Fixed points for groups actions
Fix notation. will always be a discrete countable group, although many considerations extend to general locally compact groups. An action of on by isometries is a homomorphism . Say action fixes a point if there exists such that for all , .
Definition 1 Given a class of metric spaces , say that has Property if every isometric action of on a space belonging to has a fixed point.
- Does have property ?
- Make groups with property .
- Use them.
We shall see that property is rather a source of examples and counterexamples than a proving tool.
After some generalities, we shall study the case of linear (affine) actions, where it is natural to average, and then extend the idea to certain non linear actions. A leitmotive will be Poincaré inequalities and embeddings of metric spaces.
- Fixed point properties
- Interaction between algebra and geometry
- Linear actions: Kazhdan’s property (T)
- Spectral criteria
- Metric criteria
- Averaging and random groups
- Non-linear averaging
- Uniform convexity
- Random groups with property
See the blog: I posted a bibliography last night. I will update it along the way. A good place to start is Bekka, de la Harpe and Valette’s book on property (T) and Serre’s book on Trees, amalgams, .
3. Fixed point properties
3.1. Example: The element group
The element group has fixed points whenever it acts on a uniquely geodesic space, i.e. a metric space in which every two points are connected by a unique unit speed minimizing geodesic.
Proof: Let be the non trivial element of . Choose arbitrary . If , we are done. Otherwise, let be the midpoint of the unique geodesic from to . Then .
The point of the proof is that the “average” was constructed from and using the metric alone.
3.2. Compact groups
Definition 2 Let be the class of real trees. Let be the class of Hilbert spaces (seen as metric spaces).
Note that every isometry of Hilbert space is affine.
Proposition 3 Compact groups belong to .
Proof: For infinite groups, one must assume the action is continuous (in fact, merely that the orbit maps are continuous). Use left Haar probability measure . Let be a Hilbert space. Take arbitrary . Then
is a fixed point. Indeed, given , since is affine,
Note that when , the averaging construction gives the midpoint. Furthermore, the midpoint can be viewed as the average of the measure . Formulated that way, we see a possibility to generalize the argument to those Banach spaces which admit non affine isometries. It suffice to have a purely metric notion of average of a probability measure. We shall see one later, which need not coincide with the affine average.
3.3. Actions on trees
Definition 4 Let be the class of real trees.
We first relate property to finite generation. An intermediate step is provided by cofinality.
Definition 5 Say has cofinality if is the countable union of proper subgroups.
This notion comes from set theory: certain cardinals (e.g. ) have cofinality , others (e.g. finite ones, ) do not.
Given a cofinal group , consider the following graph: vertex set is , edge set is . Cofinality implies that the obtained graph is connected. It is in fact a tree. acts on it by simplicial maps. All stabilizers are , so all -orbits are sets , they are infinite, so no fixed points. No bounded orbits either.
Note: bounded orbits iff all orbits are bounded.
Corollary 6 (Serre) If a countable group has property , then is finitely generated.
Proof: Enumerate the elements of and let be the subgroup generated by the first of them. If is not finitely generated, then all are proper, and is cofinal. Thus it does not have property .
We have proved an algebraic property of using a geometric action. This is typical of how fixed point properties work. They prevent something algebraic to happen.
3.4. Relating to
Proposition 7 If is countable and has property , then is finitely generated.
Proof: Fact 1: The metric of a tree is of negative type.
Fact 2: For any subset of a Hilbert space, any isometric -action on extends to an isometric action on the whole Hilbert space.
Combining these to facts, we see that property implies boundedness of orbits in any action on a tree.
We shall see later that . What misses now is the fact that bounded orbits in a tree implies existence of fixed point. This will follow from an averaging procedure in a tree.
3.5. Linear versus non linear actions
The above argument originates in David Kazhdan’s 4 page paper introducing property (T) in 1967. There, he considers a countable group , assumes that it is not finitely generated, enumerates,…. Then he considers
This is a unitary representation of , without fixed vectors. But the function is -invariant. So every finite subset of fixes some non zero vector in .
Theorem 8 (Kazhdan) Let be an almost simple Lie group of rank (e.g. ). Let be a lattice, i.e. a discrete subgroup of finite covolume. Let be a unitary representation of , without fixed vectors. Then there are no almost fixed vectors either.
Almost fixed vectors is weaker than the property that every finite subset has a non zero fixed vector. Kazhdan calls this property (T), we shall give a precise definition in the next section.
Instead of using unitary representations, Kazhdan could have used the isometric action of (and ) on the symmetric space , a maximal compact subgroup of . But the geometry of fundamental domains of was not fully understood at that time (it was in special cases, like , since Siegel, and possibly, Poincaré), an a proof of finite generation along these lines was completed only a bit later (by Borel and Serre), and is more complicated (see also recent preprint by Tsachik Gelander). Unitary representations can be viewed as a linearization of the situation. They are very flexible and allow for soft arguments.
Theorem 9 (Delorme, Guichardet) For locally compact and -compact groups, property (T) and property coincide.
Later, we shall give a proof based on ultrafilters.
4. Kazhdan’s property (T)
Definition 10 (Kazhdan) Let be a Hilbert space. Let be a unitary representation. Say that almost has invariant vectors if for all finite sets , for all , there is a such that and (10)
Be careful with terminology: we indeed mean “almost has invariant vectors” and not “has almost invariant vectors”.
Note: is small iff the matrix coefficient is close to the constant function uniformly on . Indeed,
So almost has invariant vectors iff in the topology of uniform convergence on compact sets, the constant function is in the closure of the matrix coefficients of .
Definition 11 (Kazhdan) Say has property (T) if for every unitary representation of , almost has invariant vectors implies that has invariant vectors.
We already saw that for discrete groups, property (T) implies finite generation.
4.2. Kazhdan constants
Lemma 12 Let have a finite generating set . It is enough to check inequality (10) with that .
Fixing , one can say that is -invariant if
One says that has almost invariant vectors if , , , which is -invariant.
Definition 13 Let have a finite generating set . Let be a Hilbert space. Let be a unitary representation. Define, for unit vectors ,
And property (T) can be reformulated as follows.
Proposition 14 For a countable discrete group and a finite generating set , the following are equivalent.
- has property (T).
- For every unitary representation , there exists such that existence of a unit vector with implies existence of non zero invariant vectors.
- There exists such that for all unitary representations , existence of a unit vector with implies existence of non zero invariant vectors.
A number as in the proposition is called a Kazhdan constant for and .
Proof: To interchange and , construct the giant unitary representation which contains at least one copy of every irreducible representation. Since is countable, each lives in a separable Hilbert space, fix one once and for all. Then unitary representations of in that space form a set, and we can sum then up.
Theorem 15 (Gelander, Zuk) has property (T). Nevertheless, it has finite generating sets with arbitrarily small Kazhdan constants.
It is not known wether has the same property. This issue is related to recent work by Helfgott, Bourgain, Gamburd on expansion of Cayley graphs of finite quotients of such groups with respect to (nearly) arbitrary generating systems.
4.3. Spectral formulation of property (T)
For unit , let us calculate
where denotes the averaging operator
viewed as an element of the group ring .
For any unitary representation , is a self adjoint operator or norm . Its spectrum is contained in , so . Saying that for all unit , is equivalent to . Invariant vectors (if any) form the -eigenspace. This proves that
Proposition 16 For a discrete group and a finite generating set , the following are equivalent.
- has property (T).
- There exists such that for all unitary representations ,
Note that the spectral criterion is one sided. We would prefer two sided gaps, since these would be equivalent to powers of tending to . To get around this annoying point, two ways.
- Replace with .
- Replace with , where is the sphere of radius (with multiplicities).
In other words, the second choice differs from the first only in that it amounts to working with the subgroup generated by , which has index at most in . We shall freely use either trick.
- For all , (17)where
We call inequality (17) a Poincaré inequality. It is nice since it is purely metric. It will make sense in a non linear setting. It is annoying that the precise value of the constant (being less than ) be essential. We shall see later situations where a mere uniform constant suffices to draw conclusions.
4.5. Decidability issues
Since acts by isometries, , and one can rewrite the Poincaré inequality (17) in the form
This is reminiscent of hypercontractivity (see Bernard Maurey’s 4th course).
View it as a statement about functions (in fact, only those functions which can be extended to actions). Take an -ball . Consider those functions such that
If (17) holds for such functions, a fortiori it holds for functions coming from orbits. In fact, to control the constant up to a factor of , one needs only check it for functions on with values in the finite set . A computer can do that.
If it fails for all , one can pass to a limiting function defined on all of , with values in some Hilbert space , arising from an orbit of an isometric -action on . Since , does not have property (T).
This is an algorithm which can enumerate all group presentations giving groups having property (T). Thus we have shown
Corollary 18 (Silberman) The set of presentations of Kazhdan groups is recursively enumerable.
Corollary 19 (Shalom) If has property (T), then is the quotient of a finitely presented group which has property (T).
Proposition 20 Property (T) is undecidable.
Proof: Since free products act on trees without fixed points, has property (T) iff is trivial. So an algorithm deciding if a presentation produces a group with property (T), applied to , would decide if is non trivial, this does not exist.