2.4 The strong Atiyah conjecture
2.4.1 Consequences for dim U (G) and R KG
Even though we formulated the strong Atiyah conjecture as a condition on the possible values of von Neumann dimensions of finitely presentedKG-modules, we will see now that it has strong implications on the structure of the ∗-regular closure as well as topological consequences. It will also become evident that the restriction to finitely presented modules is redundant.
Recall from Definition 2.1.7 that RKG denotes the ∗-regular closure of KG in U(G).
The statement of the strong Atiyah conjecture can be extended to cover finitely presented RKG-modules:
Proposition 2.4.4. Let G be a group withlcm(G)<∞ and K ⩽C a field closed under complex conjugation. Then the strong Atiyah conjecture forGholds over K if and only if every finitely presented RKG-module N satisfies
dimU(G)(U(G)⊗RKG N)∈ 1 lcm(G)Z.
Proof. If M is a finitely presented KG-module, then N := RKG ⊗KG M is a finitely presentedRKG-module with
dimU(G)(U(G)⊗KGM) =dimU(G)(U(G)⊗RK N), which proves one implication.
We can apply Corollary 1.7.9 toKG⩽RKGand any finitely presentedRKG-moduleN to obtain two finitely presentedKG-modulesN+ andN−such that N⊕ RKG⊗KGN−∼= RKG⊗KGN+. Using the additivity of dimU(G), we compute that
dimU(G)(U(G)⊗RKGN)
=dimU(G)(U(G)⊗KGN+)−dimU(G)(U(G)⊗KGN−)
which is contained in lcm(G)1 Zsince the strong Atiyah conjecture for Gis assumed to hold overK.
The following lemma shows that the von Neumann dimension is faithful for projective RKG-modules:
Lemma 2.4.5. Let P be a projective RKG-module. If dimU(G)(U(G)⊗RKGP) = 0, then P = 0.
Proof. SinceU(G)⊗RKGP is a projectiveU(G)-module, we conclude from the assumption and faithfulness of the von Neumann dimension that U(G)⊗RKG P = 0. We now use that P is in particular flat as an RKG-module, which implies that the RKG-linear map RKG ⊗RKG P → UKG ⊗RKG P induced from the injective map RKG ,→ U(G) is again injective. The codomain of the map is trivial and the domain is isomorphic to P, thus P = 0.
The following proposition and its corollary connect the strong Atiyah conjecture to the ring-theoretic structure of RKG. The statement about semisimplicity is not known to hold for the division closure DKG assuming just the strong Atiyah conjecture.
Proposition 2.4.6. Let G be a group with lcm(G) <∞ and K ⩽C a field closed under complex conjugation. If the strong Atiyah conjecture for Gholds over K, the ringRKG is semisimple. Furthermore, the parameterssandni of its Artin–Wedderburn decomposition satisfy
Xs i=1
ni ⩽lcm(G).
Proof. We first assume that RKG is semisimple and prove the second statement. Let Mn1(D1)× · · · ×Mns(Ds) be the Artin–Wedderburn decomposition ofRKG. We denote by Si some choice of a minimal left ideal of Mni(Di) for every i= 1, . . . , s as in Proposi-tion 1.6.4 and obtain from this proposiProposi-tion that as RKG-modules
RKG ∼=S1n1 ⊕ · · · ⊕Ssns.
For everyi= 1, . . . , s, theRKG-moduleSiis finitely presented by Proposition 1.6.4 (b), and hence dimU(G)(U(G)⊗RKG Si)∈ lcm(G)1 Z as a consequence of Proposition 2.4.4. Fur-thermore, since Si is projective and non-trivial, we conclude from Lemma 2.4.5 that dimU(G)(U(G)⊗RKG Si)>0. All in all, we obtain that
1 =dimU(G)(U(G)⊗RKGRKG) = Xs i=1
nidimU(G)(U(G)⊗RKG Si)⩾ Ps
i=1ni
lcm(G), and thus Ps
i=1ni ⩽lcm(G).
We now return to the proof of the first statement. SinceRKG is von Neumann regular, every finitely presentedRKG-module is projective by Proposition 1.4.3 (d). Thus, the last paragraph in fact proves the more general statement that wheneverRKG contains a direct sum of non-trivial finitely presented RKG-submodules, then the number of summands is at most lcm(G) and in particular finite.
Now assume for the sake of contradiction that RKG is not semisimple. Then some ideal J of RKG is not a direct summand, see [Gri07, Proposition 3.1]. This is impossible for finitely generated ideals by Proposition 1.4.3 (c), so we find a non-finitely generated ideal J of RKG. By repeatedly adjoining an element not contained in Ji, we obtain a chain of finitely generated ideals Ji ofRKG, where i∈N, with strict inclusions:
{0}=J0 ⊊J1 ⊊J2 ⊊· · ·⊊J.
We use Proposition 1.4.3 (c) again to conclude that everyJi is a direct summand inJi+1, with the non-trivial finitely generated projective, and hence finitely presented, complement denoted byKi. But then⊕i∈NKi is an infinite direct sum of non-trivial finitely presented RKG-submodules of RKG and we have reached a contradiction.
The following corollary shows that the strong Atiyah conjecture for a torsion-free group implies the Kaplansky zero divisor conjecture. For the class of torsion-free amenable groups, the two conjectures are equivalent by [Lüc02, Lemma 10.16].
Corollary 2.4.7. Let G be a torsion-free group and K ⩽C a field closed under complex conjugation. Then the strong Atiyah conjecture for Gholds over K if and only if the ring RKG is a division ring. If this is the case, then DKG =RKG.
Proof. We first assume the strong Atiyah conjecture forGoverK and prove that DKG = RKG is a division ring. Since lcm(G) = 1, Proposition 2.4.6 implies thats= 1andn1 = 1 in the Artin–Wedderburn decomposition ofRKG. Hence,RKG is a division ring, which is in particular division closed in every overring. As it is also a subring ofU(G)and contains KG, we obtain
DKG =D(KG,U(G)) =D(KG,RKG) =R(KG,RKG) =RKG.
IfRKGis a division ring, everyRKG-module is free and thus has integral von Neumann dimension by addivity. We conclude from the easy direction of Proposition 2.4.4 that this implies the strong Atiyah conjecture for GoverK.
The restriction on the fieldKcan in fact be dropped if one replaces the∗-regular closure in Corollary 2.4.7 by the division closure. We refer the reader to [Lüc02, Lemma 10.39]
for the slightly more technical proof.
Theorem 2.4.8. LetGbe a torsion-free group andK⩽Ca field. Then the strong Atiyah conjecture for G holds over K if and only if the ringDKG is a division ring.
Since the subgroup lcm(G)1 Z of R is discrete, the finiteness assumption on the KG-module M in Definition 2.4.2 can in fact be dropped. The proof of this fact requires a special case of the following lemma:
Lemma 2.4.9. Subquotients of finitely generatedU(G)-modules have finite von Neumann dimension.
Proof. We first show that finitely generatedU(G)-modules have finite von Neumann dimen-sion. LetM be a finitely generatedU(G)-module and choose a surjection p:U(G)n→M for some n ∈ N. Since dimU(G) is additive on exact sequences, applying it to the short exact sequence
0→ker(p)→ U(G)n→M →0 shows that
∞> n=dimU(G)(U(G)n)
=dimU(G)(ker(p)) +dimU(G)(M)
⩾dimU(G)(M).
Now let N ⩽ M be a submodule of the finitely generated U(G)-module M. Using additivity again, we obtain that
∞>dimU(G)(M) =dimU(G)(N) +dimU(G)(M/N)⩾dimU(G)(N).
The same argument then shows that quotients of N have finite von Neumann dimension.
Proposition 2.4.10. Let G be a group with lcm(G) <∞ and K ⩽C a field. Then the strong Atiyah conjecture for G holds over K if and only if every (arbitrary)RKG-module N satisfies
dimU(G)(U(G)⊗RKG N)∈ 1
lcm(G)Z∪ {∞}.
Proof. IfM is an arbitrary finitely presentedKG-module, then its von Neumann dimen-sion dimU(G)(U(G)⊗KG M) is finite by Lemma 2.4.9. Hence, from the assumption on arbitrary RKG-modules, we obtain that
dimU(G)(U(G)⊗RKG RKG⊗KGM)∈ 1 lcm(G)Z, which confirms the strong Atiyah conjecture forG overK.
We now assume that the strong Atiyah conjecture for G holds over K and consider an arbitrary RKG-moduleN. By Proposition 2.4.6, our assumption implies that the ring RKG is semisimple. Consequently, Proposition 1.6.4 implies that N ∼= S1m1 ⊕ · · · ⊕Ssms for some fixed choice {S1, . . . , Ss} of a set of representatives for the isomorphism classes of simple RKG-modules and suitable cardinal numbersm1, . . . , ms.
By Proposition 1.6.4 (b), simpleRKG-modules are finitely presented. We can therefore apply Proposition 2.4.4 to each Si and get that
bi :=dimU(G)(U(G)⊗RKG Si)∈ 1 lcm(G)Z. Using the additivity and cofinality of dimU(G), we obtain that
dimU(G)(U(G)⊗RKGSimi) =dimU(G)((U(G)⊗RKG Si)⊕mi) =mi·bi,
wheremi·bi is understood to be 0 ifbi = 0,∞ifbi 6= 0and mi is an infinite cardinal, and the result of ordinary multiplication otherwise. With the additional convention that the sum of two cardinal numbers is infinite if any of the two summands is, we finally conclude that
dimU(G)(U(G)⊗RKGN) =m1·b1+· · ·+ms·bs∈ 1
lcm(G)Z∪ {∞}.
Although our formulation of the strong Atiyah conjecture is quite algebraic, its ver-sion with coefficients in Q can in fact be formulated equivalently in terms of topological invariants:
Theorem 2.4.11. The following statements are equivalent for a group G with lcm(G)<
∞:
(a) The strong Atiyah conjecture for Gholds over Q.
(b) For every finite free G-CW-complex X and every n∈N, the n-th L2-Betti number b(2)n (X) is contained in lcm(G)1 Z.
(c) For every G-space X and every n∈ N, the n-th L2-Betti number b(2)n (X) is either infinite or contained in lcm(G)1 Z.
Proof. (a)⇒(c): Given thatRQG is a von Neumann regular ring, U(G) is flat as anRQG -module by Proposition 1.4.3 (e). We thus obtain that
Hn(U(G)⊗ZGC∗sing(X))
∼=Hn(U(G)⊗RQGRQG⊗ZGC∗sing(X))
∼=U(G)⊗RQGHn(RQG⊗ZGC∗sing(X)),
which in particular implies that
b(2)n (X) =dimU(G) Hn(U(G)⊗ZGC∗sing(X))
=dimU(G) U(G)⊗RQGHn(RQG⊗ZGC∗sing(X)) .
Given our assumption that the strong Atiyah conjecture holds for G over Q, Proposi-tion 2.4.10 allows us to conclude that the last term is either infinite or contained in
1 lcm(G)Z.
(c)⇒(b): IfX is aG-CW-complex, we have
Hn(U(G)⊗QGC∗(X))∼=Hn(U(G)⊗QGC∗cell(X))
by [Lüc98, Lemma 4.2]. The latterU(G)-module is obtained as a subquotient of the finitely generatedU(G)-moduleU(G)βn(X), whereβn(X)denotes the number of equivariantn-cells ofX, and thus has finite von Neumann dimension by Lemma 2.4.9.
(b)⇒(a): This is implied by [Lüc02, Lemma 10.5] since every free G-CW-complex is automatically proper and a G-CW-complex is cocompact if and only if it is finite.
Corollary 2.4.12. Let G be a group with lcm(G) <∞. If the strong Atiyah conjecture for G holds over Q, then Atiyah’s question has a positive answer, that is, the L2-Betti numbers b(2)n (X) are rational (or infinite) for every G-spaceX.