A couple of additional remarks that I always forget, but that get to the heart of the matter.

First, in the positive direction, the work on

Baum-Connes does have topological implications.

E.g. Guoliang Yu’s work can be used to prove the Novikov conjecture for hyperbolic groups. I think

my earlier post may give the wrong impression on this point.

But on the negative side the connections just

aren’t so good. In particular, the Novikov

conjecture passes to direct limits of groups.

The construction of groups containing expander graphs constructs them as direct limits of hyperbolic groups. So these groups which might not satisfy

Baum-Connes are already known to satisfy Novikov.

Best,

David

]]>Thanks for the detailed clarification! It sounds like a topic which I will have to learn more about at some point. For now, I’ve reworded the text slightly (which, incidentally, was not a direct quote from Avi).

]]>Baum-Connes and topology. Saying this connection yields an “application of expanders to topology” is a bit of a stretch.

The Baum-Connes conjecture is a conjecture

concerning K-theory of group C^* algebras. It conjectures that a certain map is an isomorphism between two different K groups.

The injectivity of this Baum-Connes map implies the Novikov conjecture which is a conjecture about topological invariance of certain higher signatures. This implication is robust in that knowing Baum-Connes for a particular group implies Novikov for manifolds with that group as fundamental group.

There is, however, no reverse implication. So if

Baum-Connes is false for a group, we still have

no idea about Novikov.

By a theorem of Guoliang Yu, if a group admits a uniform embedding in a Hilbert space, the Baum-Connes map is injective and so the Novikov conjecture is true. There are other variants on

Yu’s result, but this is the result relevant here.

Gromov pointed out that if one could embed a

family of expander graphs into the Cayley graph

of the group, then the group would not admit a

uniform embedding in a Hilbert space. And proposed a method for constructing such a group.

(Actually the expander graphs are only embedded

quasi-isometrically, but this is enough.) To the best of my knowledge a complete and formal account justifying the existence of these groups is not yet in available. But I think a complete account is likely to be available soon, quite possibly multiple complete accounts.

Currently, no one knows if these groups actually provide a counter-example to Baum-Connes or not.

They do provide, indirectly, counterexamples to

some generalizations of the original Baum-Connes

conjecture. This is work of Higson, Lafforgue and Skandalis that has been in print since 2002. Though I am not an expert in this particular domain, I think

it is fair to say that serious effort has been devoted to seeing if Gromov’s groups are actually counterexamples to Baum-Connes. They might be,

but new ideas are almost certainly needed.

So currently expanders really only allow us to build some groups that obstruct one method of proving a big conjecture about group C^* algebras. Therefore obstructing one method of proving a big

conjecture about topology.

These conjectures are all hard and important so it

is worthwhile knowing how not to try to prove them. But, at the moment, there is no result about

topology, positive or negative, that comes out of this story.

So maybe it will be useful to give the references I found: the result (for finite-volume, non-compact surfaces) seems to be due to R. Brooks, in “On the angles between certain arithmetically defined subspaces of “, in Annales de lâ€™institut Fourier, tome 37, no 1 (1987), p. 175-185 (available on numdam.org), see Corollary, p. 182. Here the argument of Brooks is _very_ condensed, but there is a complete write-up in lecture notes of his entitled “The spectral geometry of Riemann surfaces”, Section 2 (those notes are available as Chapter 12 of “Topology in Molecular Biology”, published by Springer, or here).

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