Constructions of non -exact groups
- Coarse properties: graphs and groups.
- Graphical presentations: from graphs to groups.
- -exact groups containing expanders.
- -exact groups having Haagerup property.
These are complete normed algebra with a conjugation . Basic examples are matrices, and more generally bounded operators on a Hilbert space.
1.1. The reduced -algebra of a group
The group algebra of a group is not a -algebra since it is not complete. In fact, there are several useful completions. The simplest is obtained as follows. The left regular representation of gives rise to an injective -homomorphism into . The reduced -algebra of is the closure of the image of this homomorphism.
Example. , the continuous functions on a circle.
Definition 1 A -algebra is exact if the functor is exact, i.e. maps short exact sequences to short exact sequences.
The minimal tensor product is the comletion of the algebraic tensor product forthe following norm: maximize norm of images under all unitary representations of the form .
One says that a -algebra is nuclear if there is only one way to complete algebraic tensor products. It turns out that is nuclear iff is amenable.
Kirchberg and Philips 2000 showed that separable exact -algebras are exactly the subalgebras of nuclear -algebras. Therefore Kirchberg raised the question of which are the -exact groups, i.e. groups such that is exact.
Hyperbolic groups. Amenable groups.
1.4. Coarse geometry motivation
Theorem 2 (Many authors, Higson-Roe, Anantharaman-Delaroche-Renault, Kirchberg-Wassermann,…) is exact iff is coarsely amenable.
This class of groups has nice properties: coarse Baum-Connes conjecture holds, hence Novikov conjecture, Kadison-Kaplansky conjecture, Borel conjecture hold.
Definition 3 A graph is coarsely amenable if , , map such that
- , .
- edges ,
- uniformly bounded supports.
Amenable graphs (hence groups) have it. Condition is stable under taking subgraphs (hence subgroups).
Exercise. Regular trees are coarsely amenable.
2. Non-coarsely amenable graphs
Lemma 4 (Willett 2011) Let be a coarse disjoint union of finite connected graphs. Assume that
- Degrees of vertices are and bounded.
- The girth of tends to infinity.
Then is not coarsely amenable.
Indeed, suppose we have maps , i.e. functions with support in . Then
On the other hand,
Hence there exists such that
Then , is has support in . Girth assumption allows to replace with a tree. Normalize , get such that but . This sounds like amenability (in fact, it is stronger, go to level sets), and indeed cannot hold for trees, contradiction.
Beware. Trees are indeed coarsely amenable, so where is the cheat?
2.1. Scheme of construction
- Find a suitable graph (as in Willett’s theorem). Where?
- Coarsely embed it into a finitely generated group. How?
In 2003, Gromov gave the first construction of this kind. Ours will be simpler, really following the above scheme.
2.2. Box spaces
The suitable graphs will be box spaces, i.e. disjoint unions of Cayley graphs of finite quotients of a group , put very far apart.
Take as a free group. This ensures that quotients have increasing girth.
Hence we get lots of non-coarsely amenable graphs, by Willett’s theorem.
2.3. Embedding graphs into groups
It will turn out that one can embed suitable graphs quasi-isometrically into groups (Finn-Sell 2015).
Gromov’s examples are expanders. Ours are not.