In this lecture we shall discuss the use of dynamical and ergodic-theoretic ideas to solve some long-standing problems arising from Lie groups and number theory. These problems come from looking at actions of Lie groups on their homogeneous spaces, and, in particular, Ad-unipotent actions. The latter actions through random and chaotic from a dynamical point of view, seem to be rigidly linked to the algebraic structure of the underlying homogeneous space: their ergodic invariant measures and orbit closures have an algebraic nature. These results have some significant applications to number theory and ergodic theory.
More specifically, let ${\bf G}$ be a second countable Lie group, $\bfGamma$ a discrete subgroup of ${\bf G}$ $\bfGamma\sbs{\bf G}=\{\bfGamma{\bf g}\colon {\bf g}\in {\bf G}\}$ and $\pi\colon {\bf G}\to \bfGamma\sbs{\bf G}$ the covering projection $\pi({\bf g})=\bfGamma{\bf g}$, ${\bf g}\in{\bf G}$. The group ${\bf G}$ acts on $\bfGamma\sbs{\bf G}$ by right translations: $x\to xg$, $x\in\bfGamma\sbs{\bf G}$, ${\bf g}\in {\bf G}$. If ${\bf U}$ is a subgroup of ${\bf G}$ and $x\in\bfGamma\sbs{\bf G}$ then the set of $x{\bf U}=\{x{\bf u}\colon{\bf u}\in{\bf U}\}$ is called the ${\bf U}$-orbit of $x$. We say that $\bfGamma$ is a lattice in ${\bf G}$ if there is a finite ${\bf G}$-invaraint measure on $\bfGamma\sbs{\bf G}$.
For a subgroup ${\bf U}$ of ${\bf G}$ a typical orbit $x{\bf U}$ in $\bfGamma\sbs{\bf G}$ is random and chaotic. We pose the following questions: (1) What are the closures of orbits $x{\bf U}$ in $\bfGamma\sbs{\bf G}$? (2) What are the ergodic ${\bf U}$-invariant Borel probability measures $\mu$ on $\bfGamma\sbs{\bf G}$? ($\mu$ is a probability measire if $\mu(\bfGamma\sbs{\bf G})=1$. A ${\bf U}$-invariant $\mu$ is ergodic if every ${\bf U}$-invariant measurable subset of $\bfGamma\sbs{\bf G}$ has $\mu$-measure zero or one).
Let us give a few natural examples. Suppose ${\bf U}=\P{\bf u}(t)\colon t\in R\}$ is a one-parameter subgroup of ${\bf G}$ a nd $x{\bf U}$ a periodic orbit. Then $x{\bf U}=\overline{x{\bf U}}$ and the normalized length measure on $x{\bf U}$ is ${\bf U}$-invariant and ergodic.
For a more general example suppose that the closure $\overline{{x{\bf U}}}$ coincides with the orbit of a larger group ${\bf H}$ containing ${\bf U}$, i.e.\ $\overline{x{\bf U}}=x{\bf H}$. In addition, it m ight happen that $x{\bf H}$ is the support of an ${\bf H}$-invariant Borel probability measure $\nu_{\bf H}$ (this happens if and only if ${\bf xHx}^{-1}\cap\bfGamma$ is a lattice in $$[\bf xHx}^{-1}$, ${\bf x}\in \pi^{-1}$, ${\bf x}\in \pi^{-1}\{x\})$ which is ergodic for the action of ${\bf U}$.
These examples motivate the following definitions.
Definition 1. A subset $A\subset\bfGamma\sbs{\bf
G}$ is called homogeneous if there exists a closed subgroup ${\bf
H}\subset{\bf G}$ and a point $x\in\bfGamma\sbs{\bf G}$ such that
$A=x{\bf H}$ and $x{\bf H}$ is the support of an ${\bf H}$-invariant
Borel probability measure $\nu_{\bf H}$.
Definition 2.. A Borel probability measure $\mu$ on
$\bfGamma\sbs{\bf G}$ is algebraic if there exist
$x\in\bfGamma\sbs{\bf G}$ and a closed subgroup ${\bf H}\subset {\bf
G}$ such that $x{\bf H}$ is homogeneous and $\mu=\nu_{\bf H}$.
It is rather exceptional for a subgroup ${\bf U}$ to have homogeneous orbit closures or algebraic ergodic measures. However, there are some ${\bf U}$ for which this happens.
Let ${\germ G}$ denote the Lie algebra of ${\bf G}$ and for ${\bf
g}\in {\bf G}$ let ${\rm Ad}_{\bdf g}\colon{\germ G}\to{\germ G}$
denote the adjoint map of ${\bf g}$ (which is the differential at the
identity of the map ${\bf h|\to {\bf g}^{-1}{\bf hg}$, $\bf h}\in{\bf
G}$). An element ${\bf g}\in{\bf G}$ is called Ad-semisimple if ${\rm
Ad__[\bf g}$ is diagonalizable over ${\bf C}$. An element ${\bf
u}\in{\bf G}$ is called Ad-unipotent if ${\rm Ad_u-{\rm Id}$ is
nilpotent. In this case ${\rm Ad_{u^r}=\sum^n_{k=1}(r^kT^k_u/k!)$ for
all $r\in{\bf Z}$ and some integer $n\geq 0$, where $T_u$ is a
nilpotent endomorphism of ${\germ G}$. This polynomial (in $r$) form
of ${\rm Ad}_{u^r}$ plays a crucial role in all of the results stated
below. A subgroup ${\bf U}\subset{\bf G}$ is Ad-unipotent if each
${\bf u}\in{\bf G}$ is Ad-unipotent.
Theorem 1. (Algebraicity of Ergodic Measures). Let
${\bf G}$ be a connected Lie group and ${\bf U}$ a connected subgroup
of ${\bf G}$ generalized by Ad-unipotent elements of ${\bf G}$. Then
given any discrete subgroup $\bfGamma$ of ${\bf G}$ every ergodic
${\bf U}$-invariant Borel probability measure on $\bfGamma\sbs{\bf G}$
is algebraic.
Theorem 2. (Homogenuity of Orbit Closures). Let
${\bf G}$ and ${\bf U}$ be as in Theorem 1. Then given any lattice
$\bfGamma$ of ${\bf G}$ and any $x\in\bfGamma\sbs {\bf G}$ the closure
of the orbit $x{\bf U}$ in $\bfGamma\sbs{\bf G}$ is homogeneous.
Theorems 1 and 2 give affirmative answers to conjectures of M. S.
Raghunathan (stated in [1] for reductive ${\bf G}$ and Ad-unipotent
${\bf U}$) and G. A. Margulis [2, Conjectures 1,2], [3, Conjectures
1,2].
Theorem 3. (Uniform Distributio of Ad-Unipotent
Flows). Let ${\bf G}$ be a connected Lie group, $\bfGamma$ a lattice
in ${\bf G}$ and ${\bf U}=\{{\bf u}(t)\colon t\in R\}$ a one-parameter
Ad-unipotent subgroup of ${\bf G}$. Then given any
$x\in\bfGamma\sbs{\bf G}$ there exists a closed subgroup ${\bf
H}\subset{\bf G}$ such that $\overline{x{\bf U}}=x{\bf H}$ is
homogeneous and $t^{-1}\int^t_0f(x{\bf
u}(s))ds\to\int_{\bfGamma\sbs{\bf G}}fd\nu_{\bf H}$ for every bounded
continuous function $f$ on $\bfGamma\sbs{\bf G}$.
Theorem 3 was conjectured by Margluis in [3, Conjectures 3 and 4].
We discuss the ideas and methods used to prove Theorems 1--3 and
further generalizations of these theorems (see [4--10]). Also we
mention earlier contributions made by other authors.
Applications to Number Theory
Theorem A1. (Margulis). Let $B(x_1,\cdots,x_n)$,
$n\geq 3$ be a real nondegenerate indefinite quadratic form in $n$
variables. Suppose that the ratio of some two coefficients of $B$ is
irrational. Then the set of values of $B$ at integer points is dense
in ${\bf R}$.
This is the content of the Oppenhein Conjecture proved by Margulis [2]
in 1986. In fact, it was Raghunathan who noticed that in order to
derive this theorem one only needs to prove a weaker version of
Theorem 2 for ${\bf G}={\rm SL}(3,{\bf R})$ and ${\bf U}={\rm
SO}(2,1)$. This is precisely what Margulis did. Since Theorem 2 is far
strongre than the case proved by Margulis, it allows to simplify his
proof. Also it allows to obtain stronger generalizations of Margulis'
Theorem and to extend it to quadratic forms over general number
fields. This was recently done by A. Borel and G. Prasad.
Applications to Ergodic Theory.
Theorem A2. (The Rigidity Theorem). Let ${\bf G}_i$
be a connected Lie group and $\bfGamma$, a lattice in ${\bf G}_i$,
containing no nontrivial normal subgroups of ${\bf G}_i$, $i=1,2$. Let
${\bf u}^{(i)}$ be an Ad-unipotent element of ${\bf G}_i$ acting
ergodically on $M_i=\bfGamma_i\sbs{\bf G}_i$ with the ${\bf
G}_i$-invariant Borel probability measure $\nu_i$. Suppose there is a
one-to-one measure preserving $\psi$ from $M_1$ onto $M_2$ such that
$\psi(x{\bf u}^{(1)})=\psi(x){\bf u}^{(2)}$ for $\nu_1$-almost every
$x\in M_1$. Then there exists a group isomorphism $\alpha$ from ${\bf
G}_1$ onto ${\bf G}_2$ such that $\psi(x{\bf h})=\psi(x{\bf
h})=\psi(x)\alpha({\bf h})$ for $\nu_1$-almost every $x\in M_1$ and
all ${\bf h}\in{\bf G}_1$. Also $\alpha(\bfGamma_1)={\bf
c}^{-1}\bfGamma_2{\bf c}$ for some ${\bf c}\in{\bf G}_2$.
This theorem says, in particular, that if the actions of the Ad-unipotent elements ${\bf u}^{(1)}$ and ${\bf u}^{(2)}$ are measure-theoretically isomoprhic then ${\bf G}_1$ must be isomorphic to ${\bf G}_2$ and $\bfGamma_1$ to $\bfGamma_2$. We proved Theorem A2 for ${\bf G}={\rm SL}(2,{\bf R})$ in 1979. Then using the same method D. Witte generalized it to any connected ${\bf G}$. But now one can deduce this theorem directly from Theorem 1. Theorem 1 also allows to show that factors and joinings of Ad-unipotent actions are all algebraic. This rigid behavior of Ad-unipotent actions puts them in a striking contrast with Ad-semisimple actions.
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