Last time, we talked about fixed point properties, how they interact with algebra ( finitely generated), Kazhdan’s property (T) and its spectral characterization. Today, we shall connect property (T) with fixed point properties. For that, we need a technical tool: ultrafilter arguments.
1. Ultrafilter arguments
Let be a sequence of real numbers. Imagine every is invited to vote yes or no to the question “is -close to ?”. If all but finitely many votes are yes, then tends to . Thus the obvious “voting scheme”
plays a key role in the notion of convergence. We intend to define convergence according to more general voting schemes. The requirements are
- is non empty.
- If , and , then .
- If , , then .
The last axiom is there to guarantee uniqueness of limit. These axiom define filters.
There is a difficulty with of obvious filter. For , half of the votes are yes, half are no, and no decision can be taken. So one should make such situations impossible.
Definition 1 Say a filter is an ultrafilter if for all , exactly one of and belongs to .
Lemma 2 For every filter on , there exist ultrafilters than contain it.
Proof: Zorn’s lemma. Indeed, ultrafilters are maximal filters.
Also, one should avoid “dictatorships”, i.e. ultrafilters that contain a singleton. These are called principal ultrafilters.
Theorem 3 (Bolzano-Weierstrass) If is a compact metric space and a non-principal ultrafilter, then for any sequence , exists.
Indeed, what the ultrafilter does is extract converging subsequences. It is particularly convenient when arranging sequences in a product space to converge. This would required extracting a converging subsequence in , and then re-extracting for to converge. The ultrafilter makes the choice once and for all.
Here is a limitation in the use of ultrafilters: One is not allowed to permute elements in a sequence. This will be harmless in our use of ultrafilters.
Usual rules of logic hold and constructions go through. For instance, ultralimits of groups are groups.
1.2. Ultralimits of pointwise metric spaces
A pointed metric space is a metric space with a base point. Given a sequence of such, define
If and , define
which exists, by Bolzano-Weierstrass.
Definition 4 Let be a non principal ultrafilter on . The -ultralimit of is
where two sequences are identified iff their vanishes.
Theorem 5 Any statement about finite sets of points in metric spaces including universal quantifiers and bounded existential quantification (e.g. such that ) holds in the ultralimit if it holds for -almost every .
Example 1 Existence of midpoints passes to ultralimits.
Indeed, the midpoint stays a bounded distance away to the base point. To conclude from this that the ultralimit of geodesic metric spaces is geodesic requires some extra convexity assumption.
Example 2 passes to ultralimits.
Indeed, amounts to the following inequality:
It is an equality for Hilbert space, and characterizes Hilbert space.
Example 3 An ultralimit of Hilbert spaces is a Hilbert space.
Example 4 An ultralimit of geodesic spaces is geodesic and .
Example 5 Every sequence of isometries such that stays bounded defines an isometry of the ultralimit. Every group homomorphism defines a group homomorphism
In fact, it is sufficient to assume that is close to being an isometry (e.g. tends to ). Also, it is sufficient to have a sequence of near homomorphisms ( close to ).
- has property .
- There exists such that for every isometric action of on a Hilbert space and every ,
Proof: 2. 1. Pick arbitrary , define . By assumption, . One checks that for all ,
This shows that , is a Cauchy sequence, the limit is a fixed point.
1. 2. By contradiction. Assume that there exist pointed Hilbert spaces with -actions such that and
Rescale the metric so that . Form the ultralimit , based at , of based at .
Since , each generator translates a bounded amount, so we get a limiting action of on such that . Also
Assume that has a fixed point on , which take to be the origin. Then and all the sit on the same sphere. Recall that is the point that minimizes
By strict convexity, unless all ‘s are equal, , contradiction. So all , , . So is a fixed point for , which has index at most . The midpoint of and is a fixed point.
1.4. Property implies property (T)
Proposition 7 (Guichardet) Property implies property (T)
Proof: Let be a group with finite generating set . Let be a unitary representation which almost has invariant unit vectors, but no invariant vectors.
By assumption, for all , there exists a unit such that . In fact, for every , , , there is a unit such that every has
Indeed, otherwise, for all unit with , there ecists such that
Iterate. This produces a sequence , , , such that . Rescale so that . Take an ultralimit. There, energy is everywhere, so no fixed points. This contradicts property .
Remark 1 Although we started with a linear action, that had a fixed point, this has disappeared from the resulting isometric action on the ultralimit.
Indeed, makes that in the rescaled space, tends to infinity.
This is not Guichardet’s initial proof. I took this proof from Gromov’s random walk in random groups paper. We could probably have used Proposition 6 instead (relating energy fro affine actions to energy for unitary representations), but I prefered to show an ultrafilter argument. Also, this argument has a wider scope, it does not require an averaging procedure.
2. Averaging in spaces
Now we pass to the nonlinear setting.
In a geodesic space, given points , , let denote a geodesic segment, and the point on that geodesic at relative distance from .
Definition 8 A geodesic metric space is is for all , , and ,
This is a uniform convexity statement about the function . This makes midpoints unique, geodesic segments between points unique.
If is a closed convex set, the function achieves a unique minimum. Indeed,
has diameter , so is non empty and a single point, denoted by . is -Lipschitz. Furthermore, any geodesic in starting at makes there an obtuse angle with (angles make sense in spaces).
Definition 9 Let be a probability measure on . Define its center of mass as the point that minimizes
Existence and uniqueness of follows from the same proof as for .
Theorem 10 (Sturm) (Jensen’s inequality). Let be a convex function. Then .
Corollary 11 Let be a group with finite generating set , acting isometrically on a space . Set
Proposition 12 Let be groups such that is compact and has a finite -invariant measure. Then has property iff does.
Proof: Assume that has property . Let act on some Hilbert space . Let be a point of fixed by . Use orbit map to push forward -invariant measure to a measure on . Then the center of mass of is -invariant.
Conversely, assume that has property . Let act on some Hilbert space . Consider the space
It is nonempty (requires partition of unity if is nondiscrete). It has a natural metric. Indeed, if , , the map is -invariant, so one can integrate it over the quotient space . The resulting metric space is an affine Hilbert space (a direct integral of Hilbert spaces), related to the induced representation. acts on it on the right, by isometries. Let be -invariant. This means that is constant, and its value is a fixed point of on .
2.4. From groups actions to random walks
Let act on with compact. As we just saw, ne can replace the study of points of under by the study of -equivariant maps . Fixed points correspond to constant functions. Let be probability measure on . A random walk on is a map to the space of compactly supported probability measures on . We furthermore assume that
- is -equivariant.
- Each is reversible with respect to , i.e. for every -invariant function ,
Here, we use Gromov’s suggestive notation .
If is -equivariant, set
and we would like to prove energy decreasing property: such that ,