# “Finite part of operator K-theory” III – repeat

(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.