolution. For the cohomology to be well defined one of course needs to know that strength-ened, relatively injective resolutions exist, but in [Gui80, III, Proposition 1.2] it is shown thatthe complex −→ E ι −→ C ( G, E ) d −→ C ( G , E ) d −→ C ( G , E ) d −→ · · · with d n ( f )( g , . . . , g n +1 ) = n +1 X i =0 ( − i f ( g , . . . , ˆ g i , . . . , g n ) (1)constitutes such a resolution. Here the “hat” denotes omission and C ( G n , E ) denotes the spaceof continuous functions from G n to E endowed with G -action given by ( g.f )( g , . . . , g n ) := g ( f ( g − g , . . . , g − g n )) . (2)When E is a Fréchet space and X is a σ -compact, locally compact, topological space whoseBorel σ -algebra is endowed with a measure µ , then one defines, for p ∈ N , L p loc ( X, E ) as thosemeasurable functions f : X → E such that Z K q ( f ( x )) p d µ ( x ) < ∞ , for every compact set K ⊂ X and every continuous seminorm q on E , and L p loc ( X, E ) is thendefined by identifying functions in L p loc ( X, E ) that are equal µ -almost everywhere. This too isa Fréchet space when endowed with the topology defined by the family of seminorms Q K,q ( f ) := p sZ K q ( f ( x )) p d µ ( x ) , where K runs through the compact subsets of X and q runs through the continuous seminormson E . As lcsc groups are σ -compact this construction applies to G and its powers and it wasshown in [Bla79, Corollaire 3.5] that the complex −→ E ι −→ L p loc ( G, E ) d −→ L p loc ( G , E ) d −→ L p loc ( G , E ) d −→ · · · , (3)with G -action and coboundary maps given by the natural analogues of (2) and (1), is also astrengthened, relatively injective resolution.2.1.1. Cohomological properties.
Recall that a lcsc group G is said to have Shalom’s property H T if for any unitary Hilbert G -module H (i.e. H is a Hilbert space endowed with the structureof a continuous G -module and each element in G acts as a unitary operator) the inclusion H G ֒ → H induces an isomorphism H n ( G, H G ) ≃ H n ( G, H ) for each n > . Note that it followsby general disintegration theory and [Bla79, Theorem 10.1] that csc nilpotent Lie groups haveproperty H T , and from this and Mal’cev theory [Mal51] it also follows that torsion free, finitelygenerated nilpotent groups have property H T ; see [Sha04, Theorem 4.1.3] for more details.The Betti numbers of G are defined as β n ( G ) := dim R H n ( G, R ) and we remark that these arefinite whenever G is a csc nilpotent Lie group or a torsion free, discrete nilpotent group. For thelatter, note that the classifying space of such groups are finite CW-complexes and the groupcohomology agrees with the cohomology of the classifying space, and for the former can be seenfor instance by passing to Lie algebra cohomology via the van Est theorem [vE55]. Moreover,when β n ( G ) < ∞ , then H n ( G, R ) is automatically Hausdorff [Gui80, III, Proposition 2.4] andhence there is no difference between reduced and ordinary cohomology. OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 5
Cup products.
For a discrete group Γ , it is well-known that its cohomology with realcoefficients H ∗ (Γ , R ) = ⊕ n > H n (Γ , R ) becomes a unital, graded commutative ring for theso-called cup product, and the same construction also works for lcsc groups, but since thisdoes not seem properly documented in the existing literature we recall the construction below.If G is lcsc groups and ξ ∈ C ( G n +1 , R ) and η ∈ C ( G m +1 , R ) , one defines their cup product ξ ⌣ η ∈ C ( G n + m +1 , R ) as ( ξ ⌣ η )( g , . . . , g n + m ) := ξ ( g , . . . , g n ) η ( g n , . . . , g n + m ) . As the cup product commutes with the G -action, in the sense that ( g.ξ ) ⌣ ( g.η ) = g. ( ξ ⌣ η ) ,it descends to a map ⌣ : C ( G n +1 , R ) G × C ( G m +1 , R ) G −→ C ( G n + m +1 )( G, R ) G . Moreover, the standard differentials satisfy a graded Leibniz rule, i.e. d n + m ( ξ ⌣ η ) = d n ( ξ ) ⌣ η + ( − n ξ ⌣ d m ( η ) , (4)from which it it follows that the cup product passes down to the level of cocycles and thatthe set of coboundaries is an ideal, so that the cup product descends to a map ⌣ : H n ( G, R ) × H m ( G, R ) −→ H n + m ( G, R ) , which turns H ∗ ( G, R ) into a unital, graded commutative ring.We will encounter more elaborate cup products in Section 4 satisfying natural analoguesof (4), so to make the argument easily available we now give the short proof of (4), in orderto leave the details later to the reader in good conscience. To prove (4), let cocycles ξ ∈ C ( G n +1 , R ) , η ∈ C ( G m +1 , R ) be given and note that d n + m ( ξ ⌣ η )( g . . . g n + m +1 ) = n + m +1 X i =0 ( − i ( ξ ⌣ η )( g . . . ˆ g i . . . g n + m +1 ) == n X i =0 ( − i ξ ( g . . . ˆ g i . . . g n +1 ) η ( g n +1 . . . g n + m +1 ) + n + m +1 X i = n +1 ( − i ξ ( g . . . g n ) η ( g n . . . ˆ g i . . . g n + m +1 )= n X i =0 ( − i ξ ( g . . . ˆ g i . . . g n +1 ) η ( g n +1 . . . g n + m +1 ) + m +1 X i =1 ( − i + n ξ ( g . . . g n ) η ( g n . . . ˆ g n + i . . . g n + m +1 ) On the other hand ( d n ( ξ ) ⌣ η + ( − n ξ ⌣ d m ( η )) ( g . . . g n + m +1 ) = = n +1 X i =0 ( − i ξ ( g . . . ˆ g i . . . g n +1 ) η ( g n +1 . . . g n + m +1 ) + ( − n m +1 X i =0 ( − i ξ ( g . . . g n ) η ( g n . . . ˆ g n + i . . . g n + m +1 )= n X i =0 ( − i ξ ( g . . . ˆ g i . . . g n +1 ) η ( g n +1 . . . g n + m +1 ) + ( − n m +1 X i =1 ( − i ξ ( g . . . g n ) η ( g n . . . ˆ g n + i . . . g n + m +1 ) , where the last equality follows since the last summand in the frist sum cancels with the firstsummand in the second; this proves (4).A direct computation shows that the cup product is continuous with respect to the topologyof uniform convergence on compacts at the level of cochains, and since we just saw that the THOMAS GOTFREDSEN AND DAVID KYED coboundaries constitute an ideal, the same is true for the their closure. Thus, the cup productalso descends to a product on H ∗ ( G, R ) .2.2. Measure equivalence.
In this section we review the necessary theory concerning (uni-form) measure equivalence for locally compact groups; we refer to [BFS13] and [KKR17] formore details on the involved notions.
Definition 2.4 ([BFS13]) . Two unimodular lcsc groups G and H with Haar measures λ G and λ H are said to be measure equivalent if there exist a standard Borel measure G × H -space (Ω , η ) and two standard Borel measure spaces ( X, µ ) and ( Y, ν ) such that:(i) both µ and ν are finite measures and η is non-zero;(ii) there exists an isomorphism of measure G -spaces i : ( G × Y, λ G × ν ) −→ (Ω , η ) , where Ω is considered a measure G -space for the restricted action and G × Y is considered ameasure G -space for the action g. ( g ′ , y ) = ( gg ′ , y ) ;(iii) there exists an isomorphism of measure H -spaces j : ( H × X, λ H × µ ) → (Ω , η ) , where Ω is considered a measure H -space for the restricted action and H × X is considereda measure H -space for the action h. ( h ′ , x ) = ( hh ′ , x ) .A standard Borel space (Ω , η ) with these properties is called a measure equivalence cou-pling between G and H , and whenever needed we will specify the additional data by writing (Ω , η, X, µ, Y, ν, i, j ) .Any measure equivalence coupling gives rise to measure preserving actions G y ( X, µ ) and H y ( Y, ν ) as well as two -cocycles ω G : H × Y → G and ω H : G × X → H . These aredefined almost everywhere by the relations i ( gω G ( h, y ) − , h.y ) = h.i ( g, y ) , for almost all g ∈ G.j ( hω H ( g, x ) − , g.x ) = g.j ( h, x ) , for almost all h ∈ H. In the definition of the actions and cocycles above, we are paying little attention to the measuretheoretical subtleties, but the diligent reader may find these worked out in detail in [KKR17,Section 2]. Note that it was was also proven in [KKR17] that one can always obtain a strict measure equivalence coupling; i.e. one in which the maps i and j are Borel isomorphisms andglobally equivariant. Definition 2.5 ([KKR17]) . A strict measure equivalence coupling (Ω , η, X, µ, Y, ν, i, j ) be-tween unimodular, lcsc groups G and H is said to be uniform if(i) for every compact C ⊂ G there exists a compact D ⊂ H such that j − ◦ i ( C × Y ) ⊂ D × X ;(ii) for every compact D ⊂ H there exists a compact C ⊂ G such that i − ◦ j ( D × X ) ⊂ C × Y .In this case G and H are said to be uniformly measure equivalent (UME), and the properties(i) and (ii) are referred to as the cocycles being locally bounded .As mentioned already, measure equivalence was introduced by Gromov as a measure theo-retic analogue of quasi-isometry, and although neither property in general implies the other,it was proven in [KKR17, Proposition 6.13] that for compactly generated , unimodular, lcscgroups, uniform measure equivalence always implies quasi-isometry, and that the converse Recall that for compactly generated groups coarse equivalence coincides quasi-isometry
OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 7 holds under the additional assumption of amenability [KKR17, Theorem 6.15]. This gener-alises earlier results for discrete groups by Shalom [Sha04] and Sauer [Sau06].If (Ω , η, X, µ, Y, ν, i, j ) is a strict UME coupling between G and H and E is a Fréchet spacethen we define L p loc (Ω , E ) as those (equivalence classes modulo equality η -almost everywhere)of measurable functions f : Ω → E such that for every C ⊂ G compact and every continuousseminorm q on E , one has Z C × Y q ( f ◦ i ( g, y )) d λ G ( g ) d ν ( y ) < ∞ , We topologise L (Ω , E ) with via the seminorms q C ( f ) := qR C × Y q ( f ◦ i ( g, y )) d λ G ( g ) d ν ( y ) and endow it with G × H - action (( g, h ) .f )( t ) := ( g, h ) .f (( g, h ) − .t ) , t ∈ Ω , ( g, h ) ∈ G × H. (5)We could of course also have defined L p loc (Ω , E ) using the map j instead of i , but since Ω isuniform this gives rise to exactly the same space. Lemma 2.6.
With the action defined by (5) the space L (Ω , E ) becomes a Fréchet G × H -module.Proof. Since Y is standard Borel we may equip it with a compact metrizable topology gen-erating the σ -algebra [Kec95, Theorem 15.6], and the space L (Ω , E ) then directly identifieswith L ( G × Y, E ) which is Frechet by [Gui80, D.2]. It suffices to show that both groupsact continuously, and by symmetry it is enough to treat the G -action. To this end, note that[Gui80, D.2.2 (vii)] gives an isomorphism of G -modules L (Ω , E ) ≃ L ( G, L ( Y, E )) , where the action on the right hand side is given by ( g.ξ )( g ′ )( y ) = g. ( ξ ( g − g ′ )( y )) , so by [Gui80,D.3.2] it suffices to show that the pointwise G -action L ( Y, E ) is continuous. To see this, notethat C ( Y, E ) is dense in L ( Y, E ) , so by [Gui80, Lemme D.8 (ii)], it is enough to show thatthe action is equicontinuous over compact sets and pointwise continuous on elements from C ( Y, E ) . To see the former, let K ⊂ G be compact and q be a continuous seminorm on E .Then since G y E is continuous, there exists a continuous seminorm q ′ on E such that for all g ∈ K and x ∈ E we have q ( g.x ) q ′ ( x ) . Hence, for g ∈ K and ξ ∈ L ( Y, E ) we also havethat Z Y q (( g.ξ )( y )) d ν ( y ) = Z Y q ( g.ξ ( y )) d ν ( y ) Z Y q ′ ( ξ ( y )) d ν ( y ) showing equicontinuity over compact sets. To see that the action is pointwise continuous on ξ ∈ C ( Y, E ) , simply note that if g n → g in G and q is a continuous seminorm on E then Z Y ( q ( g n .ξ − gξ )( y )) d ν ( y ) = Z Y ( q ( g n .ξ ( y ) − g.ξ ( y )) d ν ( y ) ν ( Y ) sup x ∈ ξ ( Y ) q ( g n .x − g.x ) −→ n →∞ , where the convergence follows from compactness of ξ ( Y ) ⊂ E and [Gui80, Lemme D.8 (iii)]. (cid:3) THOMAS GOTFREDSEN AND DAVID KYED Proof of Theorem A
The aim of the current section is to prove Theorem A, so we fix uniformly measureequivalent, unimodular lcsc groups G and H and a strict, uniform measure equivalence (Ω , η, Xµ, Y, ν, i, j ) between them, as well as Fréchet G × H -module E . Consider the space L ( G n , L (Ω , E )) endowed with G × H -action (( g, h ) .ξ )( g , . . . , g n )( t ) := ( g, h )[ ξ ( g − g , . . . , g − g n )(( g, h ) − .t )] , for ( g, h ) ∈ G × H, g , . . . , g n ∈ G, t ∈ Ω . Lemma 3.1.
The space L ( G n , L (Ω , E )) is a relatively injective G × H -module and thecomplex R G defined as −→ L (Ω , E ) −→ L (cid:0) G, L (Ω , E ) (cid:1) −→ L (cid:0) G , L (Ω , E ) (cid:1) −→ · · · (6) constitutes a strengthened, relatively injective resolution of the G × H -module L (Ω , E ) whenendowed with the standard homogeneous differentials, given by the obvious modification of theformula (1) .Proof. Throughout the proof we identify Ω with H × X through the map j defining the L -structure on Ω ; recall that under this identification the G × H -action on L (cid:0) G n +1 , L (Ω , E ) (cid:1) = L (cid:0) G n +1 , L ( H × X, E ) (cid:1) is given by the formula (( g, h ) .ξ )( g , . . . , g n )( h ′ , x ) = ( g, h ) . (cid:2) ξ (cid:0) g − g , . . . , g − g n (cid:1) ( h − h ′ ω H ( g − , x ) − , g − .x ) (cid:3) . Since X is standard Borel, [Kec95, Theorem 15.6] ensures that we can find a compact,metrizable topology on it whose open sets generate the Borel structure, and we may thereforeconsider the Frechet space L ( G n × X, E ) .Arguing as in [Gui80, n ◦ D.3.2.], we obtain that this is a Fréchet G × H -module whenendowed with the G × H -action (( g, h ) .ξ )( g , . . . , g n , x ) := ( g, h ) .ξ ( g − g , . . . , g − g n , g − .x ) . By [Bla79, Théorème 3.4], we therefore have that L (cid:0) G × H, L ( G n × X, E ) (cid:1) , with the standard G × H -action, is a relatively injective Fréchet G × H -module. A routinecalculation now shows that the map α : L (cid:0) G n +1 , L ( H × X, E ) (cid:1) −→ L ( G × H, L ( G n × X, E )) α ( ξ )( g, h )( g , . . . , g n ) := ξ ( g , . . . , g n , g )( hω H ( g − , x ) , x ) , is an isomorphism of Fréchet spaces intertwining the G × H -actions, and hence it follows also L (cid:0) G n +1 , L (Ω , E ) (cid:1) is relatively injective, as desired.Since the complex R G is simply the standard L -resolution of L (Ω , E ) considered only asa G -module, it is clearly strengthened, and hence the proof is complete. (cid:3) Proof of Theorem A.
Since the roles of G and H are symmetric in the module structure on L (Ω , E ) one may construct a strengthened, relatively injective resolution R H of L (Ω , E ) analogous to (6), whose degree n term is given by L ( H n +1 , L (Ω , E )) . Thus, both R G OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 9 and R H compute H n ( G × H, L (Ω , E )) after passing to G × H -invariants and cohomology.However, L (cid:0) G n +1 , L (Ω , E ) (cid:1) G × H = (cid:0) L ( G n +1 , L (Ω , E )) H (cid:1) G = (cid:0) L (cid:0) G n +1 , L (Ω , E ) H (cid:1)(cid:1) G , where the last equality is due to the fact that H acts trivially in the G n +1 -direction. From thiswe see that passing to G × H -invariants in R G is exactly the same as passing to G -invariantsin the L -resolution (3) of the G -module L (Ω , E ) H , and hence we obtain a topologicalisomorphism H n (cid:0) G, L (Ω , E ) H (cid:1) ≃ H n ( G × H, L (Ω , E )) . Replacing R G with R H , a symmetric argument yields that H n (cid:0) H, L (Ω , E ) G (cid:1) ≃ H n ( G × H, L (Ω , E )) , and the desired isomorphism follows.If E = R with trivial G × H -action, then by[Gui80, D.2.2 (vii)& Lemme D.9] we have anisomorphism of H -modules L (Ω , R ) G ≃ L ( G × Y, R ) G ≃ L ( G, L ( Y )) G ≃ L ( Y ) , and, similarly, an isomorphism of G -modules L (Ω , E ) H ≃ L ( X ) ; hence the last part of thestatement follows from the first. (cid:3) Remark 3.2.
Theorem A is stated for real Fréchet spaces and real cohomology, but as seenfrom the proof just given, the analogous statement over the complex numbers also holds truewith verbatim the same proof.
Remark 3.3.
The proof just given does not, directly, provide a concrete map realizing theisomorphism H n (cid:0) G, L (Ω , E ) H (cid:1) ≃ H n (cid:0) H, L (Ω , E ) G (cid:1) , but for applications, e.g. our Theo-rem B, having a concrete map is very useful, and we shall therefore now describe one such amap. By general relative homological algebra [Gui80, III, Corollaire 1.1] , any morphism χ ofcomplexes of G × H -modules from R G to R H which lifts the identity on L (Ω , E ) will inducea (topological) isomorphism H ∗ (cid:0) R G × HG , d n ↾ (cid:1) ≃ H ∗ (cid:0) R G × HH , d n ↾ (cid:1) . We now define such a map χ n : L (cid:0) G n +1 , L (Ω , E ) (cid:1) −→ L (cid:0) H n +1 , L (Ω , E ) (cid:1) by setting χ n ( ξ )( h , . . . , h n )( t ) := ξ (cid:0) π G ◦ i − ( h − .t ) , . . . , π G ◦ i − ( h − n .t ) (cid:1) ( t ) , where π G : G × Y → G denotes the projection onto the first factor. Identifying Ω with G × Y via i the map takes the form χ n ( ξ )( h , . . . , h n )( g, y ) = ξ (cid:0) ω G ( h , y ) − , . . . , ω G ( h n , y ) − (cid:1) ( g, y ) . Note that the map χ n does indeed take values in L ( H n +1 , L (Ω , E )) since Ω is uniformsuch that the cocycles are locally bounded, and from this it also follows that χ n is continuous.It is straight forward to see that χ is a chain map lifting the identity on L (Ω , E ) and fromthe cocycle identity it follows that χ n is G × H -equivariant. Proof of Theorem B
In this section we prove Theorem B. We thus assume that G and H are uniformly measureequivalent, unimodular, lcsc groups satisfying property H T whose Betti numbers are finite inall degrees, and fix a strict, uniform measure equivalence coupling (Ω , η, X, µ, Y, ν, i, j ) . By[KKR17, Proposition 2.13], and its proof, one may change the measures η, µ and ν into onesthat are ergodic for the G × H -, G - and H -actions, respectively, so we may, and shall, assumethat the original measures are ergodic. Furthermore, by rescaling the measures involved, wemay also assume that ν is a probability measure. We therefore obtain isomorphisms H n ( G, R ) = H n (cid:0) G, L ( X ) G (cid:1) (ergodicity) ≃ H n (cid:0) G, L ( X ) (cid:1) (property H T ) ≃ H n (cid:0) G, L (Ω) H (cid:1) ≃ H n (cid:0) H, L (Ω) G (cid:1) (Theorem A) ≃ H n (cid:0) H, L ( Y )) (cid:1) ≃ H n ( H, R )) (ergodicity and H T )Chasing through the isomorphisms above, using the explicit isomorphism χ n from Remark3.3, shows that κ n : L ( G n +1 , R ) −→ L ( H n +1 , R ) , given by κ n ( ξ )( h , . . . , h n ) = R Y ξ (cid:0) ω G ( h − , y ) − , . . . , ω G ( h − n , y ) − (cid:1) d ν ( y ) at the cochain levelinduces the isomorphism H n ( G, R ) ≃ H n ( H, R ) and the aim is now to prove that this isomor-phism preserves cup products. To this end, we need an auxiliary complex defined as follows: Definition 4.1.
Denote by D n the subspace of L ( H n +1 , L ( Y )) consisting of the classes(modulo equality almost everywhere) of functions ξ ∈ L ( H n +1 , L ( Y )) such that:(i) for almost all ( h , . . . , h n ) ∈ H n +1 : ξ ( h , . . . , h n ) ∈ L ∞ ( Y ) ⊂ L ( Y ) , and(ii) for all C ⊂ H n +1 compact: ess sup {k ξ ( h , . . . , h n ) k ∞ : ( h , . . . , h n ) ∈ C } < ∞ . Since L ∞ ( Y ) ⊂ L ( Y ) is an H -invariant subspace, D n becomes a (non-complete) H -submodule of L ( H n +1 , L ( Y )) . Moreover, it is easily seen that d n ( D n ) ⊆ D n +1 and hence D : ( D ) H d ↾ −→ ( D ) H d ↾ −→ ( D ) H d ↾ −→ · · · is a subcomplex of the standard L -complex computing H ∗ ( H, L ( Y )) . Definition 4.2.
For α ∈ D n , β ∈ D m and ξ ∈ L ( H n +1 , L ( Y )) we define α ⌣ ξ, ξ ⌣ α ∈ L ( H m + n +1 , L ( Y )) and α ⌣ β ∈ D n + m by α ⌣ ξ ( h , . . . , h n + m ) := α ( h , . . . , h n ) ξ ( h n , . . . , h n + m ) (7) ξ ⌣ α ( h , . . . , h n + m ) := ξ ( h , . . . , h m ) α ( h m , . . . , h n + m ) (8) α ⌣ β ( h , . . . , h n + m ) := α ( h , . . . , h n ) β ( h n , . . . , h n + m ) (9)where the products on the right hand side are the pointwise products between functions in L ∞ ( Y ) and L ( Y ) . OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 11
Lemma 4.3.
The product defined by (9) descends to a product on H ∗ ( D ) turning it into aunital ring. Similarly, the products (7) and (8) descend to the level of cohomology and reducedcohomology turning H ∗ ( H, L ( Y )) and H ∗ ( H, L ( Y )) into bimodules for H ∗ ( D ) .Proof. For f ∈ L ∞ ( Y ) , f ∈ L ( Y ) and h ∈ H one has h. ( f · f ) = ( h.f ) · ( h.f ) and h. ( f · f ) = ( h.f ) · ( h.f ) (the products being defined pointwise) and from this it follows that the cup product of H -invariant elements is again H -invariant, so that ⌣ restricts to the level of fixed points for H .Moreover, by repeating the proof of (4) mutatis mutandis, the cup products (7), (8) and (9),are seen to satisfy the obvious versions of the Leibniz rule, from which it follows that H ∗ ( D ) becomes a (graded) ring for which H ∗ ( H, L ( Y )) is a (graded) bimodule. We will omit thedetails of the latter, and instead show that H ∗ ( H, L ( Y )) becomes a H ∗ ( D ) -bimodule, sincethis proof contains the other as a special case. We need to show that if α, α ′ ∈ D n ∩ ker( d n ) and ξ, ξ ′ ∈ L ( H m +1 , L ( Y )) ∩ ker( d m ) satisfy that α − α ′ ∈ d n − ( D n − ) and ξ − ξ ′ ∈ cl (cid:0) d m − ( L ( H m , L ( Y )) (cid:1) then α ⌣ ξ − α ′ ⌣ ξ ′ ∈ cl (cid:0) d n + m − ( L ( H n + m , L ( Y )) (cid:1) . Write α − α ′ = d n − ( β ) and note that, by the Leibniz rule, we have α ⌣ ξ − α ′ ⌣ ξ ′ = ( α − α ′ ) ⌣ ξ − α ′ ⌣ ( ξ − ξ ′ )= d n − ( β ) ⌣ ξ − α ′ ⌣ ( ξ − ξ ′ )= d n + m − ( β ⌣ ξ ) − ( − n − β ⌣ d m ( ξ ) − α ′ ⌣ ( ξ − ξ ′ )= d n + m − ( β ⌣ ξ ) − α ′ ⌣ ( ξ − ξ ′ ) By assumption, there exist η k ∈ L ( H m , L ( Y )) such that lim k d m − ( η k ) = ξ − ξ ′ , so tofinish the proof it suffices to show that the cup product is pointwise continuous in the secondvariable; i.e. that if ζ k → k in L ( H m +1 , L ( Y )) then α ⌣ ζ k → k in L ( H m + n +1 , L ( Y )) .To see this, let K ⊂ H n + m +1 be given. Upon passing to a bigger compact set, we may assumethat K = Q n + mi =0 K i for some compact subsets K i ⊂ H and we now get Z K k α ⌣ ζ k k d λ ⊗ ( n + m +1) H = Z Q n + mi =0 K i k α ( h , . . . , h n ) ζ k ( h n , . . . , h n + m ) k d h · · · d h n + m sup ( h ,...,h n ) ∈ Q n − i =0 K i k α ( h , . . . , h n ) k ∞ n Y i =0 λ H ( K i ) Z Q n + mi = n K i k ζ k ( h n , . . . , h n + m ) k d h n · · · d h n + m . By definition of D n we have sup ( h ,...,h n ) ∈ Q ni =0 K i k α ( h , . . . , h n ) k ∞ < ∞ , and by assumption Z Q n + mi = n K i k ζ k ( h n , . . . , h n + m ) k d h n · · · d h n + m → k . (cid:3) Remark 4.4.
Using the usual contracting homotopy (cf. [Gui80, III, Proposition 1.4]) for theresolution ( L ( H n +1 , L ( Y )) , d n ) n ∈ N , one easily checks that ( D n , d n ) n ∈ N is a strengthenedresolution of the H -submodule L ∞ ( Y ) ⊂ L ( Y ) , but it seems less clear whether the modules D n are relatively injective or not. If this were the case, then H ∗ ( D ) would of course be nothingbut H ∗ ( H, L ∞ ( Y )) , where L ∞ ( Y ) is considered as an H -module with respect to the -norm. With the lemmas above at our disposal, we can now prove Theorem B following the strat-egy in [Sau06] almost verbatim. Recall that we have fixed a strict, ergodic UME coupling (Ω , η, X, µ, Y, ν, i, j ) and normalized the measures so that ν has total mass 1. Proof of Theorem B.
Denote by L ( Y ) the functions in L ( Y ) that integrate to zero. TheHilbert H -module L ( Y ) then splits as an (orthogonal) direct sum L ( Y ) = L ( Y ) ⊕ R , viathe map ξ (cid:0) ξ − ( R Y ξ d ν )1 Y , R Y ξ d ν (cid:1) . From this it follows that we get a decomposition of H -complexes Φ : L ( H n +1 , L ( Y )) ≃ L ( H n +1 , L ( Y )) ⊕ L ( H n +1 , R ) , by simply composing a function L ( H n +1 , L ( Y )) with the two projections p : L ( Y ) → L ( Y ) and p : L ( Y ) → R . Similarly, denote by D n the subcomplex of D n consisting ofthe (classes of) those functions who integrate to zero almost everywhere and by D n R the(classes of functions) that, almost everywhere, are are almost everywhere constant on Y ,and note that the decomposition Φ maps D n onto D n ⊕ D n R . To be extremely precise, D n consists of classes of those ξ ∈ L ( H n +1 , L ( Y )) such that [ ξ ] ∈ D n and such that foralmost all h , . . . , h n we have R Y ξ ( h , . . . , h n ) dν = 0 and D n R are the classes of those functions ξ ∈ L ( H n +1 , L ( Y )) such that [ ξ ] ∈ D n and such that for almost all h , . . . , h n we have ξ ( h , . . . , h n ) = ( R Y ξ ( h , . . . , h n ) dν )1 Y (the latter equation in L ∞ ( Y ) ⊂ L ( Y ) We thereforeobtain splittings at the level of cohomology H n ( D ) ≃ H n ( D ) ⊕ H n ( D R )H n ( H, L ( Y )) ≃ H n ( H, L ( Y )) ⊕ H n ( H, R )H n ( H, L ( Y )) ≃ H n ( H, L ( Y )) ⊕ H n ( H, R ) respecting the natural downward maps induced by the corresponding inclusions at the levelof complexes.As shown in the first paragraph of the present section, we have a continuous linear map I ∗ : H ∗ ( G, R ) −→ H ∗ ( H, L ( Y )) , which descends to an isomorphism after the passage to reduced cohomology, and maps the(class of a) continuous n -cocycle ξ ∈ C ( G n +1 , R ) G ∩ ker( d n ) to the (class of) the cocycle I ( ξ ) ∈ L ( H n +1 , L ( Y )) given by I ( ξ )( h , . . . , h n )( y ) = ξ (cid:0) ω G ( h − , y ) − , . . . , ω G ( h − n , y ) − (cid:1) . Moreover, since Ω is uniform, the the cocycle ω G is locally bounded and from this it followsthat I ( ξ ) ∈ D n , where D n is the submodule of L ( H n +1 , L ( Y )) described in Definition 4.1.Hence, I ∗ factorises as H ∗ ( G, R ) I ∗ −→ H ∗ ( D ) ι ∗ −→ H ∗ ( H, L ( Y )) , where ι is the map induced by the inclusion D n ⊂ L ( H n +1 , L ( Y )) . Denoting by ι ∗ the map obtained by composing ι ∗ : H ∗ ( D ) −→ H ∗ ( H, L ( Y )) with the natural projection H ∗ ( H, L ( Y )) → H ∗ ( H, L ( Y )) , we obtain a commutative diagram as follows: H ∗ ( G, R ) I ∗ / / H ∗ ( D ) p ∗ ( ( ι ∗ / / H ∗ ( H, L ( Y )) p ∗ ∼ / / H ∗ ( H, R ) OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 13
Here p ∗ is the map induced at the level of reduced cohomology of the map L ( Y ) → R given byintegration against ν and p ∗ is defined as p ∗ ◦ ι ∗ . Now, I ∗ is multiplicative, since already at thelevel of cochains we have I ( ξ ⌣ η ) = I ( ξ ) ⌣ I ( η ) for ξ ∈ C ( G n +1 , R ) and η ∈ C ( G m +1 , R ) ,which is seen by a direct computation. So to see that κ ∗ := p ∗ ◦ ι ∗ ◦ I ∗ is an isomorphism ofgraded rings, it suffices to show that p ∗ = p ∗ ◦ ι ∗ is multiplicative. To this end, first notice themap ι ∗ is a bimodule map with respect to the H ∗ ( D ) -bimodule structure on H ∗ ( D ) given byleft/right multiplication and the the H ∗ ( D ) -bimodule structure on H n ( H, L ( Y )) described inLemma 4.3, and its kernel is therefore a two-sided ideal. Moreover, since p ∗ is an isomorphism,we have ker( ι ∗ ) = ker( p ∗ ) , and from this we now obtain that p ∗ is multiplicative as follows:given [ ξ ] ∈ H n ( D ) and [ η ] ∈ H m ( D ) , they decompose as [ ξ ] = [ ξ ] + [ ξ ] and [ η ] = [ η ] + [ η ] for cocycles ξ ∈ D n , ξ ∈ D n R , η ∈ D m and η ∈ D m R . By definition, ι ∗ ([ ξ ]) = ι ∗ ([ η ]) = 0 and since ker( ι ∗ ) is an ideal we obtain ι ∗ ([ ξ ] ⌣ [ η ]) = ι ∗ ([ ξ ] ⌣ [ η ] + [ ξ ] ⌣ [ η ] + [ ξ ] ⌣ [ η ] + [ ξ ] ⌣ [ η ]) = ι ∗ ([ ξ ] ⌣ [ η ]) Since ξ and η takes values in essentially constant functions on Y we obtain p ∗ ◦ ι ∗ ([ ξ ] ⌣ [ η ]) ( h , . . . , h n + m +1 ) = Z Y ( ξ ⌣ η )( h , . . . , h n + m +1 ) d ν = Z Y ξ ( h , . . . , h n ) η ( h n , . . . , h n + m +1 ) d ν = Z Y ξ ( h , . . . , h n ) d ν Z Y η ( h n , . . . , h n + m +1 ) d ν = p ∗ ◦ ι ∗ ([ ξ ]) ⌣ p ∗ ◦ ι ∗ [ η ]) ( h , . . . , h n + m +1 ) The composition p ∗ ◦ I ∗ : H ∗ ( G, R ) → H ∗ ( H, R ) is therefore multiplicative and since we alreadyargued, in the beginning of this section, that this map descends to an isomorphism H ∗ ( G, R ) ≃ H ∗ ( H, R ) it follows that it too is multiplicative. (cid:3) Remark 4.5.
Of course one could also define a cup product on H ∗ ( G, C ) and the argumentsabove obviously generalise to this setting so that the analogue of Theorem B over the complexnumbers holds true as well. 5. An application
It is a well known open problem to classify connected simply connected (csc) nilpotentLie groups up to quasi-isometry, and conjecturally quasi-isometry actually coincides withisomorphism on this class of groups [Cor18]. Although isomorphism classification is wideopen in general, csc nilpotent Lie groups of dimension at most 7 are completely classified (upto isomorphism) by means of the corresponding classification of nilpotent real Lie algebras.This classification is the work by many hands, and we will here focus on Gong’s thesis [Gon98]which provides the first complete classification of all 7-dimensional, real, nilpotent Lie algebras.Regarding the quasi-isometry problem mentioned above, there are basically two main resultssupporting the conjecture. Firstly, by Pansu’s celebrated paper [Pan89], if G and H are quasi-isometric csc nilpotent Lie groups with Lie algebras g and h , respectively, then their associatedCarnot Lie algebras Car ( g ) and Car ( g ) are isomorphic. Recall that if g is a step c nilpotentLie algebra with lower central series g = g [1] > g [2] > . . . > g [ c ] > { } , then its Carnot algebra is defined by
Car ( g ) := c M i =1 g i / g i +1 , with Lie bracket given, for ¯ ξ ∈ g i / g i +1 and ¯ η ∈ g j / g j +1 , by [ ¯ ξ, ¯ η ] := [ ξ, η ] ∈ g i + j / g i + j +1 .Secondly, if G and H both contain lattices, i.e. admit rational structures [Mal51], then by[Sau06, Theorem 5.1] and [Nom54, Theorem 1] one may conclude that H ∗ ( G, R ) and H ∗ ( H, R ) are isomorphic as graded commutative rings, and this also prevents many low dimensional cscnilpotent Lie groups from being quasi-isometric; see [Cor18] for the state of the art in thisdirection. Our Theorem A and Theorem B generalise the latter result (see Section 2.1.1 forthis) and we now argue how this can be used to distinguish new csc nilpotent Lie groupsup to quasi-isometry. To this end, recall first that for a csc nilpotent Lie group G withLie algebra g , the van Est theorem [vE55] provides an isomorphism of graded commutativerings H ∗ ( G, R ) ≃ H ∗ ( g , R ) , where the latter denotes the Lie algebra cohomology og g (seee.g. [Gui80, Chapter II]). Secondly, recall that Gong’s list [Gon98] contains nine 1-parameterfamilies of real Lie algebras, and we will here focus on two of these families. We will use thesame labeling scheme as in [Gon98]; in particular, { x , . . . , x } will always denote the basiselements of a given Lie algebra, and h· · · i is used to denote the R -linear span of vectors. Asis customary, all non-specified brackets between basis elements are implicitly set to zero. The family 1357M has the following bracket relations for λ = 0 : [ x , x ] = x , [ x , x i ] = x i +2 , i = 3 , , , [ x , x ] = x , [ x , x ] = λx , [ x , x ] = (1 − λ ) x . The family has lower central series h x , x , x , x , x , x , x i > h x , x , x , x i > h x , x i > h x i , and Carnot algebra given by the relations [¯ x , ¯ x ] = ¯ x , [¯ x , ¯ x i ] = ¯ x i +1 , i = 3 , , . The family 1357N has the relations (for λ ∈ R )[ x , x ] = x , [ x , x i ] = x i +2 , i = 3 , , , [ x , x ] = λx , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x . The family has lower central series h x , x , x , x , x , x , x i > h x , x , x , x i > h x , x i > h x i , and Carnot algebra given by the relations [¯ x , ¯ x ] = ¯ x , [¯ x , ¯ x i ] = ¯ x i +1 , i = 3 , , . Thus 1357M and 1357N have the same Carnot Lie algebra for all valid values of λ , and thusthese families cannot be separated into quasi-isometry classes by Pansu’s result [Pan89]. Usingthe computer algebra system Maple (or an equivalent type of software), one can quite easilycompute the real cohomology groups of a given Lie algebra, and doing so one obtains that1357M has Betti numbers (1,3,6,8,8,6,3,1) and that 1357N has Betti numbers (1,3,5,7,7,5,3,1)for all but finitely many λ ∈ R . Thus, up to excluding finitely many values of λ , no memberof 1357M can have isomorphic real cohomology to that of any member of 1357N, and it followsthat the corresponding csc nilpotent Lie groups are not quasi isometric by Theorem A. Sinceat most a countable number of members of each family have an associated csc Lie groupwhich admits a lattice, this illustrates an instance where Theorem A can tell apart (up to the exclusion of finitely many values stems from the Gaussian elimination performed by Maple in whichcertain polynomials in λ show up as denominators. OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 15 quasi-isometry) Lie groups that are not distinguished by neither the results of Pansu, nor bythose of Sauer and Shalom [Sau06, Sha04].
Remark 5.1.
One could naively wonder if the real cohomology ring is a complete quasi-isometry invariant within the class of 7-dimensional csc nilpotent Lie groups, although thiswould be somewhat surprising since it is known that there exist non-isomorphic 5-dimensionalcsc nilpotent Lie groups with isomorphic cohomology rings [Cor18, 6.E]. This phenomenonturns out to appear in dimension 7 as well, as we will now describe by means of a concreteexample. To this end, consider the Lie algebra g := 13457 G , whose bracket relations aregiven by [ x , x i ] = x i +1 , i = 2 , , , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = − x , and g := 23457 B , defined by the bracket relations [ x , x i ] = x i +1 , i = 2 , , , [ x , x ] = x , [ x , x ] = x , [ x , x ] = − x . Using Maple, we can find a basis for the cohomology of the nilpotent Lie algebra g , whichhas Betti numbers (1,2,3,4,4,3,2,1), denoted by { e , e ij , j , i dim H j ( g , R ) , e } ,where { e ij , i = 1 , . . . , dim H j ( g , R ) } is a basis for the j ’th cohomology, and { e } and { e } are bases for the 0’th and 7’th cohomologies respectively, such that we have the followingnon-trivial products: e ∧ e = − e e ∧ e = e e ∧ e = − e e ∧ e = − e e ∧ e = − e e ∧ e = e e ∧ e = 2754 e e ∧ e = − e e ∧ e = 4 e We can likewise fix a basis for the cohomology of the algebra g := 23457 B , which has thesame Betti numbers, denoted f ij following the same scheme, which has non-zero products: f ∧ f = − f f ∧ f = f f ∧ f = − f f ∧ f = − f f ∧ f = f f ∧ f = − f f ∧ f = f f ∧ f = − f f ∧ f = f Define now a map ϕ : H ∗ ( g , R ) → H ∗ ( g , R ) defined as follows on the basis: ϕ ( e ) = f , ϕ ( e ) = f , ϕ ( e ij ) = f ij for j = 1 , , , and for all corresponding i , and put ϕ ( e ) = 2 f , ϕ ( e ) = f , ϕ ( e ) = − f ,ϕ ( e ) = 2754 f , ϕ ( e ) = − f , ϕ ( e ) = 4 f , ϕ ( e ) = 23 f . This is clearly an isomorphism of vector spaces, and the choice of coefficients, makes ϕ preservethe multiplication as well; hence ϕ is an isomorphism of graded rings. Remark 5.2.
A related question is whether it is possible to use Theorem B to determinethe first uncountable family of pairwise non-quasi-isometric csc nilpotent Lie groups thatare not distinguished by Pansu’s teorem [Pan89, Théorème 3], by considering the five 1-parameter families, I, N, N , M and N , of 7-dimensional nilpotentLie algebras whose Carnot Lie algebras are independent of the parameter. As we will shownow, this is unfortunately not the case. Using Maple, one can compute a linear basis forthe cohomology of any given finite dimensional Lie algebra, as well as multiplication tablesfor for the product. From this one sees that, within each family, the j -th Betti number β j ( g ( t )) := dim R H j ( g ( t ) , R ) , where where ( g ( λ )) λ denotes the family in question, is (exceptfor a finite number of members) independent of the parameter t . As an illustration of the typeof multiplication tables one is faced with, we include here two examples: − t − t +4 t − t +2 t − t +1 − t +2 t − t +1 − t − t +13 t − t +6 t ( t − t +4) − t − t +2 t − t +1 − t − t +4 t − t − t +1 − t +2 t − t +1 t + t +1 t +2 t +1
00 0 0 0 0 − t − t +5 t − t +6 t − t +3 t − t +1 Table 1.
Multiplication table for 3rd and 4th cohomology for the family M − t − t +32 t − t +1083 t ( t − t +6) − t − t +50 t − t +276 t − t +5046 t ( t − t +6) − t − t +158 t − t +4323 t ( t − t +6) 3 t +6 t +14 t +100 t − t +3606 t ( t − t +6)3 t +2 t +7 t +82( t − t +6) 3 t +5 t +27 t +37 t +78 t +42 t +484( t − t +4 t +6) − t +5 t +5 t +23 t − t − t +4 t +6 − t − t − t − t − t − t − t +4 t +6)11 t − t +413( t − t +6) 11 t − t +107 t − t +2466( t − t +6) − t − t +41)3( t − t +6) − t − t +45 t − t − t +6) − t − t +28 t − t ( t − t +6) 8 t − t +57 t − t +1266 t ( t − t +6) 11 t − t +1083 t ( t − t +6) − t − t − t +906 t ( t − t +6) Table 2.
Multiplication table for 3rd and 4th cohomology for the family N The tables should be read as follows: In Table 1, we have that H ( g ( t ) , R ) is 8 dimensionalwith a fixed basis e ( t ) , . . . , e ( t ) , that H ( g ( t ) , R ) is also 8 dimensional with a fixed basis e ( t ) , . . . , e ( t ) and that H ( g ( t ) , R ) is one dimensional with a fixed basis e ( t ) . The ( i, j ) thentry of the table is the coefficient of the product e i ( t ) ∧ e j ( t ) e.g. we have e ( t ) ∧ e ( t ) = − t − t + 4 t − t + 2 t − t + 1 e ( t ) from the first table. A similar reasoning applies to Table 2. OHOMOLOGICAL INDUCTION AND UNIFORM MEASURE EQUIVALENCE 17
We now explain how to construct isomorphisms of graded rings ϕ t,s : H ∗ ( g ( s ) , R ) → H ∗ ( g ( t ) , R ) . Denote the basis of H ∗ ( g ( λ ) , R ) by e ij ( λ ) following the same scheme as in Remark 5.1, butincluding the parameter. Using the fact that ϕ t,s should be linear and graded we know that ϕ t,s is given by the coefficients associated to the image of the basis elements: ϕ t,s ( e ij ( s )) = β j ( g ( t )) X k =1 α kj,i e kj ( t ) . In other words, we may consider ϕ as a block-diagonal matrix, where the blocks are β j ( g ( t )) × β j ( g ( t )) -matrices. The map ϕ being a ring homomorphism is then encoded by solutions to apolynomial system of equations in the variables α kj,i arising from the multiplication table, andone can easily check whether ϕ is bijective or not by taking the determinant of the matrixcorresponding to the solutions.The method used to obtain solutions to the system is described below, and the implemen-tation has been done in Maple 19: All equations appearing are polynomial in the variables α kj,i with certain rational functions in the parameters s or t as coefficients. First the system issimplified by inspecting the equations, and replacing every equation of the form p ( s ) α kj,i = 0 (with p a rational function in s ) with α kj,i = 0 , and similarly for equations of the form p ( t ) α kj,i α nℓ,m = 0 . Then using the zero-product rule on equations of the second type above,along with a non-singularity assumption on all the block-matrices along the diagonal, onemay in certain cases conclude which factor should be . To illustrate this, consider the fam-ily M which has β ( g ( t )) = 3 for all but finitely many values of t , and for which thecorresponding system of equations contains the equations α , α , = 0 , α , α , = 0 , α , α , = 0 Since α , = 0 , α , = 0 and α , = 0 would cause the first block-matrix to be singular, wemay conclude that α , = 0 . These zero-values are then substituted into the system whichis then reduced. The procedure is then continued until no further conclusions can be madethis way. After this, we tried solving the system, replacing s with π and t with e . In case thesystem was still too complex to solve in reasonable time, we would make qualified guesses onvalues of variables based on solutions of parts of the system. This process could be repeatedseveral times. After a full solution for s = π, t = e was obtained, we would set any remainingfree variables to 1, compute the determinants of the diagonal matrices, and substitute s and t back into the system and check whether it was still a solution. For every family, this provideda solution which was an isomorphism for all but finitely many values of the parameter. Thus,Theorem B cannot be used to tell these families apart up to quasi-isometry.Obtaining solutions to the equation systems described above carries a high degree of com-plexity. For example, the system of equations for the family 1357M consists of 1193 unique(but possibly equivalent) equations, with up to 11 summands of the form p ( t ) α kj,i α nℓ,m , wherethe coefficients p ( t ) , as already mentioned, are rational functions in t . In contrast to this, thesystem obtained from the family N consists of 94 equations, each of which contains upto 17 summands. The concrete formulas for the graded ring-isomorphism ϕ t,s : H ∗ ( g ( s ) , R ) → H ∗ ( g ( t ) , R ) are not included here due to their size; for instance for the family N the ϕ t,s is given by × -matrix, and the expression below is an example of one of its prototypicalnon-zero entries: t − s )( t +1)(6 t − t − t − s +1)( s − s +17 s − h (cid:0) − t − t − t + t (cid:1) s ++ (cid:0) − t − t − t − t (cid:1) s + (cid:0) t − − t − t − t (cid:1) s ++ (cid:0) − + t − t + t + t (cid:1) s − t − t + 2 t − − t i Maple worksheets performing this procedure with the correct guesses written down can beobtained from the authors upon request.
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E-mail address : [email protected] David Kyed, Department of Mathematics and Computer Science, University of SouthernDenmark, Campusvej 55, DK-5230 Odense M, Denmark
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