Closed manifolds with transcendental L2-Betti numbers
aa r X i v : . [ m a t h . K T ] J un Closed manifolds with transcendental L -Bettinumbers Mika¨el Pichot ∗ McGill UniversityMontreal, Canada Thomas Schick † Georg-August-Universit¨at G¨ottingenGermanyAndrzej Zuk ‡ Institute Math´ematiques de JussieuParis, France § November 19, 2018
Abstract
In this paper, we show how to construct examples of closed manifoldswith explicitly computed irrational, even transcendental L Betti num-bers, defined via the universal covering.We show that every non-negative real number shows up as an L -Betti number of some covering of a compact manifold, and that manycomputable real numbers appear as an L -Betti number of a universalcovering of a compact manifold (with a precise meaning of computablegiven below).In algebraic terms, for many given computable real numbers (in par-ticular for many transcendental numbers) we show how to construct afinitely presented group and an element in the integral group ring suchthat the L -dimension of the kernel is the given number.We follow the method pioneered by Austin [2] but refine it to getexplicit calculations which make the above statements possible. In 1974, Atiyah defined L -Betti numbers for covering spaces of closed manifolds[1]. A priori these Betti numbers are real and Atiyah asked at the end of his ∗ Mika¨el Pichot was supported by JSPS and the WPI Initiative, MEXT, Japane-mail: [email protected] † Thomas Schick was partially supported by the Courant Research Center “Higher orderstructures in Mathematics” within the German initiative of excellencee-mail: [email protected] ‡ Andrzej Zuk was supported by the Humboldt foundationemail: [email protected] § All authors were partially supported by HIM, Bonn. The main part of work was carriedout during the HIM trimester program “Rigidity”. Mika¨el Pichot, Thomas Schick, Andrzej Zuk paper to find examples where they are irrational. The question remained openand the fact that these L -Betti numbers may always be rational, and evenintegral for torsion free groups, has become known as the “Atiyah conjecture”.Under conditions on the torsion in the group, more refined conjectures have beenformulated and popularized as the “strong Atiyah conjectures” [13, Chapter 10],[4, Definition 1.1], which are satisfied for many groups.Let us observe that the discussion is concerned with two slightly differentcases: • Atiyah from the very beginning studied arbitrary normal coverings of acompact manifold M . The resulting values for the L -Betti numbers maybe very different depending on which covering of M they are associatedwith. • The most important special case, often exclusively considered in laterwork, uses the universal covering of the manifold M . This way, one definesinvariants depending only on M : these are the invariants usually referredto as the L -Betti numbers of M .The L -Betti numbers are homotopy invariants of the underlying manifold M . It follows from this that, when considering only the universal covering,i.e. the L -Betti numbers, there is in total only a countable set of possiblevalues.However, a given space can have uncountably many different normal cover-ings (corresponding to the normal subgroups of the fundamental group) so thatthe set of possible L -Betti numbers of normal coverings of compact manifoldsa priori could well be uncountable.In a recent paper [2], Tim Austin showed that the set of L -Betti numbersassociated to all possible normal coverings of compact manifolds is uncountable,and in particular contains irrational (and even transcendental) values.In the present paper, we show how to construct examples of closed mani-folds with explicitly given irrational (and transcendental) L -Betti numbers fortheir universal coverings. As explained below, we follow closely the techniquesdeveloped by Austin in [2], with refinements which allow us to make explicit di-mension calculations. Explicit calculations (and to some extend the basis of allthese developments) have been carried out previously in [3, 7, 6], which alreadylead to unexpected values of L -Betti numbers, not however to any which onecould prove to be irrational.The problem at hand has a well known purely algebraic reformulation. Theaim is to produce a finitely presented group G and an element Q in the groupalgebra Z [ G ] such that dim G ker( Q ), the von Neumann dimension of the kernelof this operator acting on l ( G ) is irrational. Then there is a standard con-struction to obtain a closed 7-dimensional manifold M with the fundamentalgroup isomorphic to G and whose third L -Betti number (computed using theuniversal covering) is equal to the dim G ker( Q ), see [13, Lemma 10.5] and [6,Proposition 6 and Theorem 7].If, instead of starting with a finitely presented group one only starts with afinitely generated group G , the standard construction will result in a manifold M with normal covering M (which is not necessarily the universal covering)such that the third L -Betti number for this covering is equal to dim G ker( Q ). losed manifolds with transcendental L -Betti numbers G which is not finitely presented but admitsa recursive presentation and thus embeds into a finitely presented group H by Higman’s theorem [9]. For a suitable element Q ∈ Q [ G ] we prove thatdim G ker( Q ) is transcendental. Clearing denominators, we can achieve that Q ∈ Z [ G ] without changing its kernel. Finally, it is a standard fact that thedimension of the kernel does not change if we let Q act on l ( H ), comparee.g. [15, Proposition 3.1].The group G will be of the form( Z ⊕ Γ2 /V ) ⋊ Γwhere V is a suitable Γ-invariant subspace of Z ⊕ Γ2 . For Γ, we will choose eitherthe free group on two generators F (as in [2]) or Z ≀ Z .The main result of [2] is to construct an uncountable family of groups G i ofthe form above and operators Q i ∈ Q [ G i ] such that the numbers dim G i (ker( Q i ))are all mutually different. It seems hard to prove that among those groupsfor which dim G i ker( Q i ) is irrational are recursively presented groups, as theirexistence is only inferred from a counting argument.In contrast, in the paper at hand we consider different operators Q for whichwe manage to explicitly compute dim G ker( Q ). Along the way, we explicitly pro-duce a recursively presented group G for which dim G ker( Q ) is transcendental.Namely, for any set of natural numbers I = { , n k } ⊂ N (listed in increasingorder 0 < n < n < . . . ) we construct a group G I as above whose presentationis determined by the set I together with Q I ∈ Q [ G I ] such thatdim G I ker( Q I ) = β + β ∞ X k =1 − dn k + k where β and β are some explicit rational numbers and d is a natural number.We prove that G I has a recursive presentation (and therefore embeds into afinitely presented group) if (and only if) I is recursively enumerable. It is nowimmediate to choose a recursively enumerable set I which leads to an irrationalor even transcendental L -dimension, e.g. by asking it to satisfy the Liouvillecondition. Recall that a real number x is a Liouville number if for any positiveinteger n , there exist integers p and q with q > < (cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12) < q n .Liouville [12] showed that such numbers are transcendental.Let us also stress the fact that we obtain these L -Betti numbers for solvablegroups (this is the reason why we use the group Γ = Z ≀ Z ), answering a questionof [2]. Note that for torsion-free solvable groups the Atiyah conjecture is known[11, Theorem 1.3].Using the explicit form of these L -dimensions of kernels for the operators weobtain, we can construct out of these for each real number r ≥ G r (ingeneral not recursively presented) and A r ∈ M n ( Z [ G r ]) with dim(ker( A r )) = r . This relies on explicit knowledge of how the kernel looks like under theoperations we employ.We will discuss possible extensions of the result which can be obtained withthe same method. In particular, with suitable modifications and additionaleffort one could produce many examples of A ∈ M n ( Z [ G ]) as above with explicitknowledge of the full spectral measure (as in [3, 7]). This spectral measure Mika¨el Pichot, Thomas Schick, Andrzej Zuk would be atomic and many of the L -dimensions of the eigenspaces would betranscendental.We also discuss more about the question which L -Betti numbers can (bymodifications of the construction) be obtained using finitely presented groups.Lukasz Grabowski [5] has independently and simultaneously, using an ap-proach which implements Turing machines directly in the integral group ring ofa suitable recursively presented group, arrived at results similar to ours. Usingyet another strategy, Franz Lehner and Stephan Wagner [10] manage in a cleverway to calculate explicitly the L -dimension of the kernel of the usual graphLaplacian (as in [3]), but for wreath products of finite groups with non-abelianfree groups, and find non-rational algebraic numbers among the values—givingyet another interesting example. Acknowledgements . The results of this paper were obtained during theTrimester Program on Rigidity at the Hausdorff Institute of Mathematics, Bonn.The authors thank HIM for the stimulating atmosphere and the generous sup-port to be able to participate in this program. The authors also thank TimAustin for fruitful discussions.
We closely follow the notations of [2]. This will hopefully help a reader interestedin reading concurrently both papers.We consider groups of the form ( Z ⊕ Γ2 /V ) ⋊ α Γ where V is some left translationinvariant subgroup of Z ⊕ Γ2 and the action α of Γ is by translations on the left.Usually, we will omit α in the notation.For the first few sections, we will only assume that Γ is generated by twoelements s , s and that it satisfies the zero divisor conjecture over Z . Thelatter means that the group ring Z [Γ] contains no non-trivial zero divisors. Itimplies in particular that the group is torsion free, so s , s are of infinite order.Later, concrete computations will be carried out notably in the case of the freegroup Γ = F and of the wreath product Γ = Z ≀ Z with natural generating sets,which of course satisfy all these conditions.We denote by Cay(Γ , S ) the right Cayley graph of Γ with respect to S = { s ± , s ± } . By a path P in Cay(Γ , S ) we mean a subset { g , g , g , . . . , g ℓ } ⊂ Γof mutually distinct consecutive elements of Cay(Γ , S ).We recall that the Pontryagin dual of Z ⊕ Γ2 is isomorphic to the infiniteproduct Z Γ2 . Sometimes we will identify an element χ ∈ Z Γ2 with the subset χ − (1) ⊂ Γ. We denote by V ⊥ = { χ ∈ Z Γ2 | h χ | v i = 0 ∀ v ∈ V } the dual of Z ⊕ Γ2 /V and recall that the Fourier transform ℓ ( Z ⊕ Γ2 /V ) ≃ L ( V ⊥ , m V ⊥ )where m V ⊥ is the Haar measure on V ⊥ , induces a spatial isomorphism betweenthe group von Neumann algebra L (( Z ⊕ Γ2 /V ) ⋊ Γ) and the cross-product vonNeumann algebra L ∞ ( V ⊥ ) ⋊ ˆ α Γ. The dual action ˆ α is on V ⊥ defined by h ˆ α g χ | u i = h χ | α g − u i for χ ∈ V ⊥ and u ∈ Z ⊕ Γ2 /V . Note that, if we think of χ ∈ Z Γ2 losed manifolds with transcendental L -Betti numbers α g χ corresponds to the subset gχ − (1), i.e. is obtainedby a left translation by g .For simplicity we sometimes denote s ( χ ) := ˆ α s χ . For F ∈ L ∞ ( V ⊥ ) wedenote by M F ∈ L ∞ ( V ⊥ ) ⋊ ˆ α Γ the twisted pointwise multiplication, defined by M F f ( χ, g ) = F (ˆ α g ( χ )) · f ( χ, g )and by T s the translation operator given by T s f ( χ, g ) = f ( χ, s − g ). Checkingthe definitions, we observe the covariant relation T s − M F T s = M F ◦ ˆ α s . (2.1) We want to construct certain operators in the rational group ring Q [ Z ⊕ Γ2 /V ⋊ Γ],viewed as acting on L ( V ⊥ , m V ⊥ ) ¯ ⊗ ℓ (Γ). They will be taken to be of the form A = X s ∈ S T s − ( M F s | V ⊥ + M F s − ◦ ˆ α s − | V ⊥ ) , (3.1)where F s : Z Γ2 → Q will depend only on finitely many coordinates around theorigin e . The operator A is self-adjoint as shown in [2, Lemma 3.1].The essential difference with the operators of [2] is that the function F s will recognize a very specific family of paths that we call “hooks” and whichsubstitute the paths “with no small horizontal doglegs” of [2, Definition 3.2].This one ingredient simplifies several computations and is what allows us tocalculate the von Neumann dimensions exactly. A path P (finite or not) in Cay(Γ , S ) is called a hook if it hasthe form P = (cid:8) gs − n , . . . , gs − , g, gs , gs s − , . . . , gs s − m (cid:9) (3.3)for some g ∈ Γ and n, m ∈ { , . . . , ∞} . If n < ∞ then gs − n is called the leftendpoint of P . We call n the length of the left leg and m the length of the rightleg . We call a path P (finite or not) a vertical segment if P = (cid:8) gs − n , . . . , g, . . . , gs m (cid:9) for some n, m ∈ Z . If h, hs ∈ P , but hs − / ∈ P we call h a lower endpoint of the hook orvertical segment P . If P is a hook and h is additionally the left endpoint, then h is called left lower endpoint .We denote by B ( g, k ) the ball of radius k around g ∈ Γ in Cay(Γ , S ). Let χ ∈ Z Γ2 . We say that χ is 1 -good if for some hook P in Cay(Γ , S ) containing e , its restriction χ | B ( e, to B ( e,
1) equals 1 on P and0 outside. We say that χ is locally good if χ is 1-good and s ( χ ) is 1-goodfor every s ∈ χ − (1) ∩ B ( e, χ is interior good if χ is locallygood and (cid:12)(cid:12) χ − (1) ∩ B ( e, (cid:12)(cid:12) = 3. We say that χ is a good end if χ is locallygood and e is a lower endpoint of the hook P above. This happens exactly if (cid:12)(cid:12) χ − (1) ∩ B ( e, (cid:12)(cid:12) = 2. Mika¨el Pichot, Thomas Schick, Andrzej Zuk
We now introduce F s : Z Γ2 → Q . If χ ∈ Z Γ2 is interior good then(i) F s ( χ ) := 1 if s − ( χ ) is interior good;(e) F s ( χ ) := 2 if s − ( χ ) is a good end;(b) F s ( χ ) := if s − ( χ ) is 1-good, but not locally good.Define F s ( χ ) := 0 otherwise. Note that this happens if χ is not interior good orif χ is, but s − χ is not 1-good.In the definition of the operator A , both F s ( χ ) and F s − ˆ α s − ( χ ) appear.For further reference, we compile a little table showing the values of these twofunctions for the different possibilities. The columns give the different propertiesof χ , the rows those of s − χ = ˆ α s − χ . The first value in each entry is F s ( χ ),the second one F s − ( s − χ ). s − χ \ χ int. good good end 1-good, notloc. good not 1-goodint. good 1; 1 0; 2 0;
0; 0good end 2; 0 0; 0 0; 0 0; 01-good, not loc. good ; 0 0; 0 0; 0 0; 0not 1-good 0; 0 0; 0 0; 0 0; 0This follows by inspecting the definition of F s . Note that F s ( χ ) dependson χ and s − χ , whereas F s − ˆ α s − ( χ ) = F s − ( s − χ ) depends on s − χ and( s − ) − s − χ = χ , which of course also explains why the second matrix weobtain is the transpose of the first.Finally note that A is a sum of operators of the form T s − M G s where G s := F s + F s − ˆ α s − itself is a linear combination of characteristic functions, and G s depends only on the 3-neighborhood of e . For later reference and conveniencewe list the relevant values of G s ( χ ) next. Note that G s ( χ ) depends on χ and s − χ ; we will give a description now. (1) G s ( χ ) = 0 if χ is not a -good because then neither χ nor s − χ is interior good.(2) (Case where χ is an end): Assume next that χ is -good and χ − (1) ∩ B ( e,
1) = { e, s } . Then G s ( χ ) = 0 if s = s because then s − χ is not -good. Moreover, G s ( χ ) = 2 if s − χ is interior good (i.e. the path extendstwo more steps) and G s ( χ ) = 0 otherwise.(3) Now assume that χ is -good and χ − (1) = { e, s − , t } . For s = s − , t , G s ( χ ) = 0 because then s − χ is not -good. Moreover,(a) (case where in one direction the path goes bad): if for s ∈ { s − , t } s − χ is not -good (i.e. the path doesn’t extend in this direction) then G s ( χ ) = 0 . Write { s − , t } = { s, s ′ } , then G s ′ ( χ ) = 0 if s ′− χ is notinterior good, and G s ′ ( χ ) = otherwise.(b) (case where in one direction, necessarily s − , the path ends). If s χ is a good end then G s − ( χ ) = 2 if χ is interior good and G s − ( χ ) = 0 if χ is not interior good (the latter situation we’ve just discussed).(c) Assume now that χ is interior good, s ∈ { s − , t } and s − χ is inte-rior good (i.e. the hook extends in two directions through e , and indirection s even two steps). Then G s ( χ ) = 2 . losed manifolds with transcendental L -Betti numbers (1) In other words: we only “move along the path”, with weight2 if one is in an interior situation or arrives at or from a good end point(with some extension of the path in all directions). We use weight if wemove to or from a point which is next to a bad point (again the path hasto extend a bit in the other direction).(2) Our definition of F s involves 1-neighborhoods rather than 10-neighborhoods.This will make calculations later easier, in particular if Γ is not the freegroup. In the framework of [2] one can economize and can reduce the sizeof the neighborhoods, albeit not to 1.(3) We emphasize that our definition of F s makes the operators A follow thehook itself, rather than its 1-neighborhoods (this convenient simplificationwill be made precise in Section 4). V ⊥ into invariant subsets This section follows pretty much [2, Section 3.2], with slight modifications thatwe indicate now.
Given χ ∈ Z Γ2 , a ball B ( g,
1) is called a good neighborhood of χ , if g − ( χ ) ∩ B ( e,
1) = { e, s } . B ( g,
1) is called a bad neighborhood if g − ( χ )is not 1-good.Having this definition, and since our notations essentially coincide, we canobtain a partition of Z Γ2 by simply copying that of [2, Section 3]. Namely, weobtain first a disjoint Borel partition Z Γ2 = C ∪ C , ∪ C , ∪ C , ∪ C , ∞ ∪ C , ∞ ∪ C ∞ , ∞ . Here C is the set of χ such that F s ( χ ) and F s − ( s − χ ) are both zero for allgenerators s . If χ / ∈ C then in particular χ is 1-good, i.e. χ − (1) ∩ B ( e, e .To which of the other sets C i,j a χ / ∈ C belongs is now determined accordingto the fate of two walkers starting at the origin and moving in opposite directionsalong this path starting at e . Indeed, for this description we identify χ with thesubset χ − (1) of Γ. Each walker will have as path a (possibly infinite) hook orvertical segment R ′ , starting at e .We have three possible disjoint ending scenarios i, j ∈ { , , ∞} for eachwalker:( ∞ ) the walker never reaches a good or a bad neighborhood, and continues hispath forever;(1) the walker reaches a good neighborhood and ends up at a lower end pointof a hook;(2) the walker reaches a bad neighborhood and stops walking. In this case welet P ′ ⊂ R ′ be the path that the given walker follows up to distance 1 ofhis stopping point. Mika¨el Pichot, Thomas Schick, Andrzej Zuk
Furthermore, in case (1) we set P ′ := R ′ to be the path followed by our givenwalker. Finally, we set R ( χ ) to be the union of the two hooks R ′ of the two walk-ers, and define P ( χ ) similarly. Note that these are hooks or vertical segmentsand that the intersection of χ with the 1-neighborhood of each endpoint of P (ifit exists) determines the ending scenario at that endpoint. Finally, let ψ ( χ ) bethe restriction of the function χ to the 1-neighborhood of R . It takes the value1 on R and 0 on all points outside R except possible on the 1-neighborhood of R \ P , the set of “bad” endpoints (if they exist).If i, j < ∞ , R and P are finite. Note that we have no way to order thewalkers a priori: only the unordered tuple of ending scenarios is significant.Next, we make a further refinement and decompose each set C i,j for i ≤ j < ∞ according to abstract triples ( P, R, ψ ) which occur in the above discussion.Let Ω i,j be the set of all such triples and set C ( P,R,ψ ) = { χ ∈ Z Γ2 | R ( χ ) = R, P ( χ ) = P, ψ ( χ ) = ψ } . We obtain:
The following is a Borel partition of Z Γ2 : Z Γ2 = C ∪ [ i,j ∈ , i ≤ j [ ( P,R,ψ ) ∈ Ω i,j C ( P,R,ψ ) ∪ C , ∞ ∪ C , ∞ ∪ C ∞ , ∞ . By intersection with V ⊥ this leads to a Borel partition of V ⊥ and therefore toan orthogonal decomposition of L ( V ⊥ ). Depending on V ⊥ , several summandsmight vanish.Later, the pile-up of eigenspaces is organised according to the followingequivalence relations on triples ( P, R, ψ ): Two triples ( P , R , ψ ) and ( P , R , ψ ) are said to be trans-lation equivalent if there exists a g ∈ Γ such that P = gP , R = gR and ψ ( gh ) = ψ ( h ) for all h ∈ U ( R , k ∈ N and X ⊂ Γ we let U ( X, k )denote the k -neighborhood of X in the Cayley graph of Γ.Equivalence classes in Ω i,j are finite (since R is finite and contains e ) andwe denote them by C ∈ Ω i,j / ∼ . Moreover, note that if e ∈ P then the set of g ∈ Γ which translates (
P, R, ψ ) to a translation equivalent pair are exactly the g ∈ P with g − ∈ P . We obtain the decomposition L ( V ⊥ ) ¯ ⊗ ℓ (Γ) ≃ H ⊕ M ≤ i ≤ j ≤ M C∈ Ω i,j / ∼ H C ⊕ H , ∞ ⊕ H , ∞ ⊕ H ∞ , ∞ (5.1)where the notation is close to the one in [2, Section 3], namely: H C = M ( P,R,ψ ) ∈C H ( P,R,ψ ) and H ( P,R,ψ ) = Im( M C ( P,R,ψ ) ) losed manifolds with transcendental L -Betti numbers H i,j = Im( M Ci,j ), H = Im( M C ).More precisely, we should have written C i,j ∩ V ⊥ , and we think of the char-acteristic function C i,j as a bounded measurable function on V ⊥ , thus actingby left multiplication on L ( V ⊥ ) and also by twisted left multiplication on L ( V ⊥ ) ¯ ⊗ l (Γ). For notational convenience, we have omitted reference to V here.Note that, because H ( P,R,ψ ) is defined by left multiplication of L ( V ⊥ ) ¯ ⊗ ℓ (Γ)with a projection in L ∞ ( V ⊥ ) ⋊ ˆ α Γ, it is a right Hilbert Z ⊕ Γ2 /V ⋊ Γ-module. Itsvon Neumann dimension is given by the measure of the subset C P,R,ψ :dim Z ⊕ Γ2 /V ⋊ Γ ( H P,R,ψ ) = m V ⊥ ( C P,R,ψ ∩ V ⊥ ) . (5.2)Corresponding statements apply to the other subspaces. For each
C ∈ Ω i,j / ∼ (with ≤ i ≤ j ≤ ) the subspace H C is A -invariant.Moreover, let V l,i,j be the following weighted graphs: a segment of length l ≥ where each interior edge has weight two, and if ( i, j ) = (1 , , both boundaryedges have weight as well, whereas if ( i, j ) = (1 , then one boundary edge hasweight while the other has weight , and if ( i, j ) = (2 , then both boundaryedges have weight .Let A l,i,j be the weighted adjacency matrices, regarded as operators on l ( V l,i,jv ) ,where V l,i,jv denotes the vertex set of the graph V l,i,j .Choose one ( P, R, ψ ) ∈ C , e.g. the one with e as the (left) lower endpoint of P ; “left” if P is a hook (and not a vertical segment).Then we have a unitary equivalence of Hilbert Z ⊕ Γ2 /V ⋊ Γ -endomorphisms A | H C ≃ id H ( P,R,ψ ) ⊗ A l,i,j where l is the length of the path P and ( i, j ) is the ending scenario of ( P, R, ψ ) .Proof. The proof is essentially the same as for the corresponding statement [2,Proposition 3.12].First observe that for g ∈ Γ the operator T g − (which is a Hilbert- Z ⊕ Γ2 /V ⋊ Γisometry) maps H ( P,R,ψ ) isometrically to H ( g − P,g − R, ˆ α g − ψ ) .This implies that A , because of its shape, maps a vector in H ( P,R,ψ ) indeedto a linear combination of vectors in H ( sP,sR,sψ ) for the generators s . However,inspection of the functions G s in Proposition 3.5 or Remark 3.6 shows that a non-zero contribution is obtained only if s ∈ P . Consequently, H C for C ∈ Ω i,j / ∼ is A -invariant. Moreover, inspection of 3.5 further shows that A maps onesummand to the other (up to identification with the unitary T g ) exactly withthe weights as described by A l,i,j ; details of the argument follow exactly as in[2, Proposition 3.12].Moreover, Proposition 3.5 also shows that the operator is zero on H ( P,R,ψ ) if | R | ≤ A |H = 0. We will concentrate now on the particular eigenvalue − Mika¨el Pichot, Thomas Schick, Andrzej Zuk
The value − is an eigenvalue for A l, , acting on l ( V l, , ) onlyfor l ≡ and the eigenspace is one dimensional in this case.The value − is never an eigenvalue for A l,i, for l ∈ N and i ∈ { , } .Proof. We first study the kernel for the l × ( l + 1)-matrix α ...α ... ...... ... β ! obtained by deleting the last row, and where α, β ∈ { , } . A simple linearrecursion shows that this kernel is 1-dimensional and spanned by the vector(2 α, − , − α , α , − , − α , . . . , x l ) , with x l = α β − l ≡ − β − l ≡ − α ) β − l ≡ . The kernel of A l,i,j is non-trivial if and only if this vector is also mapped to zeroby the last row of A l,i,j , which is simply the condition β (4 − α ) + 4 β − α = 0 l ≡ βα − β − = 0 l ≡ − β + 4(4 − α ) β − = 0 l ≡ . If i = j = 1, i.e. α = β = 2, this is satisfied if and only if l ≡ α = and either β = 1 or β = we check that the condition isnever satisfied. This finishes the proof. Z Γ2[2, Lemma 5.1] extends readily to our situation:
Given a subgroup V ≤ Z ⊕ Γ2 , a finite subset E ⊂ Γ , and ψ : E → Z , let C ( ψ ) be the set of characters χ ∈ Z Γ2 such that χ | E = ψ . If C ( ψ ) ∩ V ⊥ = ∅ , then m V ⊥ ( C ( ψ )) = 1 |{ ψ ′ ∈ Z E : C ( ψ ′ ) ∩ V ⊥ = ∅}| . Proof.
Given ψ , ψ : E → Z , it is enough to show that m V ⊥ ( C ( ψ )) = m V ⊥ ( C ( ψ ))whenever both C ( ψ ) and C ( ψ ) intersect V ⊥ . Take χ i ∈ C ( ψ i ) ∩ V ⊥ . Thentranslation by χ − χ sends C ( ψ ) ∩ V ⊥ to C ( ψ ) ∩ V ⊥ and preserves themeasure m V ⊥ .We also remark that if C ( ψ ) ∩ V ⊥ = ∅ , then certainly m V ⊥ ( C ( ψ )) = 0.Given a finite subset F ⊂ Γ and a subgroup Λ ⊂ Γ, we define a left invariantsubgroup V F, Λ of Z ⊕ Γ2 in the following way: V F, Λ = span Z { gF − gtF | g ∈ Γ , t ∈ Λ } , (7.2) losed manifolds with transcendental L -Betti numbers F is the characteristic function of the set F .Setting χ ( F ) := P v ∈ F χ ( v ), we have V ⊥ F, Λ = { χ ∈ Z Γ2 | χ ( gF ) = χ ( gtF ) , ∀ g ∈ Γ , t ∈ Λ } . (7.3)We will need furthermore that for certain finite subsets F, G ⊂ Γ as consid-ered in below the family { gF | g ∈ G } is linearly independent over Z . (7.4)Since over Z any linear combination of elements in { gF | g ∈ G } is of the form X g ∈ H gF = (cid:0) X g ∈ H g (cid:1)(cid:0) X f ∈ F f (cid:1) ; H ⊂ G, this linear independence is guaranteed if the group algebra Z [Γ] has no non-trivial zero divisor, namely that the zero divisor conjecture for Γ holds over Z .This is satisfied for the examples we discuss in this paper (namely Z ≀ Z or thefree group) in fact there are no known torsion–free counterexamples. Let
E, F be subsets of Γ and Λ ≤ Γ be a subgroup. We saythat E has the extension property relative to ( F, Λ) if whenever ψ : E → Z satisfies ψ ( gF ) = ψ ( gtF ) ∀ g ∈ Γ , t ∈ Λ such that gF ∪ gtF ⊂ E (7.6)then exists χ ∈ V ⊥ F, Λ such that χ | E = ψ .In other words, if the obvious set of conditions on ψ is satisfied on E , then ψ extends to an element of V ⊥ F, Λ . Given E, F, Λ , Γ as in Definition 7.5, we letΩ
F,E be the set Ω
F,E := { gF ⊂ E | g ∈ Γ } . We denote by Ω
F,E / Λ the set of classes of the equivalence relation ∼ Λ on Ω F,E given by right multiplication by Λ on Γ, namely gF ∼ Λ g ′ F ⇐⇒ ∃ t ∈ Λ : gtF = g ′ F ⇐⇒ g − g ′ ∈ Λ . This equivalence holds as Γ is torsion free and F is finite.The following is a generalization of [2, Corollary 5.9] with the same proof. Let
E, F be finite subsets of Γ and Λ ≤ Γ be a subgroup. Assumethat E has the extension property relative to ( F, Λ) . Then |{ ψ ∈ Z E | C ( ψ ) ∩ V ⊥ F, Λ = ∅}| = 2 | E |− K where K = | Ω F,E | − | Ω F,E / Λ | and with C ( ψ ) as in Lemma 7.1.Proof. Since E has the extension property relative to ( F, Λ), the subset { ψ ∈ Z E | C ( ψ ) ∩ V ⊥ F, Λ = ∅} coincides with { ψ ∈ Z E | ψ ( gF ) = ψ ( g ′ F ) whenever gF, g ′ F ∈ Ω F,E , gF ∼ Λ g ′ F } . Mika¨el Pichot, Thomas Schick, Andrzej Zuk
The latter is the annihilator of the finite dimensional subspace V E,F, Λ := span Z { gF − g ′ F | gF, g ′ F ∈ Ω F,E , gF ∼ Λ g ′ F } . By (7.4) this span is the direct sum over the equivalence classes C ∈ Ω F,E / Λ of V C := span Z { gF − g ′ F | gF, g ′ F ∈ C } and again by (7.4) dim Z ( V C ) = | C |− K := dim Z ( V E,F, Λ ) = X C ∈ Ω F,E / Λ | C | − | Ω F,E | − | Ω F,E / Λ | . Therefore, dim Z { ψ ∈ Z E : C ( ψ ) ∩ V ⊥ F, Λ = ∅} = | E | − K, hence the result. Assume, in the situation of Section 5, that V ⊥ = V ⊥ F, Λ for Λ ⊂ Γ a subgroup and F ⊂ Γ finite as above. Moreover, given ψ : U ( R, → Z ,assume that it extends to V ⊥ and that U ( R, satisfies the extension propertyfor F . Then dim Z ⊕ Γ2 /V ⋊ Γ ( H P,R,ψ ) = 2 −| U ( R, | + K , (7.9) where K is the number of equivalence classes of subsets gF of U ( R, ( g ∈ Γ ),with gF ∼ gtF for t ∈ Λ .Proof. This is a direct consequence of Lemma 7.1 and Lemma 7.7.
We need a sufficiently general condition for deciding when a set E has theextension property relative to ( F, Λ). The following criterion is an analog of [2,Lemma 5.5].
Suppose that s is of infinite order, F ⊂ { s k | k ∈ Z } ⊂ Γ is finiteand that the subset B ⊂ Γ is horizontally connected , i.e. ∀ g ∈ Γ { gs k | k ∈ Z } ∩ B is a connected segment.Then B has the extension property (as in Definition 7.5) relative to ( F, Λ) for any subgroup Λ of Γ .Proof. Let B n be an increasing sequence of subsets of Γ such that B = B andsuch that B n +1 is obtained from B n by adding an element at distance 1 from B n , B n +1 is horizontally connected and the union of B n is Γ. We constructa sequence of functions χ n such that χ = ψ and χ n +1 | B n +1 = χ n and thecondition (7.6) is true with E = B n . This implies the existence of the extension.Of course it suffices to give a construction of χ .Suppose that we add to B one element h to get B = B ∪ { h } and we wantto construct χ . If gF ⊂ B and h ∈ gF then by horizontal connectivity of B and the special shape of F the element h is an end-point of F .Then, again by horizontal connectivity of B it is not possible that h ∈ g ′ F for some other g ′ with g ′ F ⊂ B . losed manifolds with transcendental L -Betti numbers t ∈ Λ \ { e } such that also gtF ⊂ B then necessarily gtF ⊂ B andby (7.6) χ ( h ) is imposed by χ ( h ) = χ ( gF \ { h } ) − χ ( gtF ) . We only need to show that this is independent of the choice of t . Indeed, if for t, t ′ ∈ Λ both gtF, gt ′ F ⊂ B then, as t ′ t − ∈ Λ and because B satisfies condition(7.6), χ ( gtF ) = χ ( gt ′ F ).If no pair gF, gtF as considered above exist, we can choose χ ( h ) at will,e.g. χ ( h ) := 0, as no additional condition has to be satisfied for (7.6) to holdfor E = B . V I and the effect on the eigenspaces As before, we consider a group Γ generated by s , s , but from now on we willmostly concentrate on the case of the free group or of Z ≀ Z . For n ∈ N = { , , . . . } , define t n := s n s s − n . For I ⊂ N ,define Λ I := h t i | i ∈ I i ≤ Γ.If F ⊂ Γ is finite, define V F,I := span Z { g F − gt F | g ∈ Γ , t ∈ Λ I } ⊂ Z ⊕ Γ2 so its Pontryagin dual is V ⊥ F,I = { χ ∈ Z Γ2 | χ ( gF ) = χ ( gtF ) , ∀ g ∈ Γ , t ∈ Λ } as in Definition (7.3). Finally, we specialize to F l = { s − , e, s } and set G I := Z ⊕ Γ2 /V F l ,I ⋊ Γ. For
Γ = Z ≀ Z = h s , s | [ s k s s − k , s ] = 1 ∀ k ∈ Z i (9.3) the elements t n are the free abelian generators of a free abelian subgroup, equalto the kernel of the obvious projection to Z = h s i .For Γ the free group on free generator s , s , the elements t n are the freegenerators of a free subgroup.In particular, if / ∈ I then Λ I intersects h s i , the subgroup generated by s ,only in the trivial element.Proof. For Z ≀ Z this is part of the structure theory of the wreath product: thebase Z ⊕ Z is a free abelian group with generators s n s s − n for n ∈ Z . The group h s i = Z acts on the base by the obvious permutation of the basis elements, thesemi-direct product is Z ≀ Z .For the free group, the assertion follows from an easy normal form calcula-tion, which is carried out in detail in [2, Lemma 5.2]. Assume that P = (cid:8) gs − n , . . . , g, gs , gs s − , . . . , gs s − m (cid:9) isa hook as in (3.3) with n, m ∈ N and I ⊂ N is given. Assume moreover that Γ is either the free group on free generators s , s or Γ = Z ≀ Z as in (9.3) .Then xt = y for x = y ∈ P and t ∈ Λ I exactly if t = t k for some generator t k with k ∈ I such that k ≤ n and k ≤ m , x = gs − k , y = gs s − k (or t = t − k , x = gs s − k , y = gs − k ). Mika¨el Pichot, Thomas Schick, Andrzej Zuk
Proof.
Obviously, the t k , x, y we have given satisfy all the conditions in bothcases.We next show that the conditions are necessary. Write x = gs ǫ x s a x and y = gs ǫ y s a y in P , with ǫ x , ǫ y ∈ { , } and a x , a y ∈ Z . If x − y ∈ Λ I , then inparticular x − y = s − a x s ǫ y − ǫ x s a y is mapped to the trivial element under theprojection to the infinite cyclic group h s i which maps s to 1. It follows that a y = a y , so x − y = s − a x s ǫ y − ǫ x s a x . In the two cases we study, Λ I is containedeither in the free or the free abelian groups on generators s v s s − v ( v ∈ Z ). Theassertion now follows. As before, assume Γ generated by s , s is either Z ≀ Z orthe free group on s , s . Set F l = { s − , e, s } and let R ⊂ Γ be a hook, ψ : U ( R, → Z the characteristic function of R . Then ψ extends to V ⊥ F l ,I .Moreover, gF l , hF l ⊂ U ( R, are equivalent if and only if g, h ∈ R and there is t ∈ Λ I with g = ht .Proof. By a normal form argument, we know that whenever gF l ⊂ U ( R, g ∈ R . By Lemma 8.1 we only have to check that, whenever xF l and yF l for x, y ∈ R are equivalent, then ψ ( xF l ) = ψ ( yF l ). By Proposition 9.4 (andnormal form in Γ), if xF l and yF l are equivalent then | xF l ∩ U ( R, | = 1 = | yF l ∩ U ( R, | : the intersection would have different cardinality only if x (or y ) was part of the “bend” of the hook, i.e. xs or xs − ∈ R , but then x isonly equivalent to itself. Finally ψ ( xF l ) ≡ | xF l ∩ U ( R, | (mod 2), therefore ψ ( xF l ) = ψ ( yF l ) and the proposition follows. Adopt the situation of Proposition 9.5. Assume that n and m are the length of the left and of the right leg of R , respectively. Then, with K := | I ∩ { , . . . , min( m, n ) }| , dim ( Z ⊕ Γ2 /V Fl,I ) ⋊ Γ ( H R,R,ψ ) = 2 − n + m ) − K . (9.7) Proof.
Because of Proposition 9.5 and Lemma 8.1, we can directly apply Corol-lary 7.8.By normal form, we know that inside U ( R,
1) there are no relations and that | R | = n + m + 2, hence | U ( R, | = 3( n + m + 2) + 2. Moreover, by Propositions9.5 and 9.4, the correction term K is exactly as given.We conclude this section by showing that in the situation of Proposition 9.5the sets C , ∞ , C , ∞ , and C ∞ , ∞ , as defined in Section 4, are negligible withrespect to m V ⊥ Fl,I . Given g ∈ Γ , let D g be the measurable subset of Z Γ2 defined by D g = { χ ∈ Z Γ2 | χ ( gs − k ) = 1 and χ ( gs − k s a ) = 0 , ∀ k ≥ , a = ± } . Then m V ⊥ Fl,I ( D g ) = 0 .Proof. Given g ∈ Γ and an integer N ≥ D g,N = { χ ∈ Z Γ2 | χ ( gs − k ) = 1 and χ ( gs − k s a ) = 0 , ∀ ≤ k ≤ N, a = ± } losed manifolds with transcendental L -Betti numbers F = F l = { s − , e, s } and E = { s − k s − , s − k , s − k s | k = 0 , . . . , N − } .The number of ways to embed shifted copies of F into E is | Ω E,F | = N . ByLemma 7.1 and 7.7 we have µ V ⊥ Fl,I ( D g,N ) ≤ N −| Ω E,F || = 12 N . Since D g = \ N ≥ D g,N , we obtain µ V ⊥ Fl,I ( D g ) ≤ − N for all N ≥ Keeping the notations above, we have m V ⊥ Fl,I ( C , ∞ ) = m V ⊥ Fl,I ( C , ∞ ) = m V ⊥ Fl,I ( C ∞ , ∞ ) = 0 . In particular, H , ∞ = { } = H , ∞ = H ∞ , ∞ . Proof.
Indeed, C , ∞ ∪ C , ∞ ∪ C ∞ , ∞ ⊂ S g ∈ Γ D g , so we may apply Lemma 9.8.The second claim follows directly from the corresponding version of (5.2). Lemma 9.8 and its corollary above can be extended to almostarbitrary subgroups V F, Λ , where F ⊂ Γ is a finite subset, Λ ≤ Γ is a subgroupas in Section 7, and E ⊂ Γ is any set having the extension property with respectto ( F, Λ). For instance it is enough to have that | F | ≥ gF ( g ∈ Γ) which are included in E are pairwise disjoint (as the proof of Lemma9.8 shows).
10 Explicit calculation of the von Neumann di-mension of the eigenspace
We continue with the situation of Section 9. We deal only with the eigenvalue − A and we set Q = A + 2. Fix I := { , n , n , . . . } ⊂ N with n := 2 < n < n < . . . and such that n k ≡ ∀ k . Choose Γ = Z ≀ Z or Γ free on two generatorsand set G I = ( Z ⊕ Γ2 /V F l ,I ) ⋊ Γ as in Definition 9.1. Construct A ∈ Q [ G I ] asabove. Then dim G I ker( A + 2) = β + β ∞ X k =1 − n k + k . Here, β and β are explicitly given rational numbers, compare (10.4) .In particular, these numbers show up as L -Betti numbers of normal cover-ings of compact manifolds with covering group G I .Proof. We use Proposition 5.3 to decompose A . By Corollary 9.9 and Lemma 6.1the only contributions to the eigenvalue − H C if C ∈ Ω , / ∼ Mika¨el Pichot, Thomas Schick, Andrzej Zuk and if the length of the associated hook R is congruent 1 modulo 3, and the L -dimension of the eigenspace is thendim G I ( H R,R,ψ ) = 2 − l + l ) − | I ∩{ ..., min { l ,l }}| , (10.2)where R is the hook, ψ is the characteristic function of the hook in its 1-neighborhood and l , l are the lengths of the left and right leg of the hook,respectively. Note that the length of the hook is l + l + 1, so we get a contri-bution exactly if l + l is divisible by 3.Write I = { , n , n , . . . } with 2 < n < n < . . . . We have to add thesummand (10.2) for each 1 ≤ l , l with l + l divisible by 3 (each such cor-responding to one class of hook passing through e ). To facilitate the effect of | I ∩ { , . . . , min { l , l }}| , we choose the disjoint decomposition of the ( l , l )-plane into subsets V k := U k \ U k +1 , where U k = { ( l , l ) | l , l ≥ n k } , such that | I ∩ { , . . . , min { l , l }}| = k on V k .We obtain (with convention n = 2)dim G I ker( A + 2) = 2 − ∞ X k =0 k X ( l , l ) ∈ V k l + l ≡ − l + l ) = 2 − ∞ X k =0 k X ( l , l ) ∈ U k l + l ≡ − l + l ) − X ( l , l ) ∈ U k +1 l + l ≡ − l + l ) . (10.3)Recall that all n k are congruent 2 modulo 3. We distinguish the cases l =3 r + r with r = 0 , , l = 3 r + 2 − r to get l + l ≡ U k X ( l , l ) ∈ U k l + l ≡ − l + l ) = X r =0 ∞ X r =0 − n k +3 r + r ) ∞ X r =0 − n k +3 r +2 − r ) = 3 · − n k +2 (1 − − ) − . Substituting this in (10.3) we getdim G I ker( A + 2) = 32 (1 − − ) ∞ X k =0 (cid:18) k − n k −
12 2 k +1 − n k +1 (cid:19) = 32 (1 − − ) − + 32 (1 − − ) ∞ X k =1 − n k + k . (10.4) losed manifolds with transcendental L -Betti numbers
11 Arbitrary real numbers as L -Betti numbersfor normal coverings Our main point about the explicit formulas for L -Betti numbers is two-fold: onthe one hand, we want to show that every positive real number is an L -Bettinumber. This is the goal of the current section.Secondly, we want to show that we get transcendental L -Betti numbers for universal coverings, which translates algebraically that we have to use finitelypresented groups. This will be done in the last sections.Now we show how, starting from the L -Betti numbers we explicitly obtainin Theorem 10.1, one can construct (again explicitly) more groups and elementsin their group rings to finally get the following theorems. For every real number r ≥ their is a finitely generated group Γ r , an l ∈ N and a r ∈ M l ( Z Γ r ) such that dim Γ r (ker( a r )) = r and from a dyadic expansion r = P λ j j with λ j ∈ { , } we obtain (in princi-ple) an “explicit” description of Γ r and a r .Moreover, there is a compact manifold M with a normal covering ˜ M (withcovering group Γ r ) such that b (2)3 ( ˜ M ; Γ r ) = r. To prove this from the previous constructions, we review a couple of con-structions for which we can control the L -Betti numbers in terms of L -Bettinumbers of the ingredients. Let Γ , Γ be two groups, l , l ∈ N and a j ∈ M l j ( Z [Γ j ]) for j = 1 , . Form the “block sum” a := a ⊕ a ∈ M l + l ( Z [Γ × Γ ]) , where we tacitly identify Γ j with its image in Γ := Γ × Γ and identify a j withits image under the induced map. Then dim Γ (ker( a )) = dim Γ (ker( a )) + dim Γ (ker( a )) . Proof.
This is well known and essentially clear. First of all, by the inductionprinciple (e.g. [15, Proposition 3.1]), dim Γ (ker( a j )) = dim Γ j (ker( a j )) for j =1 ,
2, where we think of a j either as living over Z [Γ] or over Z [Γ j ].Secondly, the kernel of a (as block sum) is the direct sum of the kernels of a and of a (in l (Γ) l + l ). As the von Neumann dimension is additive for directsums, the assertion follows. Let Γ , Γ be two groups, l , l ∈ N and a j ∈ M l j ( Z [Γ j ]) for j = 1 , . Assume that a and a are non-negative (if necessary, replace them by a ∗ j a j ). Form the “tensor sum” a := a ⊗ id + id ⊗ a ∈ M l · l ( Z [Γ ] ⊗ Z [Γ ]) , thinking of Z [Γ] = Z [Γ ] ⊗ Z [Γ ] acting on l (Γ × Γ ) = l (Γ ) ⊗ l (Γ ) . Then dim Γ (ker( a )) = dim Γ (ker( a )) · dim Γ (ker( a )) . Mika¨el Pichot, Thomas Schick, Andrzej Zuk
Proof.
This lemma is also well known and follows from the fact that in thissituation ker( a ) = ker( a ) ⊗ ker( a ). A detailed argument for a special case canbe found in the proof of [3, Theorem 4.1].For the sake of completeness, let us give a more explicit proof here. If p j isthe orthogonal projection onto ker( a j ) for j = 1 , N Γ, induced up from N Γ j ), we claim that in this situation p := p ⊗ p is theprojection onto the kernel of a . As ( a ⊗ id + id ⊗ a )( p ⊗ p ) = 0, the imageof p is contained in the kernel of a .Now, (1 − p ) ⊗ p + p ⊗ (1 − p ) + (1 − p ) ⊗ (1 − p ) is an orthogonaldecomposition of 1 − p . On the image of (1 − p ) ⊗ p , which is equal toim(1 − p ) ⊗ im( p ), a coincides with a ⊗ id which is > p ⊗ (1 − p ).On the image of (1 − p ) ⊗ (1 − p ) which coincides with im(1 − p ) ⊗ im(1 − p ), a coincides with a ⊗ id + id ⊗ a , and both summands are >
0. Altogether, onthe complement of im( p ) a > a ) = im( p ).Finally, we have to compute the Γ-trace of p . Let e , . . . , e l be the stan-dard basis vectors of l (Γ ) l and f , . . . , f l be the standard basis vectors of l (Γ ) l (the characteristic function of the neutral element in the correspondingcomponent).Then { e i ⊗ f j } i =1 ,...,l ; j =1 ,...,l will be the standard basis for l (Γ × Γ ) l · l .Consequentlytr Γ ( p ) = l X i =1 l X j =1 h p ⊗ p ( e i ⊗ e j ) , e i ⊗ e j i l (Γ ) ⊗ l (Γ ) = l X i =1 h p ( e i ) , e i i l (Γ ) · l X j =1 h p ( f j ) , f j i l (Γ ) = tr Γ ( p ) · tr Γ ( p )This proves the claim. Let U ⊂ R ≥ be a subset of the non-negative real numberswith the following properties(1) U is closed under multiplication with and addition of non-negative rationalnumbers;(2) U is additively closed: if r, s ∈ U then also r + s ∈ U ;(3) there are rational numbers a, q ∈ Q ≥ , q > and d ∈ N such that for everyincreasing sequence ≤ n < n < . . . the number a + q P ∞ k =1 k − dn k ∈ U .Then U = R ≥ .Proof. Choose m ∈ N such that b := 2 dm − q > a is a multiple of q . Adding therational number b − a and multiplying with the rational number 2 q − we seethat all real numbers of the form2 · dm + ∞ X k =1 k − dn k ∈ U ; 0 ≤ n < . . . . (11.5) losed manifolds with transcendental L -Betti numbers d by D := dm and using only sequences where each n k is a multipleof m , and multiplying with suitable powers of 2, we see that all real numbers ofthe form ∞ X k =0 k − Dn k ; 0 ≤ n < n < . . . (11.6)belong to U .Because U is closed under multiplication with non-negative rational numbersit suffices to show that U contains some non-empty open interval.Moreover, because U is additively closed and closed under multiplicationwith powers of 2, it suffices to show that U contains every real number of theform r = X n ∈ I − Dn ; I ⊂ N (11.7)since an arbitrary real number between 0 and 1 is a sum of at most D multiples(by 2 k with 0 ≤ k < d ) of numbers of the form (11.7).Fix therefore I ⊂ N . We now describe 2 D − numbers of the form (11.6) withsum equal to r .Instead of writing down the formulas, we describe the digits of these numbersin dyadic expansion. Note that the relevant feature of any number of the form(11.6) is that the consecutive digits occur at places which are multiples of D (asis true for r ), but each new digit shifted one further “to the left”.The first 2 D − dyadic digits of r each give one (the first) digit of the 2 D − numbers to be constructed. The next digit of r (the summand 2 − Dn v with v = 2 D − + 1), shifted by 2 D − to the right, gives the second digit of each ofthe r numbers to be constructed. Note that this is a summand of the form2 · − D ( n v +1) . Note also that the sum of these 2 D − summands is exactly2 − Dn v , i.e. the corresponding digit of r . The next two digits are used, shiftedby 2 D − in the first or last 2 D − , respectively, of our numbers to be constructed.The same reasoning as before shows that these summands have the right formand add up to the right digits of r . The next 4 digits, shifted by 2 D − , are usedin one quarter each, i.e. 2 D − , of our numbers to be constructed.We continue this construction inductively until we arrive at 2 D − digitswhich are not to be shifted at all. Then we cyclically continue this patterninductively.The result are by construction the 2 D − numbers, each of the form (11.6),which therefore belong to U and which add up to r .As explained above, this implies the assertion. Every non-negative real number is an L -Betti number of somecovering of a compact manifold.Proof. By a standard reduction, it suffices that for every r ∈ R ≥ there is afinitely generated group Γ, d ∈ N and A ∈ M n ( Z Γ) such that dim Γ (ker( A )) = r .However, the main result of this paper asserts that for I = { , n , n , . . . } with n = 2 < n < n < . . . all congruent 2 modulo 3 and for certain β , β ∈ Q > , whenever r = β + β ∞ X k =1 k · − n k d there is a finitely generated Γ r and a r ∈ Z [Γ r ] such that dim Γ r (ker( a r )) = r .0 Mika¨el Pichot, Thomas Schick, Andrzej Zuk
Using in addition Lemmas 11.2 and 11.3 the set of von Neumann dimensionsof kernels satisfies the assumptions of Proposition 11.4. The corollary follows.
12 Structure of the groups G I To show that there are also universal coverings with transcendental L -Bettinumbers—equivalently matrices over the group ring of a finitely presented groupwith transcendental L -dimension of the kernel, we have to analyze the groupsused in Theorem 10.1 more precisely.Recall from Definition 9.1 that, starting with Γ = h s , s i either free orΓ = Z ≀ Z and given a subset I ⊂ N , fixing F l = { s − , e, s } , we have groups G I := (cid:0) Z ⊕ Γ2 /V F l ,I (cid:1) ⋊ Γ . Consider the basis { δ g | g ∈ Γ } of Z ⊕ Γ2 = Z [Γ] where δ g is the characteristicfunction of { g } ⊂ Γ.Set u := P h ∈ F l δ h . Recall that V F l ,I is the Z [Γ]-submodule of Z [Γ] gen-erated by the elements of the form P h ∈ F l ( δ h − δ th ) = u − tu, t ∈ Λ I , so as Z -vector space it is generated by elements of the form X h ∈ F l ( δ gh − δ gth ) = gu − gtu, g ∈ Γ , t ∈ Λ I . (1) The subgroup V F l ,I is generated as Z [Γ] -module by theelements w g := gu − u = P h ∈ F l ( δ h − δ gh ) with g = t n , n ∈ I .(2) The translates of u by powers of s , i.e. { gu | g ∈ h s i} satisfy the followingproperty: if the support of P nk =1 s a k u with a < a < · · · < a n belongs to { s l , s l +11 , . . . , s r } then l + 1 ≤ a and a n ≤ r − .In particular, the gu with g ∈ h s i form a linearly independent subset ofthe vector space Z [ h s i ] ⊂ Z [Γ] .(3) Write y ∈ V F l ,I as sum y = P i g i ( t i u − u ) with g i ∈ Γ , t i ∈ Λ I and withminimal number of such summands.Fix a left coset g h s i of h s i and assume that the support of y (consideredas a function on Γ ) is contained in g { s l , s l +11 , s l +21 , . . . , s r } with l ≤ r .Then, in the (minimal) sum y = P i g i ( t i u − u ) , if g i or g i t i ∈ g h s i , thenthey already lie in g { s l +11 , . . . , s r − } .In particular, if the support of y does intersect the coset g h s i , then r − l ≥ . On the other hand, if the support of y does not intersect the coset g h s i ,then none of the g i t i and g i belongs to the coset g h s i .(4) If g ∈ Γ with w g = gu − u ∈ V F l ,I then g ∈ Λ I .Proof. By definition, V F l ,I is generated as Z [Γ]-module by the w g with g ∈ Λ I .However, w g + gw g ′ = X h ∈ F l ( δ h − δ gh + δ gh − δ gg ′ h ) = w gg ′ . As Λ I by definition is generated by { t n | n ∈ I } , (1) follows. losed manifolds with transcendental L -Betti numbers s is infinite cyclic, (2) is a well known statement about Z [ Z ]:if x := P nk =1 s a k u with a < · · · < a n , then the value of x at s a − is non-zero,so l ≤ a −
1, and similarly r ≥ a n + 1.To prove (3), fix an arbitrary element y = P i g i t i u − g i u as in (3). Considernow all the summands such that g i t i or g i ∈ g h s i . These are exactly thesummands in y contributing with one or two summands of the form gs a i u whosesupport is contained in g { s l , . . . , s r } ⊂ g h s i . By (2), all the summands with a i ≤ l have to appear pairwise to cancel each other out. But if we woulde.g. have g i = g j for i = j then we could write g i t i u − g i u + g j t j u − g j u =( g i t j )( t − j t i ) u − ( g i t j ) u with t − j t i ∈ Λ I and g i t j ∈ Γ, thus being able to write y with fewer summands, violating the minimality for the expression of y . Thesame reasoning rules out that g i t i = g j with i = j , where we obtain g i t i u − g i u + g j t j u − g j u = g i ( t − i t j ) u − g i u . Finally, terms with g i t i = g i by minimalityalso don’t appear.Similarly, we can rule out that for a minimal expression g i = gs a i or g i t i = gs a i with a i > r − g i t i or g i ∈ h s i then the summandsof the form gs a i u in y do not cancel and therefore y intersects the coset g h s i .To prove (4) observe that the support of w g does intersect only the cosets h s i and g h s i . Written with minimal number of summands as in (3) therefore gu − u = X t i s a i u − s a i u with t i ∈ Λ I (12.2)such that we have equal cosets t i h s i = g h s i . If g ∈ h s i , then Λ I containsa non-trivial power of s and therefore by Lemma 9.2 contains s , so g ∈ Λ I .Otherwise, Λ I ∩ h s i = { e } and by (2) and minimality, the above expression(12.2) for gu − u consists of exactly one summand gu − u = ts a u − s a u whichfinally implies a = 0 and g = t ∈ Λ I . The group G I has a recursive presentation if and only if I is recursively enumerable, i.e. if there is a Turing machine listing exactly allelements of I .Proof. Assume that I is recursively enumerable.Using Lemma 12.1, a presentation of G I is given by the generating set s , s , τ =: δ e with the following relations: • τ = 1 • g − τ g =: δ g commutes with h − τ h =: δ h for each g, h ∈ Γ. • Q x ∈ F l δ gx δ gt n x is trivial for each n ∈ I and each g ∈ Γ. • If Γ = Z ≀ Z , in addition we need the relations of this group: s n s s − n commutes with s for each n ∈ Z .As it is easy to list all elements of Γ, starting with the Turing machine for I wecan produce a Turing machine listing all these relations, i.e. this presentation isrecursive.Assume, on the other hand, that there is a Turing machine producing allthe relations in G I . In particular, it will list all the words w t n = Q x ∈ F l δ x δ t n x for the n ∈ I . Because the word problem in Γ and in Z [Γ] is solvable, we can2 Mika¨el Pichot, Thomas Schick, Andrzej Zuk recognize these words and determine the n ∈ I from them. On the other hand,by Lemma 12.1 if b / ∈ I (i.e. t b / ∈ Λ I ) then w t b / ∈ V F l ,I i.e. w t b and therefore b is not listed. In other words, this algorithm produces exactly the elements of I ,and hence I is recursively enumerable. The group G I does have solvable word problem if and only if I is recursive, i.e. there is a Turing machine listing the elements of I and anotherone listing those of the complement of I .Proof. Assume that G I has a solvable word problem. This means that, ifwe write down w t n = Q x ∈ F l δ x δ t n gx we can decide whether w t n = e or not,i.e. w t n ∈ V F l ,I or not. By Lemma 12.1 this means that we can decide whether t n ∈ Λ I or not, i.e. whether n ∈ I .Let us now suppose that I is recursive. There is a (computable) normal formfor each element of Z ⊕ Γ2 ⋊ Γ, written as the product of an element of Z ⊕ Γ2 and ofan element of Γ. It follows that, since Γ has solvable world problem, the wordproblem in G I is solvable if and only if it is solvable in the normal subgroup Z ⊕ Γ2 /V F l ,I .This is equivalent to solving whether an element x ∈ Z ⊕ Γ2 belongs to V F l ,I .This can be done as follows (provided I is recursive):The function x is finitely supported on Γ with values in Z . Consider, as inthe proof of Lemma 12.1, its restriction to the coset C := g h s i . Assume this re-striction is non-zero and form the (minimal) support interval { gs − , gs , . . . , gs d } as in Lemma 12.1 (we choose g in its coset appropriately). Because we can solvethe word problem in Γ, we can compute all these non-empty support intervalsfor the different cosets of h s i .By Lemma 12.1 if d ≤ x / ∈ V F l ,I .Otherwise we now check, using that I is recursive together with Lemma9.2, whether there is t ∈ Λ I such that gt is in the interior of another supportinterval, by checking whether g − g ′ ∈ Λ I h e i for the finitely many cosets g ′ h s i intersecting the support of x non-trivially.If this is not the case, then by Lemma 12.1 x / ∈ V F l ,I . Otherwise, subtract gu − gtu (which is an element of V F l ,I ) from x and continue as above.This decreases the sum of the lengths of the support intervals. Therefore,after finitely many steps, either we observe that x / ∈ V F l ,I or the support isempty, i.e. x ∈ V F l ,I .
13 Finitely presented groups
There is an explicitly given finitely presented group G andelement A ∈ Z [ G ] such that dim G ker( A ) is transcendental.Consequently, there is a compact manifold M such that an L -Betti numberof the universal covering is transcendental.Proof. In Theorem 10.1 we give an explicit construction of G and A such thatdim G ker( A ) = β + β P ∞ k =1 − dn k + k for every subset I = { n < n < . . . } ⊂ N .Moreover, if I is recursively enumerable, e.g. I = { k ! | k ∈ N } then thecorresponding group G has a recursive presentation by Theorem 12.3. If we usethis set I , then the resulting P ∞ k =1 − dk !+ k is transcendental as it is a Liouvillenumber [12]. losed manifolds with transcendental L -Betti numbers G by a finitely presented supergroup. To produce such a finitelypresented group which contains the recursively presented groups G , we use Hig-man’s theorem [9]. How to explicitly construct the supergroup and its presen-tation is shown nicely in [14, Chapter 12, p. 450 ff]. The finitely presented groups in Theorem 13.1 are obtained viaapplication of Higman’s embedding theorem. Unfortunately, although the re-cursively presented groups used as input for this theorem can be arranged tobe solvable, this can not be expected for the resulting finitely presented group(indeed, the method of proof will produce groups which contain non-abelian freesubgroups). Moreover, the construction in principle is explicit, but in practicethe finite presentation obtained will be extremely cumbersome.Some of the examples of Grabowski [5] are much more explicit and givesolvable (hence amenable) groups.
Using the method of proof of Proposition 11.4 one can obtainmany transcendental numbers which occur as L -Betti numbers of universalcoverings of manifolds, or equivalently as kernel-dimensions for elements in thegroup ring of finitely presented groups. In particular, one can obtain all numbersof the form P n ∈ I − n for a subset I ⊂ N which is recursive. In this case,moreover, we can arrange that the groups in question have a solvable wordproblem by Theorem 12.4. Grabowski obtains all numbers P n ∈ I − n where I is recursivelyenumerable. We obtain all P ∞ k =1 − dn k + k for I = { n < n < . . . } recursivelyenumerable. Itai and Dror Bar-Natan explained to us that these two classesof groups do not coincide . Variations of the constructions will yield yet othervalues.It is clear that there are all together only countably many possible L -Bettinumbers using the integral group ring of finitely presented groups (as the set ofisomorphism classes of these groups is countable).It is an open question how this set exactly looks like. In [8] it is implicitlydiscussed that for any L -Betti number r of the universal covering of a finiteCW-complex there is a Turing machine which produces a sequence of rationalnumbers whose limit superior is r .In [8], it is also shown that an L -Betti number obtained from a finitelypresented group with solvable word problem which is of L -determinant class(as introduced in [15], and satisfied for all the groups we constructed in thisarticle) is of the form P n ∈ I − n for a recursive subset I ⊂ N . Consequently,these are precisely the L -Betti numbers obtained with groups which have asolvable word problem and satisfy the determinant conjecture. The construction we have described here allows for many modi-fications. Essentially, we can make an operator A which accepts local patterns Let J = { n < n < . . . } be an infinite recursively enumerable, but not recursive set. Set I := { n k } which is then also recursively enumerable. But T I := { n k + k } is not recursivelyenumerable: otherwise, as k ≤ n k << n k one could recover from the 2 n k + k also k (and n k ). But the information that n k is the k -th smallest element of J allows us, by waiting until k − J are listed, to determine exactly the elements of J which are smallerthan n k and eventually to decide which numbers are in J and which are not —contradictingthat J is not recursive. Mika¨el Pichot, Thomas Schick, Andrzej Zuk in the Cayley graph of Γ. One interesting modification would be to only accept1-neighborhoods of hooks with a thickened neighborhood of the ends.Then one could replace in the definition of the quotient groups the set F l bya slightly larger set F = { e, s , s , s } . Its translates only fit into the relevantset (the 1-neighborhood of the hook with thickened ends) at the ends. Thisway one could arrange to have identifications in such subsets only if the twolegs of the hook have equal length, and to have exactly one identification in thiscase. Nothing else changes, but the final sum corresponding to the calculationof Theorem 10.1 gives β ′ + β ′ ∞ X k =1 − dn k . It is then easy to see that, using recursively presented groups, we can get allnumbers P ∞ k =1 − n k with I = { n < n < . . . } recursively enumerable. Con-sequently, these numbers are also obtained as L -Betti numbers of universalcoverings of compact manifolds. We can go even one step further with our modifications andinstead of hooks with two vertical legs work with hooks with left leg verticalas before, but right leg horizontal { g, gs , . . . , gs d } . Again, one looks at the1-neighborhood of such hooks, but with thickened ends.Finally, one uses F = { s − , s − , e, s , s , s , s − } in the form of a cross. Thentranslates of F fit only into our neighborhoods of the hook if they are placed inthe end.Instead of the subgroups Λ I one works with subgroups Λ ′ I generated by s n s n for n ∈ I . At least if Γ is free, this subgroup is free on these generators andhas appropriate properties corresponding to those of Λ I we used above, and theextension lemma works for the cross F (use the proof of [2, Lemma 5.5]).We can then arrange the local patterns at the two ends (which are locallydifferent: one is horizontal, the other vertical) to differ for those which contributeto the eigenvalue − not possible. It follows that,instead of an identification which increases the weight of the contribution to thespectrum, those paths where both legs have length n k ∈ I do not contribute atall. Carrying out the calculations, we obtain for I recursively enumerable an L -Betti number (for a recursively presented group) of the form β ′′ − β ′′ ∞ X k ∈ I − dk ; with β ′′ , β ′′ ∈ Q , d ∈ N . We haven’t checked, but expect, that the same works with Γ = Z ≀ Z . References [1] M. F. Atiyah. Elliptic operators, discrete groups and von Neumann alge-bras. In
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