# arXiv:math/0605596v2 [math.DS] 4 Dec 2007 THE ERGODIC THEORY OF LATTICE SUBGROUPS ALEXANDER GORODNIK

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### Transcript of arXiv:math/0605596v2 [math.DS] 4 Dec 2007 THE ERGODIC THEORY OF LATTICE SUBGROUPS ALEXANDER GORODNIK

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THE ERGODIC THEORY OF LATTICE SUBGROUPS

ALEXANDER GORODNIK AND AMOS NEVO

Abstract. We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice Γ in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice.

For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Γ, provided certain natural spectral, geometric and reg- ularity conditions are satisfied by the group G, the lattice Γ, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quan- titative mean ergodic theorem in L2(G/Γ) for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports. We demonstrate the new explicit error terms that we obtain by a variety of examples.

Contents

1. Main results : Semisimple Lie groups case 3 1.1. Admissible sets 3 1.2. Ergodic theorems on semisimple Lie groups 4 1.3. The lattice point counting problem in admissible domains 6 1.4. Ergodic theorems for lattice subgroups 8 2. Examples and applications 12

Date: Final version, September 2007. 1991 Mathematics Subject Classification. Primary 22D40; Secondary 22E30,

28D10, 43A10, 43A90. Key words and phrases. Semisimple Lie groups, algebraic groups, lattice sub-

groups, ergodic theorems, maximal inequality, equidistribution, spectral gap, spher- ical functions.

The first author was supported in part by NSF Grant. The second author was supported in part by the Institute for Advanced Study

and an ISF grant. 1

http://arxiv.org/abs/math/0605596v2

2 ALEXANDER GORODNIK AND AMOS NEVO

2.1. Hyperbolic lattice points problem 12 2.2. Counting integral unimodular matrices 13 2.3. Integral equivalence of general n-forms 15 2.4. Lattice points in S-algebraic groups 17 2.5. Examples of ergodic theorems for lattice actions 17 3. Definitions, preliminaries, and basic tools 20 3.1. Maximal and exponential-maximal inequalities 20 3.2. S-algebraic groups and upper local dimension 22 3.3. Admissible and coarsely admissible sets 22 3.4. Absolute continuity, and examples of admissible averages 25 3.5. Balanced and well-balanced families on product groups 27 3.6. Roughly radial and quasi-uniform sets 29 3.7. Spectral gap and strong spectral gap 31 4. Statement of results : general S-algebraic groups 32 4.1. Ergodic theorems for admissible sets 32 4.2. Ergodic theorems for lattice subgroups 36 5. Proof of ergodic theorems for G-actions 38 5.1. Iwasawa groups and spectral estimates 38 5.2. Ergodic theorems in the presence of a spectral gap 41 5.3. Ergodic theorems in the absence of a spectral gap, I 47 5.4. Ergodic theorems in the absence of a spectral gap, II 49 5.5. Ergodic theorems in the absence of a spectral gap, III 52 5.6. The invariance principle, and stability of admissible averages 60 6. Proof of ergodic theorems for lattice actions 63 6.1. Induced action 63 6.2. Reduction theorems 65 6.3. Strong maximal inequality 67 6.4. Mean ergodic theorem 70 6.5. Pointwise ergodic theorem 75 6.6. Exponential mean ergodic theorem 76 6.7. Exponential strong maximal inequality 80 6.8. Completion of proofs of ergodic theorems for lattices 82 6.9. Equidistribution in isometric actions 83 7. Comments and complements 85 7.1. Explicit error term 85 7.2. Exponentially fast convergence versus equidistribution 86 8. Appendix : volume estimates and volume regularity 87 8.1. Admissibility of standard radial averages 88 8.2. Convolution arguments 94 8.3. Admissible, well-balanced, boundary-regular families 96 8.4. Admissible sets on principal homogeneous spaces 100 8.5. Tauberian arguments and Hölder continuity 102 References 108

THE ERGODIC THEORY OF LATTICE SUBGROUPS 3

1. Main results : Semisimple Lie groups case

1.1. Admissible sets. Let G be a locally compact second countable (lcsc) group, and Γ ⊂ G a lattice subgroup. Consider the following four fundamental problems in ergodic theory that present themselves in this context, namely :

(1) Prove ergodic theorems for general families of averages on G, (2) Solve the lattice point counting problem (with explicit error

term) for any lattice subgroup Γ and for general domains on G, (3) Prove ergodic theorems for arbitrary actions of a lattice sub-

group Γ, (4) Establish equidistribution results for isometric actions of the

lattice Γ.

Our purpose in the present paper is to give a complete solution to these problems for non-compact semisimple algebraic groups over ar- bitrary local fields, and any of their lattices. Our results apply also to lattices in products of such groups, and thus also to S-algebraic groups and their lattices. In fact, many of our arguments hold in greater gen- erality still, and we will elaborate on that further in our discussion below. However, for simplicity of exposition we will begin by describ- ing the main results, as well as some of their applications, in the case of connected semisimple Lie groups. We start by introducing the following definition, which describes the

families βt that will be the subject of our analysis. Fix any left-invariant Riemannian metric on G, and let

Oε = {g ∈ G : d(g, e) < ε}.

Let mG denote a fixed left Haar measure on G.

Definition 1.1. An increasing family of bounded Borel subsets Gt, t > 0, of G will be called admissible if there exists c > 0 such that for all t sufficiently large and ε sufficiently small

Oε ·Gt · Oε ⊂ Gt+cε, (1.1) mG(Gt+ε) ≤ (1 + cε) ·mG(Gt). (1.2)

Let us briefly note the following facts (see Prop. 3.13 and Prop. 5.24 below, as well as the Appendix for the proof).

(1) Admissibility is independent of the Riemannian metric chosen to define it.

(2) Many of the natural families of sets in G are admissible. In particular the radial sets Bt projecting to the Cartan-Killing

4 ALEXANDER GORODNIK AND AMOS NEVO

Riemannian balls on the symmetric space are admissible. Fur- thermore, the sets {g ; log ‖τ(g)‖ < t} where τ is faithful linear representation are also admissible, for any choice of linear norm ‖·‖.

(3) Admissibility is invariant under translations, namely if Gt is admissible, so is gGth, for any fixed g, h ∈ G.

It is natural to define also the corresponding Hölder conditions. As we shall see below, whenever a spectral gap is present, the assumption of admissibility can be weakened to Hölder admissibility.

1.2. Ergodic theorems on semisimple Lie groups. We define βt to be the probability measures on G obtained as the restriction of Haar measure to Gt, normalized by mG(Gt). The averaging operators associated to βt when G acts by measure-

preserving transformations of a probability space (X, µ) are given by

π(βt)f(x) = 1

mG(Gt)

∫

Gt

f(g−1x)dmG(g) .

Assume G is connected semisimple with finite center and no compact factors. Then

(1) The family βt (and Gt) will be called (left-) radial if it is invari- ant under (left-) multiplication by some fixed maximal compact subgroup K, for all sufficiently large t. Standard radial averages are those defined in Definition 3.18.

(2) The action is called irreducible if every non-compact simple factor acts ergodically.

(3) The action is said to have a strong spectral gap if each sim- ple factor has a spectral gap, namely admits no asymptotically invariant sequence of unit vectors (see §3.6 for a full discussion).

(4) The sets Gt (and the averages βt) will be called balanced if for every simple factor H and every compact subset Q of its complement, βt(QH) → 0. Gt will be called well-balanced if the convergence is at a specific rate (see §3.5 for a full discussion).

Our first main result is the following pointwise ergodic theorem for admissible averages on semisimple Lie groups.

Theorem 1.2. Pointwise ergodic theorems for admissible aver-

ages. Let G be a connected semisimple Lie group with finite center and no non-trivial compact factors. Let (X, µ) be a standard Borel space with a probability-measure-preserving ergodic action of G. Assume that Gt is an admissible family.

THE ERGODIC THEORY OF LATTICE SUBGROUPS 5

(1) Assume that βt is left-radial. If the action is irreducible, then βt satisfies the pointwise ergodic theorem in L

p(X), 1 < p 0 depends explicitly on the spectral gap (and the family Gt).

The conclusion holds also in actions of G with a spectral gap, provided the averages satisfy the additional necessary condition of being well-balanced (see §§3.5, 3.7 for the definitions).

Regarding Theorem 1.2(1), we remark that the proof of pointwise convergence in the case of reducible actions without a spectral gap is quite involved, and we have thus assumed in that case that the aver- ages are standard radial, well-balanced and boundary-regular to make the analysis tractable. However, the reducible case will be absolutely indispensable for us

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