# The Ergodic Theory of Lattice Subgroups (AM-172)

Alexander Gorodnik
Amos Nevo
Pages: 136
https://www.jstor.org/stable/j.ctt7rwxj

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-xiv)
4. Chapter One Main results: Semisimple Lie groups case
(pp. 1-10)

The present chapter is devoted to describing the main results in the case of connected semisimple Lie groups, which is fundamental in what follows.

We start by introducing the notion of admissibility, which describes the families of averaging sets${G_t}$that will be the subject of our analysis.

LetGbe a connected semisimple Lie group with finite center and no nontrivial compact factors. Fix a left-invariant Riemannian metric onGand denote the associated invariant distance bydand the Haar invariant measure by${m_G}$. Let

${\mathcal{O}_\varepsilon}=\{g\;\in\;G:\;d(g,\;e)\;<\;\varepsilon\}$.

Definition 1.1. An increasing family of bounded Borel subsets${G_t}$,$t>0$, of...

5. Chapter Two Examples and applications
(pp. 11-18)

Let us now consider some examples and applications of the results stated in Chapter 1 and compare our results to some precedents in the literature.

We begin by applying Theorem 1.5 to the classical lattice point–counting problem in hyperbolic space. Let us call a lattice subgroup$\Gamma$tempered if the spectrum of the representation of the isometry group in$L_0^2({G {\left/ { { \ }} \right.}\Gamma })$istempered, namely, the representation is weakly contained in the regular representation of the isometry group, or equivalently (see [CHH]) for a dense family of functions, the corresponding matrix coefficients are in${L^{2 + \varepsilon }}(G)$for every$\varepsilon \; > \;0$.

Corollary 2.1. Let...

6. Chapter Three Definitions, preliminaries, and basic tools
(pp. 19-32)

In this chapter we introduce the necessary tools and definitions which will allow us to develop ergodic theorems on general locally compact second countable groups in a systematic fashion. We refer the reader to §1.5, where the motivation for some of the concepts appearing below is explained.

LetGbe an lcsc group with a left-invariant Haar measure${m_G}$. Let(X, B, μ)be a standard Borel space with a Borel-measurable G-action preserving the probability measureμ. There is a natural isometric representation${\pi_X}$ofGon the spaces${L^p}(\mu)$,1 ≤ p ≤ ∞, defined by

$({\pi_X}(g)f)(x)\;=\;f({g^{-1}}x),\quad\;g\;\in\;G,\;\;f\;\in\;{L^p}(\mu)$

To each...

7. Chapter Four Main results and an overview of the proofs
(pp. 33-46)

In the present chapter we will state our main results, namely, the ergodic theorems for actions of G and for actions of Γ in the presence of a spectral gap and in the absence of a spectral gap. We will also give an overview of the proofs of these results, as well as an overview of the proofs of the volume regularity results that are the subject of Chapter 7.

Let us now formulate the two basic ergodic theorems for actions of an S-algebraic group G, which we will prove in the following chapter. As usual, it is the maximal...

8. Chapter Five Proof of ergodic theorems for S-algebraic groups
(pp. 47-70)

In the present chapter we will prove the ergodic theorems for admissible or Hölderadmissible averages on anS-algebraic group G stated in Theorems 4.1 and 4.2. We will distinguish two cases, namely, whether the action ofGonXhas a spectral gap or not. As noted already in Chapter 4, the arguments that will be employed below in these two cases are quite different, but both use spectral theory in a material way. Therefore we will begin by recalling the relevant facts from spectral theory. In order to consider allS-algebraic groups, it is convenient to work in the...

9. Chapter Six Proof of ergodic theorems for lattice subgroups
(pp. 71-92)

We now turn to consideration of a discrete lattice subgroup Γ of an lcsc groupGand to the problem of establishing ergodic theorems for actions of the lattice, given the validity of ergodic theorems for actions ofG. We will begin by discussing the induced action and then develop a series of reduction arguments of increasing precision to achieve this goal. In §6.8 we put all the arguments together and finalize the proofs of all the ergodic theorems stated for G and Γ, and in §6.9 we prove the equidistribution result.

The existence of a lattice implies thatG...

10. Chapter Seven Volume estimates and volume regularity
(pp. 93-112)

The present chapter is devoted to establishing regularity properties of the standard averages and more general ones, as well as to establishing conditions sufficient for the averages to be balanced or well balanced. We will also discuss boundaryregularity and differentiability properties of volume functions for some metrics, particularly,CAT(0)-metrics.

We begin with a proof of Theorem 3.15, whose statement we recall.

Theorem 3.15.For an S-algebraic group$G=G(1)\,\cdots\,G(N)$as in Definition 3.4, the following families of sets${G_t}\subset\;G$are admissible, where${a_i}$are any positive constants, and$t\in\;{\mathbb{R}_+}$when S contains at least one infinite place and$t\;\in\;{\mathbb{N}_+}$otherwise.

1....

11. Chapter Eight Comments and complements
(pp. 113-116)

In the present chapter we give a formula for the error term in the lattice point problem, present an example of an action where exponentially fast almost sure convergence holds but equidistribution fails, and comment on the existence of balanced averages.

Let us state the following explicit error estimate in the lattice point–counting problem in admissible domains, which follows from the results above. It is convenient to denote by${\Sigma_{{p^+}}}$the subset of the positive-definite spherical functions on an Salgebraic groupGthat are in${L^{p+\varepsilon}}(G)$for every$\varepsilon\;>\;0$. If

$\int_G{|{\kern1pt}\phi{\kern1pt}{|^{p+\varepsilon}}}d{m_G}(g)\;\leqslant\;C(\varepsilon)<\;\infty$

is independent of$\phi\;\in\;{\Sigma_{{p^+}}}$, we say that the...

12. Bibliography
(pp. 117-120)
13. Index
(pp. 121-121)