(I posted this yesterday but it seems to have vanished into the ether – I am trying again.)

This series of posts addresses the preprint “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) by Guoliang Yu and Shmuel Weinberger. In my previous post I gave the description of their main conjecture (let’s call it the Finite Part Conjecture) and showed how it would follow from the Baum-Connes conjecture (or, simply, from the statement that the Baum-Connes assembly map was an injection).

The point of the paper, though, is that the Finite Part Conjecture can be verified for many groups for which the Baum-Connes conjecture is presently out of reach (and including some for which versions of the Baum-Connes conjecture are known to fail). In particular there is a “finitization” argument, similar to those deployed by Yu in other contexts, which requires only “local” information on the given group.

**Definition:** Let \(\mathfrak H\) be some class of (discrete) groups (e.g. the class of finite groups, or the class of groups coarsely embeddable into Hilbert space). A group \(G\) *belongs locally to \(\mathfrak H\)* if, for each finite subset \(F \subseteq G\), there are a group \(H\in {\mathfrak H}\) and an injective map \(\phi\colon F\to H\) which is *homomorphic* in the sense that

\[ \phi(g_1)\phi(g_2)=\phi(g_1g_2) \]

whenever \(g_1\), \(g_2\) and \(g_1g_2\) all belong to \(F\).

**Theorem** (Weinberger-Yu) The Finite Part Conjecture is a local property of groups. In other words, if \(\mathfrak H\) is a class of groups for which the Finite Part Conjecture is true, and \(G\) belongs to \(\mathfrak H\) locally, then the Finite Part Conjecture is true for \(G\).

*Proof*. Find an exhausting sequence \(F_k\) of finite subsets of \(G\), closed under formation of inverses, and homomorphic injections \(\phi_k\colon G\to H_k\) where \(H_k\in{\mathfrak H}\). One can arrange that \(\phi_k\) preserves the order of all finite order elements (make it homomorphic and injective on the larger finite set which contains \(F_k\) together with all the powers of its finite order elements.) Any element of the group ring \( {\mathbb C}G\) is spanned by \(F_k\) for \(k\) large enough, and this gives a linear map

\[ \Phi\colon {\mathbb C} G \to A:= \left(\prod_{k=1}^\infty C^*(H_k)\right)\Big/ \left(\bigoplus_{k=1}^\infty C^*(H_k)\right) \]

which is in fact a \(*\)-homomorphism and thus passes to the maximal \(C^*\)-algebra of \(G\). Now use the induced map on K-theory,

\[\Phi_*\colon K_0(C^*(G)) \to K_0(A) \]

to detect large subgroups of \(K_0(C^*(G))\); the K-theory of \(A\) can be estimated using the obvious six term exact sequence together with the fact that all the groups \(H_k\) satisfy the Finite Part Conjecture.

In my next posts I’ll try to explain how the authors use this idea to prove the Finite Part Conjecture for a large class of groups.