For my fourth (and last) post related to the “Conférence Internationale Géométrie Ergodique” (held at the Math. Department of Orsay – Univ. Paris 11 from 23th to 27th May), I’ll transcript below my notes for the lecture of Yves Guivarc’h. The title of his talk was:
1. Spectral gap and strong ergodicity properties for groups of affine transformations on tori and nilmanifolds
1.1. General Setting
Let be a measure-preserving action of a countable group, i.e., is a probability space, and is a countable group of measurable transformations such that for all . Recall that is -ergodic whenever or whenever for every (i.e., any -invariant subset has either zero or full -measure). Equivalently, is -ergodic whenever the regular -representation on the Hilbert space of -integrable functions on with zero mean has no -invariant vector.
Given a probability measure on the (countable) group , we say that is symmetric when for all . In the sequel, we will exclusively deal with symmetric probability measures on whose support generates , i.e.,
Definition 1 We say that as above has a spectral gap (or SG property for short) if the -representation doesn’t weakly contain the identity representation in the sense that there is no sequence with and as for every .
In terms of a probability measure on as above, the spectral gap property is equivalent to the following property. Consider the Markov operator given by
associated to the random walk on defined by .
Then, it is possible to show that SG property is equivalent to where is the spectral radius of . In particular, the SG property implies that exponentially fast as for any (a nice stochastic property).
As an illustration of these notions, Y. Guivarc’h mentioned the following examples:
Example 1 Let be the (flat) -dimensional torus equipped with the normalized Lebesgue measure . Consider ) be the subgroup of automorphisms of generated the matrices
Recall that is freely generated by and , and it consists exactly of the matrices in equal to the identity modulo . Consider the probability measure
on . Since is a free group on two generators, one can show that
Actually, a result of H. Kestensays that the spectral radius of the Markov operator of the probability measure
on a free group generated by is
Example 2 Let , , be the normalized Haar measure on . Assume that the closure of is non-amenable. Then, as it was shown by Y. Shalom, the SG property is valid.
Example 3 For any , if satisfies Kazhdan’s property (T), then the SG property holds.
Example 4 Let , be the normalized Haar measure of , and be a dense in such that the coefficients of the matrices in are algebraic. Then, as it was shown by J. Bourgain and A. Gamburd, SG property is valid.
During this post, we mainly consider or, more generally, where is a nilpotent connected and simply connected Lie group and is a lattice in , is the normalized Haar measure, and is a countable group of affine transformations of . Recall that an automorphism of is (the application induced by) an automorphism of the group such that , and an affine transformation of is an application of the form
where and is an automorphism of . In particular, we have a natural projection , .
In the sequel, we denote by the image of under the natural projection . Also, we denote by the derived group of and by the maximal torus factor of , and is the induced (affine) action of on .
The main results of today’s post are the following ones (obtained by B. Bekka and Y. Guivarc’h in this recent preprint here).
For the sake of his talk, Y. Guivarc’h said that he was not going to prove these results in full generality: indeed, as we will see later, he will only illustrate some techniques in the prototypical examples of (3-dimensional) Heisenberg nilmanifolds. More precisely, consider the Heisenberg group
and the integer lattice
The Heisenberg nilmanifold can be seen as a circle bundle over the 2-dimensional (maximal) torus
Remark 1 In the general case , B. Bekka and Y. Guivarc’h don’t whether an estimate such as (1) holds. In particular, in their paper, they get SG property in the general case by non-effective (indirect) methods.
Observe that, in the case of Heisenberg nilmanifold and , we get that Theorem 4 is a consequence of (1) above: in one direction, SG property of trivially implies SG property of , and in the other direction, SG property of means (for instance, this is the case if is non-amenable) and, hence, by (1), , i.e., SG property of holds.
Concerning the applications of these theorems, the following result is essentially a corollary of Theorem 4:
Corollary 5 In the context of Theorem 4:
- acts strongly mixing on if and only if acts strongly mixing on ;
- SG property is equivalent to strong ergodic (SE) property (in the sense of K. Schmidt).
- SG property is equivalent to stable spectral gap (stable SG) property (in the sense of S. Popa).
- SG property is equivalent to the uniqueness of -invariant mean on .
Here, strong ergodicity in the sense of K. Schmidt means that any asymptotically -invariant sequence of Borel subsets of has zero or full measure asymptotically, i.e., if as for any , then one has as . It is known that SG proprety always implies SE property, but the converse is false in general (as it was shown by K. Schmidt). However, this Corollary says that there is no counterexample to “SG property if and only if SE property” amongst countable affine actions on compact nilmanifolds.
Also, stable spectral gap in the sense of S. Popa means that the natural diagonal action of on has spectral gap.
Now, we pass to the presentation of some ideas of the proof of (1).
2. Ideas of proofs
We begin with a brief discussion of metaplectic representations (a.k.a. Weil representations). Let and . Since
is the center of , we have that preserves , so that we have a natural homomorphism (where is the maximal torus factor of . In particular, the unitary representation of on can be decomposed
where is the maximal torus factor (introduced in the previous section), and are isotypical component equivalent to a finite multiple of an irreducible Heisenberg representation (whose description will be given in a moment). Notice that the representation on factors through the natural map into the standard representation of on .
Let us start the (sketch of) proof of (1) in the particular case of a subgroup . Since in this case fixes the center of , one has that preserves . So, the unitary representation on admits a decomposition similar to the one above.
Coming back to the unitary irreducible representations , it is possible to show that , , where is a projective unitary representation of and , , is an unitary irreducible representation of the Heisenberg group on . Also, the (metaplectic/Weil) representation lifts to a double cover of known as the metaplectic group (but we’re going to use this today).
Now one invokes a result of R. Howe and C. Moore on decay of matrix coefficients of metaplectic/Weil representations: for Schwartz functions, is . This implies that the matrix coefficients of are and the matrix coefficients of are . By an argument of R. Howe, one can check that implies , where is a countable set. In other words, is contained in a sum of countably many copies of . This allows to compare with the regular representation of on . The outcome of this comparison is:
Here, given an unitary representation on a Hilbert space , we denote by the Markov operator , . Also, we used that, by definition, is the norm of the Markov operator associated to .
In this way, we conclude that for every . Denoting by , one obtain that
holds in the case . By analyzing the operator separately and by using the fact that
one sees that the inequality (1)holds in the case .
Closing his talk, Y. Guivarc’h mentioned en passant some results about SG property and amenability in the case (see Corollary 3 above).
Let be a locally compact group acting on a space equipped with an almost invariant probability measure (in the sense that for every ).
Definition 6 We say that is co-amenable whenever there is an -invariant mean on .
The relationship between co-amenability and the spectral gap property is given by the following proposition:
In the sequel, we will consider the case of where is a finite dimensional real vector space, is a Lebesgue measure on and . Recall that, in this situation, is co-amenable if and only if there is an invariant probability on .
By a celebrated result of H. Furstenberg, if is co-amenable, then is not totally irreducible, and is relatively compact.
Next, we recall the following definition:
Definition 8 Given , let be the largest vector subspace of such that is amenable, where is the closure of the stabilizer of in . Also, we denote by , where is the Zariski closure of in , and is the connected component of containing the identity element.
In this way, one can show the following result (see Lemma 11 of B. Bekka and Y. Guivarc’h paper):
Proposition 9 is the largest vector subspace such that the stabilizer of in is distal (i.e., acts on in a distal way).
This proposition is used in the proof of the following result (see Proposition 13 of B. Bekka and Y. Guivarc’h paper) related to the study of non-amenability properties:
- (a) if is co-amenable then
- (b) if has some invariant mean on , then (i.e., “there are useful non-trivial rational subspaces”).
At this point, Y. Guivarc’h runned out of time, so he ended his talk with the following few words about the proof of Theorem 2: essentially, the key idea is to use the fact that the absence of SG property for would give us an invariant mean on , that is, is co-amenable (see Proposition 7 above). By Proposition 9 and Proposition 10 (a), (b), acts in a distal way on the non-trivial rational subspace and hence Theorem 2 follows.