Approximating l 2 -Betti numbers of an amenable covering by ordinary Betti numbers

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1 Comment. Math. Helv. 74 (1999) /99/ $ /0 c 1999 Birkhäuser Verlag, Basel Commentarii Mathematici Helvetici Aroximating l 2 -Betti numbers of an amenable covering by ordinary Betti numbers Beno Eckmann Abstract. It is shown that the l 2 -Betti numbers of an amenable covering of a finite cell-comlex can be aroximated by ordinary Betti numbers of the finite Følner subcomlexes. This is a new roof, using simle homological arguments, of a recent result of Dodziuk and Mathai. Mathematics Subject Classification (1991). 20, 55, 57. Keywords. Amenable grous, covering saces, l 2 -homology, Betti numbers. 0. Introduction Let Y be an infinite amenable covering of a finite cell-comlex X with covering transformation grou G. Thenthe l 2 -Betti numbers β (Y ) can be aroximated by the average ordinary Betti numbers of the finite subcomlexes Y j of a Følner exhaustion of Y. This has been roved by Dodziuk and Mathai [D-M]. The urose of the resent aer is to give a simle homological roof of that result. It consists in examining the l 2 -homology ma H (Y j ) H (Y ) induced by the inclusion Y j Y. 1. Følner sequence 1.1. We consider a discrete infinite amenable grou G and a free cocomact G-sace Y. By this we mean a cell comlex Y on which G oerates freely by ermutation of the cells, with finite orbit comlex X = Y/G. Then Y is a covering of X with covering transformation grou G. Since G is a factor grou of the fundamental grou of X, andxis a finite comlex, G is necessarily finitely generated. In short Y is called an infinite amenable covering of X It is known (Cheeger-Gromov [C-G], see also [E] or [D-M]) that in such a situation there exists in Y a Følner sequence (or Følner exhaustion) Y j,j=1,2,3,...

2 Vol. 74 (1999) Aroximating l 2 -Betti numbers 151 Here is its descrition in the form we will need later. For each closed -cell σ in X we choose an arbitrary lift ˆσ in the corresonding G-orbit. The union of all ˆσ, 0, together with its toological closure ( i.e. adding if necessary boundary cells of the ˆσ ) is a closed fundamental domain D for the G-action in Y. The Y j form an increasing sequence of finite subcomlexes of Y with union Y ; each Y j is a union of N j distinct translates x ν D, ν =1,2,..., N j,x ν G, of D. Let further Ẏj be the toological boundary of Y j and Ṅj the number of translates of D which meet Ẏj. From the combinatorial Følner criterion [F] for amenability it follows easily that the sequence Y j can be chosen such that N j /N j 0forj. 2. l 2 -chains, restricted trace 2.1. The cellular -chains of Y with R coefficients constitute a free RG module C (Y ); as basis we can take the lifts (see 1.2) ˆσ i of the -cells σi of X, i = 1, 2,..., α, where α is the number of -cells of X. Each -cell of Y can be uniquely written as xˆσ,x G, i i =1,..., α, and in each orbit the G-action is by left translation As Y is an infinite comlex, one considers besides the ordinary -chains also l 2 -chains, i.e. square-summable real linear combinations of the cells of Y. They constitute a Hilbert sace C (2) (Y ) where all the cells xˆσ i as above form an orthonormal basis. We sometimes omit Y and simly write C (2). The induced action of G on C (2) is isometric For any Hilbert subsace H of C (2), not necessarily G-invariant, there is the orthogonal rojection Φ: C (2) C (2) with image H. We consider the following restricted trace of Φ referring to a finite subcomlex Y j of Y consisting of N j translates of the fundamental domain D. Here amenability is not required; it is in 3.4 only that Y j will refer to a Følner sequence in Y. Let Π j be the rojection C (2) C (2) with image C (2) (Y j ). Since Y j is a finite comlex, we have C (2) (Y j )=C (Y j ); thus Π j is rojection on a finite dimensional R-subsace of C (2) form the R-trace whose basis consists of all cells x ν ˆσ i with ν N j. One can d j (H) =trace R Π j Φ It will be examined for some secial subsaces H. Note that it can be exressed

3 152 B. Eckmann CMH by scalar roducts in C (2) as α N j d j (H) = < Φ(x ν ˆσ i ),x νˆσ i >+ <Φ(τ ),τ >. τ i=1 ν=1 where the τ are cells in Ẏj not of the form x ν ˆσ i Proerties of d j : 1) Since Φ is idemotent and self-adjoint, the scalar roducts above are equal to < Φ(x ν ˆσ i ), Φ(x ν ˆσ i ) > and < Φ(τ ), Φ(τ ) > resectively and thus 0: The restricted trace d j (H) isnon-negative. 2) Note that one always has d j (H) dim R Π j (H) since tr R (Π j Φ) Π j Φ dim R im(π j Φ) dim R Π j (H). If in articular H is a subsace of C (Y j )thend j is the same as the trace of the rojection of C (Y j )toh. Since these are finite-dimensional vector saces, the trace is = dim R H. 3) If H decomoses orthogonally into H 1 + H 2 then d j (H) =d j (H 1 )+d j (H 2 ). Just note that then Φ = φ 1 + φ 2 where φ i is the rojection onto H i,i=1,2and relace Φ in the scalar roducts above. 4) In case H is G-invariant the rojection Φ is G-equivariant and < Φ(x ν ˆσ i ), x ν ˆσ i > is equal to < Φ(ˆσi ), ˆσi >. But Σα i=1 < Φ(ˆσi ), ˆσi > is just the von Neumann dimension dim G H (see e.g. [L] or [E2]). Thus in that case d j (H) =N j dim G H lus an error term T j coming from the boundary cells τ which is dim R C (Ẏj). 3. Maing H (Y j ) into H (Y ) 3.1. In the following, homology H is to be understood as reduced l 2 -homology (cycles modulo the closure of boundaries). It can be reresented by the orthogonal comlement of the sace of boundaries in the -cycle sace, i.e. by harmonic chains (boundary = 0 and coboundary δ = 0). In this sense we will consider H (Y )as a Hilbert subsace of C (2) (Y )andh (Y j ) as a subsace of C (Y j ) Since the boundary oerator in C (2) commutes with the G-action, the homology grou H (Y ) considered as a subsace of C (2) is G-invariant. According to 2.4, 4)wehave d j (H (Y)) = N j dim G H (Y )+T j =N j β (Yrel.G)+T j,

4 Vol. 74 (1999) Aroximating l 2 -Betti numbers 153 where β denotes the l 2 -Betti number and T j is the error term from 2.4,4). As for H (Y j ), we have by 2.4, 2) the ordinary -th Betti number of Y j. d j (H (Y j )) = dim R H (Y j )=β (Y j ), 3.3. The inclusion of Y j in Y induces a bounded linear ma φ : H (Y j ) H (Y ). Let K be the kernel of φ, andk its orthogonal comlement in H (Y j ); and I the image of φ, andi its orthogonal comlement in H (Y ). We will look closer at these harmonic subsaces of C (Y j )andc (2) (Y ) resectively in order to get estimates for the values of d j. We recall that commutes with the inclusion of Y j in Y but in general not with the the restriction of Y to Y j, and that for δ things are the other way around. In articular a harmonic chain in Y j need not be harmonic in Y, but can be made harmonic by adding a well-defined element of the closure of boundaries We decomose the -chains c C (2) as c = ċ + c where all -cells of ċ intersect the toological boundary Ẏj and c does not contain any such cell. This yields an orthogonal decomosition of C (2) into Ċ and C. Wenowusethe amenability of the covering and assume that Y j is a term of the Følner sequence. Then dim R Ċ N j α. 1) If c K, with c = δc =0inY j,thenc C (2) +1 (Y ). If we assume ċ =0, c =c C, then δ commutes with the inclusion, i.e. δc =0inY. But since cocycles are orthogonal to the closure of the sace of boundaries, it follows that c =0. ThusK C =0,andK is isomorhic to a subsace of Ċ. Therefore d j (K )=dim R K dim R Ċ Ṅjα. 2) As for d j (I ) it does not exceed dim RR where R =res j I and res j is the restriction from Y to Y j. The chains c I fulfill c = δc = 0. Moreover <c,z>=0forall-cycles z in Y j since φ(z) =z+b, with b C (2) +1.Forr R the same holds excet ossibly for r =0. Butifr=ċ+c as above, and if we assume ċ =0then r = 0. From <r,z>=0forall-cycles z in Y j it follows that r is a coboundary in Y j, r = δs. Thus<r,r>=<r,δs>=< r,s>= 0, whence r =0andR C = 0. As before this imlies dim R R Ṅjα and we get d j (I ) dim RR N j α K is isomorhic as a (finite-dimensional) vector sace to I.Theird j need not be equal, but we show that their difference fulfills an inequality similar to

5 154 B. Eckmann CMH those above. The isomorhism is given by adding to each c K a well defined element b(c) C (2) +1 (Y ), in order to get a harmonic chain in Y. If, in articular, c K C then δc =0inY, whence c I. Thus K C is a subsace of I which remains unchanged under Π j. This imlies that d j (I ) d j (K C )= dim R K C and dim R K d j (I ) dim R K /K C. But K /K C is isomorhic to (K + C )/C which is contained in C (2) isomorhic to Ċ. Thus its dimension is N j α whence d j (K ) d j (I ) Ṅjα. /C 3.6. Finally we have β (Y j ) N j β (Y rel.g) =d j (H (Y j )) d j (H (Y )) + T j = d j (K ) d j (I )+(d j(k ) d j(i )) + T j where T j is the error term in 2.4. By3.4 and 3.5 and since T j N j α this yields which goes to 0 with j.thus 1 N j β (Y j ) β (Y rel.g) 4α N j N j lim j 1 N j β (Y j )=β (Yrel.G). This is the aroximation statement mentioned in the introduction. References [C-G] J. Cheeger and M. Gromov, L 2 -cohomology and grou cohomology, Toology 25 (1986), [D-M] Jozef Dodziuk and Varghese Mathai, Aroximating L 2 -invariants of amenable covering saces: a combinatorial aroach, Prerint. [E] B. Eckmann, Amenable grous and Euler characteristic, Comment. Math. Helv. 67 (1992), [E2] B. Eckmann, Projective and Hilbert modules over grou algebras, and finitely dominated saces, Comment. Math. Helv. 71 (1996), [F] E. Følner, On grous with full Banach mean value, Math. Scand. 3 (1995), [L] W. Lück, Aroximating L 2 -invariants by their finite-dimensional analogues, GAFA 4 (1994),

6 Vol. 74 (1999) Aroximating l 2 -Betti numbers 155 Beno Eckmann Forschungsinstitut für Mathematik Eidg. Technische Hochschule 8092 Zürich Switzerland (Received: February 2, 1998)