Posted by: matheuscmss | October 10, 2011

Conférence internationale Géométrie Ergodique (Orsay 2011) IV

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 {(\Gamma, X,\nu)} be a measure-preserving action of a countable group, i.e., {(X,\nu)} is a probability space, and {\Gamma} is a countable group of measurable transformations {\gamma:X\rightarrow X} such that {\gamma\nu=\nu} for all {\gamma\in\Gamma}. Recall that {\nu} is {\Gamma}-ergodic whenever {\nu(A)=0} or {1} whenever {\gamma(A)=A} for every {\gamma\in\Gamma} (i.e., any {\Gamma}-invariant subset {A\subset X} has either zero or full {\nu}-measure). Equivalently, {\nu} is {\Gamma}-ergodic whenever the regular {\Gamma}-representation {\pi(\nu)} on the Hilbert space {L^2_0(X,\nu)} of {L^2}-integrable functions on {(X,\nu)} with zero mean has no {\Gamma}-invariant vector.

Given {\mu} a probability measure on the (countable) group {\Gamma}, we say that {\mu} is symmetric when {\mu(\{\gamma^{-1}\})=\mu(\{\gamma\})} for all {\gamma\in\Gamma}. In the sequel, we will exclusively deal with symmetric probability measures {\mu} on {\Gamma} whose support generates {\Gamma}, i.e., {\langle\textrm{supp}\,\mu\rangle=\Gamma}

Definition 1 We say that {(\Gamma,X,\nu)} as above has a spectral gap (or SG property for short) if the {\Gamma}-representation {\pi} doesn’t weakly contain the identity representation {\textrm{Id}_{\Gamma}} in the sense that there is no sequence {f_n\in L^2_0(X,\nu)} with {\|f_n\|_{L^2}=1} and {\|f_n-\pi(\gamma)f_n\|_{L^2}\rightarrow0} as {n\rightarrow\infty} for every {\gamma\in\Gamma}.

In terms of a probability measure {\mu} on {\Gamma} as above, the spectral gap property is equivalent to the following property. Consider the Markov operator {P_{\mu}:L^2_0(X,\nu) \rightarrow L^2_0(X,\nu)} given by

\displaystyle P_{\mu}\phi(x)=\sum\limits_{\gamma\in\Gamma}\mu(\{\gamma\})\phi(\pi(\gamma)x)

associated to the random walk on {\Gamma} defined by {\mu}.

Then, it is possible to show that SG property is equivalent to {r(P_{\mu})<1} where {r(P_{\mu})=\lim\limits_{n\rightarrow\infty}\|P_{\mu}^n\|^{1/n}} is the spectral radius of {P_{\mu}}. In particular, the SG property implies that {P_\mu^n\phi\rightarrow 0} exponentially fast as {n\rightarrow\infty} for any {\phi\in L^2_0(X,\nu)} (a nice stochastic property).

As an illustration of these notions, Y. Guivarc’h mentioned the following examples:

Example 1 Let {X=\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2} be the (flat) {2}-dimensional torus equipped with the normalized Lebesgue measure {\nu}. Consider {\Gamma\subset \textrm{Aut}(\mathbb{T}^2) (= SL(2,\mathbb{Z})}) be the subgroup of automorphisms of {X} generated the matrices

\displaystyle a=\left(\begin{array}{cc}1 & 2 \\ 0 & 1\end{array}\right), \quad b=\left(\begin{array}{cc}1 & 0 \\ 2 & 1\end{array}\right).

Recall that {\Gamma=\langle a,b\rangle} is freely generated by {a} and {b}, and it consists exactly of the matrices in {SL(2,\mathbb{Z})} equal to the identity modulo {2}. Consider the probability measure

\displaystyle \mu = \frac{1}{4}(\delta_{a}+\delta_{b}+\delta_{a^{-1}}+\delta_{b^{-1}})

on {\Gamma}. Since {\Gamma=\langle\textrm{supp}\,\mu\rangle} is a free group on two generators, one can show that

\displaystyle r(P_{\mu})=\frac{\sqrt{3}}{2}<1.

Actually, a result of H. Kestensays that the spectral radius {r(P_{\mu})} of the Markov operator {P_{\mu}} of the probability measure

\displaystyle \mu=\frac{1}{2k}\sum\limits_{i=1}^k(\delta_{a_i}+\delta_{a_i^{-1}})

on a free group {F_k} generated by {a_1,\dots,a_k} is

\displaystyle r(P_{\mu})=\frac{\sqrt{2k-1}}{k}.

Example 2 Let {\Gamma\subset SL(2,\mathbb{R})}, {X=SL(2,\mathbb{R})/SL(2,\mathbb{Z})}, {\nu} be the normalized Haar measure on {X}. Assume that the closure of {\Gamma} is non-amenable. Then, as it was shown by Y. Shalom, the SG property is valid.

Example 3 For any {(X,\nu)}, if {\Gamma} satisfies Kazhdan’s property (T), then the SG property holds.

Example 4 Let {X=SU(2)}, {\nu} be the normalized Haar measure of {SU(2)}, and {\Gamma\subset SU(2)} be a dense in {SU(2)} such that the coefficients of the matrices in {\Gamma} are algebraic. Then, as it was shown by J. Bourgain and A. Gamburd, SG property is valid.

During this post, we mainly consider {X = \mathbb{T}^d} or, more generally, {X=N/\Delta} where {N} is a nilpotent connected and simply connected Lie group and {\Delta} is a lattice in {N}, {\nu} is the normalized Haar measure, and {\Gamma\subset \textrm{Aff}(X)} is a countable group of affine transformations of {X}. Recall that an automorphism {\tau} of {N/\Delta} is (the application induced by) an automorphism {\tau} of the group {N} such that {\tau(\Delta)=\Delta}, and an affine transformation of {N/\Delta} is an application {\phi} of the form

\displaystyle x\Delta\mapsto \phi(x\Delta):=\alpha\cdot\tau(x)\Delta

where {\alpha\in N} and {\tau} is an automorphism of {N/\Delta}. In particular, we have a natural projection {p_{aut}:\textrm{Aff}(X)\rightarrow\textrm{Aut}(X)}, {\phi\mapsto \tau}.

In the sequel, we denote by {\Gamma^{aut}} the image of {\Gamma} under the natural projection {p_{aut}}. Also, we denote by {N^1=[N,N]} the derived group of {N} and by {\overline{X} = X/N^1\Delta} the maximal torus factor of {X=N/\Delta}, and {\overline{\Gamma}} is the induced (affine) action of {\Gamma} on {\overline{X}}.

The main results of today’s post are the following ones (obtained by B. Bekka and Y. Guivarc’h in this recent preprint here).

Theorem 2 SG property is not valid in the case {X=\mathbb{T}^d} if and only if there exists a non-trivial {S\subset\mathbb{T}^d} {\Gamma^{aut}}-invariant subtorus such that the image of {\Gamma^{aut}} on {\textrm{Aut}(\mathbb{T}^d/S)} is virtually Abelian.

Corollary 3 In the case {X=\mathbb{T}^d}, if {\Gamma^{aut}} is {\mathbb{Q}}-irreducible on {\mathbb{R}^d} (i.e., there is no non-trivial {\Gamma^{aut}}-invariant rational subspace of {\mathbb{R}^d}) and {\Gamma^{aut}} not virtually Abelian, then {(\Gamma,X,\nu)} has the SG property.

Theorem 4 More generally, in the case {X=N/\Delta}, {\Gamma\subset \textrm{Aff}(X)}, we have that {(\Gamma,X,\nu)} has SG property if and only if {(\overline{\Gamma}, \overline{X}=N/N^1\Delta,\overline{\nu})} has SG property. Here, {\overline{\Gamma}} and {\overline{\nu}} are obtained from {\Gamma} and {\nu} via the natural map {X\rightarrow\overline{X}}.

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

N=H_3(\mathbb{R}):= \left\{\left(\begin{array}{ccc}1&x&z \\ 0&1&y \\ 0&0&1\end{array}\right):x,y,z\in\mathbb{R}\right\},

and the integer lattice

{\Delta=H_3(\mathbb{Z}):=\left\{\left(\begin{array}{ccc}1&a&c \\ 0&1&b \\ 0&0&1\end{array}\right):a,b,c\in\mathbb{Z}\right\}}.

The Heisenberg nilmanifold {X:=N/\Delta=H_3(\mathbb{R})/H_3(\mathbb{Z})} can be seen as a circle bundle over the 2-dimensional (maximal) torus

{\overline{X}=\left(\begin{array}{ccc}1& \mathbb{R}/\mathbb{Z} &0 \\ 0&1&\mathbb{R}/\mathbb{Z} \\ 0&0&1\end{array}\right)}.

In particular, given {\Gamma\subset\textrm{Aff}(X)}, we can think that {\overline{\Gamma}} is a subset of the affine group of the 2-dimensional torus {\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2}, namely, a semi-direct product of {SL_2(\mathbb{Z})=\textrm{Aut}(\mathbb{T}^2)} and {\mathbb{Z}^2}. In this language, we will see that for every {\Gamma\subset Aut(X)}

\displaystyle r(P_{\mu})\leq r_0(\overline{\Gamma})^{1/4} \ \ \ \ \ (1)

where {r_0(\overline{\Gamma})} is the spectral radius of the Markov operator associated to {(\overline{\Gamma},\overline{X},\overline{\nu})}.

Remark 1 In the general case {\Gamma\subset \textrm{Aff}(X)}, 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 {X=H_3(\mathbb{R})/H_3(\mathbb{Z})} and {\Gamma\subset Aut(X)}, we get that Theorem 4 is a consequence of (1) above: in one direction, SG property of {(\Gamma,X,\nu)} trivially implies SG property of {(\overline{\Gamma},\overline{X},\overline{\nu})}, and in the other direction, SG property of {(\overline{\Gamma},\overline{X},\overline{\nu})} means {r_0(\overline{\Gamma})<1} (for instance, this is the case if {\overline{\Gamma}} is non-amenable) and, hence, by (1), {r(P_{\mu})<1}, i.e., SG property of {(\Gamma,X,\nu)} 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:

  • {\Gamma} acts strongly mixing on {X=N/\Delta} if and only if {\overline{\Gamma}} acts strongly mixing on {\overline{X}=N/N^1\Delta};
  • 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 {\Gamma}-invariant mean on {L^{\infty}(X)}.

Here, strong ergodicity in the sense of K. Schmidt means that any asymptotically {\Gamma}-invariant sequence {(A_n)} of Borel subsets of {X} has zero or full measure asymptotically, i.e., if {\nu(\gamma A_n\Delta A_n)\rightarrow 0} as {n\rightarrow\infty} for any {\gamma\in\Gamma}, then one has {\nu(A_n)(1-\nu(A_n))\rightarrow 0} as {n\rightarrow\infty}. 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 {\Gamma} on {X\times X} 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 {X=H_3(\mathbb{R})/H_3(\mathbb{Z})} and {\Gamma\subset Aut(X)}. Since

{[H_3(\mathbb{R}),H_3(\mathbb{R})] = \left\{\left(\begin{array}{ccc}1&0&z\\ 0&1&0 \\ 0&0&1\end{array}\right):z\in\mathbb{R}\right\}}

is the center of {H_3(\mathbb{R})}, we have that {Aut(X)} preserves {[H_3(\mathbb{R}),H_3(\mathbb{R})] H_3(\mathbb{Z})}, so that we have a natural homomorphism {Aut(X)\rightarrow Aut(\overline{X})} (where {\overline{X}=H_3(\mathbb{R})/[H_3(\mathbb{R}),H_3(\mathbb{R})] H_3(\mathbb{Z})\simeq \mathbb{T}^2} is the maximal torus factor of {X}. In particular, the unitary representation of {Aut(X)} on {L_0^2(X)} can be decomposed

\displaystyle L_0^2(X) = \bigoplus\limits_{m\in\mathbb{Z}-\{0\}}\mathcal{H}_m \oplus L_0^2(\mathbb{T}^2)

where {\mathbb{T}^2=\overline{X}} is the maximal torus factor (introduced in the previous section), and {\mathcal{H}_m} are isotypical component equivalent to a finite multiple of an irreducible Heisenberg representation {\pi_m} (whose description will be given in a moment). Notice that the {Aut(X)} representation on {L_0^2(\mathbb{T}^2)} factors through the natural map {Aut(X)\rightarrow Aut(\mathbb{T}^2)=SL_2(\mathbb{Z})} into the standard representation of {SL_2(\mathbb{Z})} on {L_0^2(\mathbb{T}^2)}.

Let us start the (sketch of) proof of (1) in the particular case of a subgroup {\Gamma\subset Aut(X)}. Since in this case {\Gamma} fixes the center of {H_3(\mathbb{R})}, one has that {\Gamma} preserves {\mathcal{H}_m}. So, the unitary {\Gamma} representation on {L_0^2(X)} admits a decomposition similar to the one above.

Coming back to the unitary irreducible representations {\pi_m}, it is possible to show that {\pi_m(\gamma)=\omega(\gamma)\otimes\phi_m(\gamma)}, {\gamma\in\Gamma}, where {\omega} is a projective unitary representation of {SL_2(\mathbb{R})} and {\phi_m\left(\begin{array}{ccc}1&x&z \\ 0&1&y \\ 0&0&1\end{array}\right)f(s) = e^{2\pi i m z} e^{2\pi x(s-y/2)} f(s-y)}, {f\in L^2(\mathbb{R})}, is an unitary irreducible representation of the Heisenberg group {H_3(\mathbb{R})} on {L^2(\mathbb{R})}. Also, the (metaplectic/Weil) representation {\omega} lifts to a double cover of {SL_2(\mathbb{R})} known as the metaplectic group {Mp_2(\mathbb{R})} (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 {f,g} Schwartz functions, {\gamma\mapsto \langle \omega(\gamma)f,g\rangle} is {L^{6+\varepsilon}(SL_2(\mathbb{R}))}. This implies that the matrix coefficients of {\omega^{\otimes 4}} are {L^2(SL_2(\mathbb{R}))} and the matrix coefficients of {\pi_m^{\otimes 4}} are {\ell^2(\Gamma)}. By an argument of R. Howe, one can check that {\pi_m^{\otimes 4}\in \ell^2(\Gamma)} implies {\pi_m^{\otimes 4}(\Gamma)\subset \bigoplus\limits_{i\in I} \ell^2_i(\Gamma)}, where {I} is a countable set. In other words, {\pi_m^{\otimes 4}(\Gamma)} is contained in a sum of countably many copies of {\ell^2(\Gamma)}. This allows to compare {\pi_m^{\otimes 4}} with the regular representation {\lambda_{\Gamma}} of {\Gamma} on {\ell^2(\Gamma)}. The outcome of this comparison is:

\displaystyle \|\pi_m(\mu)\|^4 = \|\pi_m^{\otimes 4}(\mu)\|\leq r_0(\mu).

Here, given an unitary {\Gamma} representation {U} on a Hilbert space {V}, we denote by {U(\mu)} the Markov operator {U(\mu):V\rightarrow V}, {U(\mu)v=\sum\limits_{a\in\Gamma}\mu(\{a\})U(a)v}. Also, we used that, by definition, {r_0(\Gamma)} is the norm of the Markov operator {\lambda_{\Gamma}(\mu)} associated to {\lambda_{\Gamma}}.

In this way, we conclude that {\|\pi_m(\mu)\|\leq r_0(\mu)^{1/4}} for every {m}. Denoting by {\mathcal{H} = \bigoplus\limits_{m\in\mathbb{Z}-\{0\}}\mathcal{H}_m}, one obtain that

\displaystyle r(P_{\mu}|_{\mathcal{H}})\leq r_0(\overline{\Gamma})^{1/4}

holds in the case {\Gamma\subset Aut(X)}. By analyzing the operator {P_{\mu}|_{L_0^2(\mathbb{T}^2)}} separately and by using the fact that

\displaystyle L_0^2(X) = \bigoplus\limits_{m\in\mathbb{Z}-\{0\}}\mathcal{H}_m \oplus L_0^2(\mathbb{T}^2)

one sees that the inequality (1)holds in the case {\Gamma\subset Aut(X)}.

Closing his talk, Y. Guivarc’h mentioned en passant some results about SG property and amenability in the case {X=\mathbb{T}^d} (see Corollary 3 above).

3. Co-amenability

Let {G} be a locally compact group acting on a space {E} equipped with an {G} almost invariant probability measure {\lambda} (in the sense that {g\lambda\sim\lambda} for every {g\in G}).

Definition 6 We say that {(G,E,\lambda)} is co-amenable whenever there is an {G}-invariant mean on {L^{\infty}(E,\lambda)}.

The relationship between co-amenability and the spectral gap property is given by the following proposition:

Proposition 7 Let {\mu\in\mathcal{M}^1(G)} be a probability measure on {G} with {\langle\textrm{supp}(\mu)\rangle = G}. If {r_0(\mu)=1} (i.e., SG property doesn’t hold) then {(G,E,\lambda)} is co-amenable.

In the sequel, we will consider the case of {E=V-\{0\}} where {V} is a finite dimensional real vector space, {\lambda=l} is a Lebesgue measure on {V} and {G\subset GL(V)}. Recall that, in this situation, {(G,V-\{0\},l)} is co-amenable if and only if there is an invariant probability {\mu} on {\mathbb{P}(V)}.

By a celebrated result of H. Furstenberg, if {G\subset SL(V)} is co-amenable, then {G} is not totally irreducible, and {G} is relatively compact.

Next, we recall the following definition:

Definition 8 Given {H\subset GL(V)}, let {V(H)} be the largest vector subspace {W} of {V} such that {\overline{H}_W} is amenable, where {\overline{H}_W} is the closure of the stabilizer of {W} in {H}. Also, we denote by {H^0:=H\cap (Zc(H))^0}, where {Zc(H)} is the Zariski closure of {H} in {GL(V)}, and {(Zc(H))^0} is the connected component of {Zc(H)} 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 {V(H)} is the largest vector subspace {W\subset V} such that the stabilizer {H^1_W} of {W} in {H^1} is distal (i.e., {H^1_W} acts on {W} 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:

Proposition 10

  • (a) if {(H,V-\{0\})} is co-amenable then {V(H)\neq 0}
  • (b) if {\Gamma=H\subset SL_d(\mathbb{Z})} has some invariant mean {m} on {\mathbb{Z}^d-\{0\}}, then {m(V(\Gamma)\cap\mathbb{Z}^d)=1} (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 {\Gamma} would give us an invariant mean on {\mathbb{Z}^d-\{0\}}, that is, {\Gamma=H} is co-amenable (see Proposition 7 above). By Proposition 9 and Proposition 10 (a), (b), {\Gamma^0=H^0} acts in a distal way on the non-trivial rational subspace {V(H)} and hence Theorem 2 follows.


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