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Embeddings into Hilbert spaces, and -algebras
The leading idea is that -algebras provide a useful tool in the study of purely metric problems. We shall not follow the history which went in the opposite direction: a topological problem (Novikov conjecture) was cast in a -algebraic framework (Baum-Connes conjecture), and it was discovered that purely metric issues were relevant for the -algebra problem.
Plan of the lectures
- Groups as metric spaces
- Coarse embeddings into Hilbert spaces
- Novikov conjecture
- Yu’s Property (A)
- -algebras associated with groups and metric spaces
- -algebraic characterizations of Property (A)
1. Groups as metric spaces
Topology urges us to understand finitely presented groups, i.e. fundamental groups of compact manifolds or simplicial complexes. To a large extent, the notions we shall encounter only depend on metrics carried by these groups.
1.1. Word length
is a finitely generated group, a finite symmetric generating set, . For define its word length
Then defines a left invariant distance on .
Example 1 free abelian group, ().
Then distance coincides with -norm.
Example 2 free group on generators , ().
Say a word is reduced on the alphabet if obvious simplifications or never occur. Then is the set of reduced words, with group law given by concatenation followed by reduction. By definition, is the number of letters needed to spell out .
Example 3 discrete Heisenberg group, ().
is the set of upper triangular integer matrices with ‘s on the diagonal,
Every group element can be uniquely written in the form , . Nevertheless, as a group and as a metric space, is very different from . For instance, , since .
Exercise. Check that an other choice of finite generating system leads to an equivalent metric, i.e. there exists a constant such that
1.2. Cayley graphs
The mysterious distance can be viewed as the length metric induced by a graph having as a vertex set.
Definition 1 Let be a group, a finite symmetric subset. The Cayley graph is the undirected graph with set of vertices , set of edges
In other words, two vertices , are adjacent iff .
Proposition 2 1. is vertex-transitive. In fact, acts simply transitively on vertices (on the left).
2. is -regular (every vertex has neighbours).
3. is connected iff generates .
Proof: A vertex is connected to the unit element iff is a product of finitely many elements of .
From now on, we assume that is generating. Then equals the distance between and as vertices in .
Example 4 , . Then is the infinite simplicial line.
Example 5 , . Then is the infinite square grid in the plane.
Example 6 , the infinite dihedral group, i.e. the group of isometries of the simplicial line. Let where is the reflection across and is the reflection across . Then is isomorphic to the simplicial line.
In a row, group elements are .
Example 7 , . Then is the -regular tree.
1.3. Amenability for graphs
Here is our first example of a property of groups which only depends on its metric. We first define it for graphs.
Let be a locally finite, connected graph. For a finite set of vertices, let
Definition 3 The Cheeger or isoperimetric constant of is
measures the quality of viewed as a communication network. means that every finite set of emitters can transmit information to a proportion of channels at least .
Definition 4 Say that is amenable if either is finite or .
Example 8 (grid in Euclidean -space). Then .
Indeed, let be a subcube of size . Then , .
Example 9 , the -regular tree. Then .
Proof: First assume that is a finite subtree. Let us prove by induction on that . This is obvious when . Induction step: add an edge to to get , with , . Then .
If is an arbitrary finite subset of , let be its connected components. Then
Exercise. Show that .
Solution. Let denote the sphere of integer radius and be the ball of radius . Then , and
thus . Letting tend to implies that .
1.4. Amenability for groups
Definition 5 (F\o lner, 1950’s). Let be a finitely generated group. Say is amenable if , finite, there exists a finite subset such that
Such sets are called -F\o lner sets.
Theorem 6 Assume is finitely generated. Then the following are equivalent.
i) is amenable.
ii) Every Cayley graph of is amenable.
iii) Some Cayley graph of is amenable.
Proof: ii)iii) is clear.
i)ii). Let be a Cayley graph. Take and a F\o lner set for . Then
So
thus .
iii)i). Assume is an amenable graph. Fix a finite set and . Find such that , where . For , write , . Then
showing that is an amenable group.
Example 10 , are amenable, is not.
2. Coarse embeddings
Definition 7 Let and be two metric spaces, and a map. Say that is a coarse embedding if there exist control functions and tending to infinity at such that , ,
Special choices for lead to subclasses,
- quasi-isometric embeddings if ,
- bi-Lipschitz embeddings if ,
- isometric embeddings if .
Example 11 Different choices of finite generating system lead to a bi-Lipschitz embedding .
Example 12 Inclusion is a bi-Lipschitz embedding.
Example 13 The (discontinuous, non injective) integer part map , is a quasi-isometric embedding.
Example 14 Let be a metric space. Let , where . This is an isometric embedding.
Example 15 Let be a tree. There is a coarse embedding of into the Banach space . When , it is an isometric embedding.
Proof: Fix a vertex . For , define as the characteristic function of the set of edges of the unique path from to . Then is supported on the unique path from to , so
Exercise. Let be a connected graph, viewed as a metric space. Let be a coarse embedding. Show that the control function can be chosen to be linear.
Solution. Let , be vertices of , . Then there exists a sequence of vertices such that . By the triangle inequality
where .
The following Proposition appears in Bekka-Chérix-Valette, [BCV].
Proposition 8 Let be a finitely generated amenable group. Then admits an equivariant coarse embedding into , .
Equivariant means with respect to some isometric action of on .
Proof: Let be an increasing sequence of finite subset of whose union is . Since is ameanable, it admits a F\o lner sequence relative to and , i.e.
Let us embed in . Let be the characteristic function of , normalized to that its -norm be equal to . Let be the left regular representation of on , i.e. acts on an function by . Define ,
Given , there exists such that . Then
so the series defining converges, since .
Define an affine isometric action of on as follows.
By construction, the action does not fix the origin , and by triangle inequality,
To get a lower bound, set
This tends to infinity, since for every , the set is finite.
References
[BCV] Bekka, Mohammed; Chérix, Pierre-Alain; Valette, Alain, Proper affine isometric actions of amenable groups. Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), 1–4, London Math. Soc. Lecture Note Ser., 227, Cambridge Univ. Press, Cambridge, 1995.