“Finite part of operator K-theory” IV

Continuing this series (earlier posts here, here and here) on the paper of Weinberger and Yu, I’m expecting to make two more posts: this one, which will say something about the class of groups for which they can prove their Finite Part Conjecture, and one more, which will say something about what can be done with the conjecture once one knows it.

Section 3 of the paper contains a proof of the Finite Part Conjecture for groups that are finitely embedabble in Hilbert space.   (This means “finitely embeddable in the class $$\mathfrak H$$ of groups that are coarsely embeddable in Hilbert space, as in the previous post; finitely embeddable is a slight weakening of the notion locally embeddable that we used there.)  Given that the FPC is a local property of groups, its enough to prove it for groups coarsely  embeddable in Hilbert space.  Here one has a result from the old paper of Skandalis, Tu and Yu relating coarse Baum-Connes to groupoids.

Theorem (STY Theorem 6.1) If a countable group $$\Gamma$$ admits a coarse embedding into Hilbert space, then the Baum-Connes assembly map

$\require{AMSsymbols} K^\Gamma_*(\underline{E}\Gamma;A) \to K_*(A\rtimes\Gamma)$

is an injection for any separable $$\Gamma$$-$$C^*$$-algebra $$A$$.

The authors call this injectivity of the Baum-Connes assembly map the “Strong Novikov Conjecture”.  I indicated in post 2 an argument (based on the homological characterization of the Baum-Connes map in low degrees) that this formulation of the SNC implies the FPC.  The first part of Section 3 of the Weinberger-Yu paper contains a different (and more elaborate) proof of this implication going through cyclic cohomology, Schatten-class-coefficient group algebras, etc – in the end though boiling down to the simple trace argument we discussed in post 1. I haven’t yet figured out whether I believe that this more elaborate discussion is needed.   Anyhow, one way or the other, we find that coarsely embeddable groups satisfy FPC, and then “finitization” implies that finitely embeddable groups do so also.

By the way, the section of the paper where these calculations occur is titled “Applications”!  It does contain some applications, which I’ll get to in my next, but it also contains the main “meat” of the actual proof.

So which groups are finitely embeddable?  In particular, is “finitely embeddable” genuinely weaker than “coarsely embeddable”?  W-Y address these questions in Section 5 and give several examples.

Example 1 A residually finite group is FEH (finitely embeddable in Hilbert space).  Moreover, the same applies to an extension of a residually finite group by a torsion-free group.  In particular, if G is virtually torsion-free (that is, has a torsion-free subgroup of finite index) then it is FEH.

Example 2 The property FEH is preserved by suitable direct limits: those that are finitely injective, meaning that for any finite subset $$F$$ of one of the groups appearing in the direct limit sequence, the maps of the sequence, applied to the successive images of $$F$$, are ultimately injective.   (This would be implied by all the maps in the sequence being globally injective, but it is much weaker.)  In particular, Gromov’s examples of non coarsely embeddable groups are limits of this sort (where the individual groups in the sequence are hyperbolic and therefore coarsely embeddable).  This shows that FEH is a true generalization of coarse embeddability.

Example 3 Let $$9M,x_0)$$ be a connected analytic manifold and let $$G$$ be a group of analytic diffeomorphisms of $$M$$ preserving the base-point $$x_0$$. Then there are homomorphisms $$\phi_k$$ from $$G$$ to the (linear) groups of automorphisms of the $$k$$-jets at $$x_0$$; these groups are coarsely embeddable by the results of Guentner, Higson and Weinberger (on the Novikov conjecture for linear groups).  On any finite subset of $$G$$, the map $$\phi_k$$ is injective for $$k$$ large enough (because any finite set of distinct analytic maps is distinguished by finitely many jets).  Thus $$G$$ is FEH.

References

Guentner, Erik, Nigel Higson, and Shmuel Weinberger. 2005. “The Novikov Conjecture for Linear Groups.” Publications Mathématiques De l’Institut Des Hautes Études Scientifiques 101 (1) (June 1): 243–268. doi:10.1007/s10240-005-0030-5.

Skandalis, G., J.–L. Tu, and G. Yu. 2002. “Coarse Baum–Connes Conjecture and Groupoids.” Topology 41: 807–834.