This is a sequel to an earlier post on the Weinberger-Yu paper referenced below. Weinberger and Yu state their main conjecture as follows. Let \(G\) be a discrete group.
Conjecture 1.1. If \(\{g_1, · · · , g_n\}\) is a collection of non-identity elements in G with distinct
finite orders, then
(1) \(\{[p_{g_1}], · · · , [p_{g_n}]\}\) generates an abelian subgroup of \(K_0(C^*(G))\) having rank \(n\);
(2) any nonzero element in this abelian subgroup is not in the image of the assembly map \(\mu \colon K^G_0 (EG) \to K_0(C^∗(G))\), where \(EG\) is the universal space for proper and free \(G\)-actions.
Recall that, for \(g\in G\) of finite order \(n\), \(p_g\) is the projection in the group algebra defined by averaging the powers of \(g\), that is \(p_g = \frac{1}{n}\sum_{k=0}^{n-1}g^k\).
The authors then add: “In fact, we can state a stronger conjecture in terms of K-theory elements coming from finite subgroups and the number of conjugacy classes of nontrivial finite order elements. Such a stronger conjecture follows from the strong Novikov conjecture but would not survive inclusion into large groups.” In this post I want to expound this perhaps slightly mysterious paragraph.
The Baum-Connes assembly map (for groups with torsion) runs from the equivariant K-homology of the space \(\underline{E}G\), the universal space for proper \(G\)-actions, to the K-theory of the group \(C^*\)-algebra of \(G\). In low dimensions, it is well known that this map can be described by using a Chern character – see the papers of Baum-Connes and Matthey referenced below. In particular, Matthey’s theorem 1.1 includes a diagram of assembly map which incorporates an injective homomorphism
\[ \beta_i^{(t)}\colon H_i(G;FG) \to K_i^G(\underline{E}G)\otimes{\mathbb C} \]
for \(i=0,1,2\), where \(FG\) is the collection of finitely supported complex-valued functions on the finite order elements of \(G\), on which \(G\) acts by conjugation. In particular, \(H_0(G;FG) \) is simply the vector space spanned by the conjugacy classes of finite order elements. From the Baum-Connes conjecture (in fact, from the injectivity of the Baum-Connes assembly map) it would therefore follow that \(K_0(C^*(G))\otimes{\mathbb C}\) contains a summand of rank equal to the number of conjugacy classes of finite order elements of \(G\). This is, of course, at least equal to the number of distinct orders of finite order elements, since conjugate elements of \(G\) have the same order. Thus we would obtain part (i) of the authors’ conjecture (in a strengthened form) form the injectivity of the BC assembly map (which is what they are referring to as the ‘strong Novikov conjecture’ above). Their part (ii) would also follow from BC injectivity comparing the homological version of the LHS of the Baum-Connes assembly map with the corresponding homological version of the LHS of the ordinary assembly map (involving \(EG\) rather than \(\underline{E}G\) ).
Weinberger and Yu don’t stop at this point for two reasons, which I think are related.
(a) They want a conjecture which will “survive inclusion into large groups”. What they mean by this is that if \(G\) is a subgroup of some larger group \(G’\), they want a lower bound not just for the rank of \(K_0(C^*G)\) but also for \(K_0(C^*G’)\). Now “the number of conjugacy classes of finite order elements” does not behave monotonically under inclusion of subgroups – non-conjugate elements in \(G\) can become conjugate in \(G’\) and in fact if \(g_1,g_2\in G\) have the same order then there will always be an HNN extension \(G’\) in which they become conjugate – but “the number of distinct orders of finite order elements” obviously does behave monotonically in this situation.
(b) Related to this is the method of the authors’ proof of the conjecture which appears to involve embedding finite subsets of \(G\) in larger groups or spaces. The point is that by some kind of decomposition procedure, which incorporates the flexibility to increase the size of the group, one can prove Conjecture 1.1 even in some situations where the injectivity of the Baum-Connes map itself seems to be out of reach.
I’ll try to say more about the proof next time.
References
Baum, Paul, and Alain Connes. 1988. “Chern Character for Discrete Groups.” In A Fête of Topology, 163–232. Boston, MA: Academic Press.
Matthey, Michel. 2004. “The Baum–Connes Assembly Map, Delocalization and the Chern Character.” Advances in Mathematics 183 (2) (April 1): 316–379. doi:10.1016/S0001-8708(03)00090-2.
Weinberger, Shmuel, and Guoliang Yu. 2013. “Finite Part of Operator K-Theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-Rigidity of Manifolds”. ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744