The following diagram visualizes the implications and equivalences between the various variants of the Atiyah conjecture. Variants and implications marked with an asterisk next to their reference are introduced or proved, respectively, in this thesis.
Center-valued Atiyah conjecture (2.5.1) dimuU(G)(U(G)⊗KGM)∈LK(G)
Strong Atiyah conjecture (2.4.2) dimU(G)(U(G)⊗KGM)∈ lcm(G)1 Z
Weak Atiyah conjecture (2.3.2) dimU(G)(U(G)⊗KGM)∈Q
Algebraic Atiyah conjecture (3.1.1)L K0(KF)→K0(RKG)
Rank of K0(RKG) (3.3.8*) rkZ(K0(RKG)) =|conK(G)f,cf|
Rationalized algebraic Atiyah conjecture (3.1.5*) LK0(KF)⊗ZQ→K0(RKG)⊗ZQ (3.1.4*)
(2.5.2) (3.3.8*)
(3.3.8*)
(3.1.6*)
The structure of the ring R KG
We have seen in the previous chapter that the structure of the ∗-regular closure RKG is determined in large parts by suitable variants of the Atiyah conjectures. If G is torsion-free, than the story ends here, with RKG being a division ring. However, if G contains non-trivial torsion, there is more to say about this ring.
The aim of the current chapter is to further studyRKG and its zerothK-group in the presence of torsion. In Section 3.1, we discuss another variant of the Atiyah conjecture, the algebraic Atiyah conjecture, that has been introduced by Jaikin-Zapirain and is formulated entirely in terms of the K-theory of RKG and the group rings of finite subgroups of G.
Relying on results of Knebusch, Linnell, and Schick, we prove that this conjecture is equivalent to the center-valued Atiyah conjecture. Even though it is not new from this point of view, its K-theoretic formulation turns out to be rather useful when studying inheritance properties. In Section 3.2, we thus benefit from using the algebraic Atiyah conjecture in an analysis of center-valued Atiyah conjecture’s behavior under a change of the coefficient field. Based on a base change result of Jaikin-Zapirain for the strong Atiyah conjecture for sofic groups, we prove that the center-valued Atiyah conjecture for such groups over Q implies that over any field with sufficiently many roots of unity or transcendental extensions thereof.
In the second part of the chapter, we turn to structural results on RKG that do not depend on the strong Atiyah conjecture. Building on work of Lück, we show in Section 3.3 that the rank ofK0(RKG)always admits a lower bound in terms of generalized conjugacy classes of elements of finite order in G. This lower bound matches the rank predicted by the center-valued Atiyah conjecture. Finally, in Section 3.4, we discuss an open question of Handelman about the unit-regularity of ∗-regular rings for the specific case of RKG. We prove that for a sofic group G, the ringRKG is always unit-regular if K has infinite transcendence degree overQ, thereby providing a partial answer to a question by Ara and Goodearl.
3.1 The algebraic Atiyah conjecture
In his survey article [Jai19b], Jaikin-Zapirain introduces the following variant of the Atiyah conjecture:
Definition 3.1.1 ([Jai19a, Conjecture 6.2]). Let G be a group and K ⩽ C a subfield closed under complex conjugation. We say that the algebraic Atiyah conjecture for G holds over K if the map M
F⩽G
|F|<∞
K0(KF)→K0(RKG) is surjective. We call this map the algebraic Atiyah map.
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By the universal property of the colimit, the algebraic Atiyah map factors as M
F⩽G
|F|<∞
K0(KF)↠colim
F⩽G
|F|<∞
K0(KF)→K0(KG)→K0(RKG),
where the first map is always surjective. It follows that the algebraic Atiyah conjecture for Gholds over K if and only if the composition of the second and third map is surjective.
Remark3.1.2. If the groupGsatisfies theK-theoretic Farrell–Jones conjecture, the second map above is an isomorphism. In this case, the algebraic Atiyah conjecture for G over K is thus equivalent to the surjectivity of the map K0(KG) → K0(RKG). The latter condition notably no longer directly involves the finite subgroups of G. In this sense, it can be viewed as a generalization of Corollary 2.4.7 to groups with torsion.
Lemma 3.1.3. Let G be a group andK ⩽Ca subfield closed under complex conjugation.
Then DKG is a∗-subring ofU(G).
Proof. Consider the subringsRi⩽U(G)defined in Remark 1.7.7 in the situationR:=KG and S := U(G). Since K is closed under complex conjugation, the ring R0 = KG is a
∗-subring of U(G). Now assume that Ri is a ∗-subring of U(G) for some i⩾0. The set Ui := {x−1 | x ∈ Ri, x ∈ U(G)×} is closed under ∗ since the anti-automorphism maps units to units. Thus, also the ring Ri+1, which is generated byRi and Ui, is a ∗-subring of U(G). By induction, all subrings Ri ⩽ U(G) are ∗-subrings, and therefore also their directed unionDKG.
The following theorem answers a question of Jaikin-Zapirain raised in [Jai19a, 6.1].
Theorem 3.1.4. Let G be a group with lcm(G)<∞ and K⩽C a subfield closed under complex conjugation. The algebraic Atiyah conjecture for G holds over K if and only if the center-valued Atiyah conjecture for G holds overK.
Proof. Assume that G satisfies the algebraic Atiyah conjecture. We adapt the proof of [KLS17, Theorem 3.7] to use RKG instead of DKG and K-theory instead ofG-theory.
Let M be a finitely presented KG-module. The RKG-module M := RKG ⊗KG M is again finitely presented and, since RKG is von Neumann regular, also projective by Proposition 1.4.3 (d). It thus represents a class in K0(RKG), which by assumption is an integer linear combination of classes in K0(KFi) for finitely many finite subgroups Fi ⩽ G, i = 1, . . . , k. Recall that KF is semisimple for a finite group F, which implies that K0(KF) is generated by classes that are represented by idempotents inKF. Using the assumption, we thus deduce that there are idempotents x+i , x−i ∈ KFi, i = 1, . . . , k, which may be0 or1and where the same finite subgroup can appear multiple times, such that
M⊕ Mk
i=1
RKGx−i ∼= Mk
i=1
RKGx+i .
By inducing up further to U(G) and applying the dimension function dimuG for finitely presentedU(G)-modules, we obtain
dimuU(G)(U(G)⊗KGM) =dimuU(G)
U(G)⊗RKGM
=dimuU(G) Mk
i=1
U(G)x+i
−dimuU(G) Mk
i=1
U(G)x−i
= Xk i=1
dimuU(G)(U(G)x+i )− Xk
i=1
dimuU(G)(U(G)x−i )∈LK(G).
Now assume that G satisfies the center-valued Atiyah conjecture over K. By Theo-rem 2.5.3 (c), DKG is semisimple and in particular von Neumann regular. Since K ⩽C is closed under complex conjugation, Lemma 3.1.3 implies that it is even ∗-regular. But KG ⩽DKG ⩽ RKG ⩽ U(G) by construction and RKG is the smallest ∗-regular subring of U(G) containingKG, hence DKG =RKG and the algebraic Atiyah conjecture forGis implied by Theorem 2.5.3 (c).
Given its purely K-theoretic formulation, the algebraic Atiyah conjecture lends itself to being considered rationally:
Definition 3.1.5. Let G be a group and K ⩽C a subfield closed under complex conju-gation. We say that therationalized algebraic Atiyah conjecture for Gholds over K if the
map M
F⩽G
|F|<∞
K0(KF)⊗ZQ→K0(RKG)⊗ZQ
is surjective. We call this map the rationalized algebraic Atiyah map.
Theorem 3.1.6. LetGbe a group andK ⩽Ca subfield closed under complex conjugation.
Then the rationalized algebraic Atiyah conjecture for G over K implies the weak Atiyah conjecture for Gover K.
Proof. LetM be a finitely presented KG-module. As in the proof of Theorem 3.1.4, the RKG-module M :=RKG⊗KGM represents a class in K0(RKG). Since we assumed the algebraic Atiyah map to be rationally surjective, we can find n ∈ Z such that n[M] ∈ K0(RKG) lies in its image. As in the proof of Theorem 3.1.4, we thus find finitely many finite subgroupsFi ⩽Gand idempotents x−i , x+i ∈KFi, wherei= 1, . . . , k, such that
Mn⊕ Mk
i=1
RKGx−i ∼= Mk
i=1
RKGx+i . We now apply the classical von Neumann dimension and obtain
n·dimU(G)(U(G)⊗KGM)
=dimU(G)(U(G)⊗KGMn)
=dimU(G)(U(G)⊗RKG Mn)
=dimU(G) Mk
i=1
U(G)x+i
−dimU(G) Mk
i=1
U(G)x−i
= Xk
i=1
dimU(G)(U(G)x+i )− Xk i=1
dimU(G)(U(G)x−i )
= Xk
i=1
dimU(G)(U(G)⊗KFiKFix+i )− Xk i=1
dimU(G)(U(G)⊗KFiKFix−i ).
Since finite groups satisfy the strong Atiyah conjecture over K, we conclude that dimU(G)(U(G)⊗KGM)∈ 1
n·lcm{|Fi| |i= 1, . . . , k}Z. In particular, it is a rational number.