“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.

 

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