Theorem 11.4. In fact, we prove the following invariance principle:

######  Theorem 15.1.

Let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice. Let
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$ be an action and let
$M^{\alpha}$ denote the suspension space with induced $G$-action. Let $\mu$ be
an ergodic, $A$-invariant Borel probability measure on $M^{\alpha}$ whose
projection to $\mathrm{SL}(n,\mathbb{R})/\Gamma$ is the Haar measure.

Then, if $\dim(M)\leq n-2$ the measure $\mu$ is $G$-invariant. Moreover, if
$\alpha$ preserves a volume form $\mathrm{vol}$ and if $\dim(M)\leq n-1$ then
the measure $\mu$ is $G$-invariant.

Note that Theorem 15.1 does not require that $\Gamma$ be cocompact.666However,
in the case that $\Gamma$ is nonuniform, the space $M^{\alpha}$ is not compact
and some care is needed to define Lyapunov exponents; in particular, we must
specify a Riemannian metric on $M^{\alpha}$. A Riemannian metric on
$M^{\alpha}$ adapted to this setting is constructed in [AWBFRHZW-
latticemeasure]. Theorem 11.4 follows immediately from Theorem 15.1: since $A$
is abelian (in particular amenable) and the space of probability measures on
$M^{\alpha}$ projecting to the Haar measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ is nonempty, $A$-invariant, and weak-$*$
compact, the Krylov-Bogolyubov theorem implies there is an $A$-invariant Borel
probability measure $\mu$ on $M^{\alpha}$ projecting to the Haar measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$. Since the Haar measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ is $A$-ergodic, we may moreover assume that
$\mu$ is $A$-ergodic. Theorem 15.1 implies $\mu$ is $G$-invariant and Theorem
11.4 then follows from the following elementary claim.

######  Claim 15.2.

The $\Gamma$-action $\alpha$ on $M$ preserves a Borel probability measure if
and only if the induced $G$-action on $M^{\alpha}$ preserves a Borel
probability measure (which necessarily projects to the Haar measure on
$G/\Gamma$).

Indeed, if $\mu$ is a $G$-invariant measure on $M^{\alpha}$ then conditioning
on the fiber of $M^{\alpha}$ over $e\Gamma\in G/\Gamma$ gives an
$\alpha$-invariant measure on $M$ viewed as the fiber of $M^{\alpha}$ over
$e\Gamma$. On the other hand, if $\hat{\mu}$ is an $\alpha$-invariant measure
on $M$ then, writing $m_{G}$ for the Haar measure on $G$, we have
$m_{G}\times\hat{\mu}$ is a (right) $\Gamma$-invariant and (left)
$G$-invariant measure on $G\times M$ and hence descends to a (finite)
$G$-invariant measure on $M^{\alpha}$.

######  Remark 15.3.

For more general semisimple Lie groups $G$ we have the following theorem which
follows from the proof of Theorem 15.1. In this setting, we take $A$ to be a
maximal split Cartan subgroup; that is, $A$ is a maximal, connected, abelian
subgroup of $\mathbb{R}$-diagonalizable elements.

######  Theorem 15.1′.

Let $G$ be a simple Lie group and let $\Gamma\subset G$ be any lattice. Let
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$ be an action and let
$M^{\alpha}$ denote the suspension space with induced $G$-action. Let $\mu$ be
an ergodic, $A$-invariant Borel probability measure on $M^{\alpha}$ whose
projection to $G/\Gamma$ is the Haar measure.

Then, if $\dim(M)<\mathrm{rank}(G)$ then the measure $\mu$ is $G$-invariant.
Moreover, if $\alpha$ preserves a volume form $\mathrm{vol}$ and if
$\dim(M)\leq\mathrm{rank}(G)$ then the measure $\mu$ is $G$-invariant.

######  Remark 15.4.

In fact, Theorem 15.1 and 15.1′ hold for actions by $C^{1}$-diffeomorphisms.
This can be shown by the invariance principle of Avila and Viana [MR2651382]
(see the discussion in Remark 9.6). We present below a proof that uses
(mildly) the $C^{1+\beta}$ hypotheses as this motivates the proof of
Proposition 15.5 (which allows us to establish an analogue of Theorem 15.1′
for manifolds of higher critical dimension) in the next section which requires
the higher regularity of the action.

We proceed with the proof of Theorem 15.1 which is adapted from [Cantat]. This
argument is somewhat simpler than the argument in [1608.04995, AWBFRHZW-
latticemeasure] (though is special for the case $\mathrm{SL}(n,\mathbb{R})$).
The main simplification was observed by S. Hurtado. This argument is exploited
in [BDZ] to obtain results for actions by $C^{1}$ diffeomorphisms.

###### Proof of Theorem 15.1.

Let $\mu$ be an ergodic, $A$-invariant Borel probability measure on
$M^{\alpha}$ whose projection to $\mathrm{SL}(n,\mathbb{R})/\Gamma$ is the
Haar measure.

Recall that $A\simeq\mathbb{R}^{n-1}$. In the non-volume-preserving case,
since $\dim(M)\leq n-2$ there are at most $n-2$ fiberwise Lyapunov exponents.
In particular, the intersection of the kernels of the fiberwise Lyapunov
exponents is a subspace of $A$ whose dimension is at least $1$. In the volume-
preserving case, there are at most $(n-1)$ fiberwise Lyapunov exponents;
however, these satisfy the linear relation they necessarily sum to zero since
the cocycle is cohomologous to an $\mathrm{SL}^{\pm}(n-1,\mathbb{R})$-valued
cocycle (recall 12.1) whence for every $g\in G$,

$0=\int\log|\det({Dg}{\restriction_{{F}}})|\ d\mu=\sum\lambda_{i,\mu}^{F}.$

Thus, if $\dim(M)\leq n-1$ and if $\alpha$ is a volume-preserving action, then
the intersection of the kernels of all fiberwise Lyapunov exponents again has
dimension at least 1. In particular, in either case we may find a nonzero
$s_{0}\in A$ such that

$\lambda^{F}_{i,\mu}(s_{0})=0\text{ for every fiberwise Lyapunov exponent
$\lambda^{F}_{i,\mu}$.}$ (15.1)

Recall that entropy can only decrease under a factor. Thus

$h_{\mu}(s_{0})\geq h_{\mathrm{Haar}}(s_{0})$

where $h_{\mathrm{Haar}}(s_{0})$ denotes the entropy of translating by $s_{0}$
on $\mathrm{SL}(n,\mathbb{R})/\Gamma$ with respect to the Haar measure.

Recall we interpret the roots $\beta$ of $\mathrm{SL}(n,\mathbb{R})$ as the
(non-zero) Lyapunov exponents for the $A$-action on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ with respect to any $A$-invariant measure
and hence also as Lyapunov exponents for the $A$-action on the fiber bundle
$M^{\alpha}$ transverse to the fibers and tangential to the local $G$-orbits.
See discussion in Section 13.2. Let $N_{+}\subset G$ be the subgroup generated
by all root subgroups $U^{\beta}$ with $\beta(s_{0})>0$. Similarly, let
$N_{-}\subset G$ be the subgroup generated by all root subgroups $U^{\beta}$
with $\beta(s_{0})<0$. The orbits of $N_{+}$ and $N_{-}$ in
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ correspond, respectively, to the unstable
and stable manifolds for the action of translation by $s_{0}$ on $G/\Gamma$.
Since $s_{0}$ is in the kernel of all fiberwise Lyapunov exponents, each
tangent space $F(x)$ to the fibers of $M^{\alpha}$ is contained in the neutral
Lyapunov subspace $E^{c}_{s_{0}}(x)$ for the action of $s_{0}$ on
$(M^{\alpha},\mu)$ for almost every $x$. Thus, the orbits of $N_{+}$ and
$N_{-}$ in $M^{\alpha}$ also correspond, respectively, to the unstable and
stable manifolds for the action of $s_{0}$ on $M^{\alpha}$.

We have that

$h_{\mathrm{Haar}}(s_{0})=\sum_{\beta(s_{0})>0}\beta(s_{0})=h_{\mathrm{Haar}}(s_{0}^{-1})=\sum_{\beta(s_{0})<0}(-\beta(s_{0})).$

In particular, from the choice of $s_{0}$, the Margulis–Ruelle inequality
(Theorem 8.6(1)), and the Ledrappier–Young Theorem (8.3) (page 8.3)

$\sum_{\beta(s_{0})>0}\beta(s_{0})=h_{\mathrm{Haar}}(s_{0})\leq
h_{\mu}(s_{0})=h_{\mu}(s_{0}\mid N_{+})\leq\sum_{\beta(s_{0})>0}\beta(s_{0}).$

It follows that

$h_{\mu}(s_{0}\mid N_{+})=\sum_{\beta(s_{0})>0}\beta(s_{0}).$

By Theorem 9.5, it follows that $\mu$ is $N_{+}$-invariant. Similarly we have
that $\mu$ is $N_{-}$-invariant.

In particular, $\mu$ is invariant by the subgroups $N_{-}$, $N_{+}$, and $A$
of $G$. To end the proof, we claim the following standard fact: the subgroups
$N_{-}$ and $N_{+}$ generate all of $\mathrm{SL}(n,\mathbb{R})$. It follows
from the claim that the measure $\mu$ is $G$-invariant.

To prove the claim, it is best to work with Lie algebras. Let
$\mathfrak{n}_{+},$ $\mathfrak{n}_{-}$, and $\mathfrak{a}$ be the Lie algebras
of $N_{-}$ and $N_{+}$, and $A$, respectively. Let $\mathfrak{h}$ be the Lie
algebra generated by $\mathfrak{n}_{+}$ and $\mathfrak{n}_{-}$. For any
$X\in\mathfrak{a}$ we have

$[X,\mathfrak{h}]=\mathfrak{h}$

since $\mathfrak{a}$ normalizes each root space $\mathfrak{g}^{\beta}$. For
roots $\beta,\hat{\beta}$ with $\beta(s_{0})\neq 0$ and
$\hat{\beta}(s_{0})\neq 0$ we have

$[\mathfrak{g}^{\hat{\beta}},\mathfrak{g}^{\beta}]\subset\mathfrak{h}$

by definition. For roots $\beta,\hat{\beta}$ with $\beta(s_{0})>0$ and
$\hat{\beta}(s_{0})=0$ we have

$[\mathfrak{g}^{\hat{\beta}},\mathfrak{g}^{\beta}]=\mathfrak{g}^{\beta+\hat{\beta}}\subset\mathfrak{h}$

since either $\mathfrak{g}^{\beta+\hat{\beta}}=0$ (if $\beta+\hat{\beta}$ is
not a root) or $(\hat{\beta}+\beta)(s_{0})=\beta(s_{0})>0$ (if
$\beta+\hat{\beta}$ is a root). Similarly, for roots $\beta,\hat{\beta}$ with
$\beta(s_{0})<0$ and $\hat{\beta}(s_{0})=0$ we have

$[\mathfrak{g}^{\hat{\beta}},\mathfrak{g}^{\beta}]\subset\mathfrak{h}.$

It follows that $\mathfrak{h}$ is an ideal of the Lie algebra
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{R})$ of $\mathrm{SL}(n,\mathbb{R})$. But
$\mathfrak{sl}(n,\mathbb{R})$ is simple (i.e. has no nontrivial ideals). Since
$\mathfrak{h}\neq\\{0\\}$, it follows that
$\mathfrak{h}=\mathfrak{sl}(n,\mathbb{R})$ and the claim follows. ∎

$\bigast$15.2. Advanced invariance principle: nonresonance implies invariance.
Theorem 15.1 gives the optimal dimension count in Theorem 11.4 for actions by
lattices $\Gamma$ in $\mathrm{SL}(n,\mathbb{R})$. However, for lattices in
other simple Lie groups, the critical dimension in Theorem 15.1′ falls below
the critical dimension expected for the analogous versions of Theorem 11.4 and
Theorem 11.6. For instance, the group $G=\mathrm{Sp}(2n,\mathbb{R})$, the
group of $(2n)\times(2n)$ symplectic matrices over $\mathbb{R}$, has rank $n$.
Theorem 15.1′ implies that for any lattice $\Gamma\subset G$ and any compact
manifold $M$ with $\dim(M)\leq n-1$, any action
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$ preserves a Borel
probability measure. However, the main result of [AWBFRHZW-latticemeasure]
shows for a lattice $\Gamma$ in $\mathrm{Sp}(2n,\mathbb{R})$ that any action
$\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$ preserves a Borel probability
measure when $\dim(M)\leq 2n-2$. To obtain the optimal critical dimensions, it
is necessary to use a more advanced invariance principle developed in
[AWBFRHZW-latticemeasure] and based on key ideas from [AWB-GLY-P3].

Recall that we interpret roots $\beta^{i,j}\colon A\to\mathbb{R}$ as the
nonzero Lyapunov exponents for the action of $A\simeq\mathbb{R}^{n-1}$ on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ (for any $A$-invariant measure on
$G/\Gamma$.) Each root $\beta^{i,j}$ has a corresponding root subgroup
$U^{i,j}\subset\mathrm{SL}(n,\mathbb{R})$. Given an ergodic, $A$-invariant
measure $\mu$ on $M^{\alpha}$ we also have fiberwise Lyapunov exponents
$\lambda^{F}_{1,\mu},\lambda^{F}_{2,\mu},\dots,\lambda^{F}_{p,\mu}\colon
A\to\mathbb{R}$ for the restriction of the derivative of the $A$-action on
$(M^{\alpha},\mu)$ to the fiberwise tangent bundle $F\subset TM^{\alpha}$ in
$M^{\alpha}$. Then, the roots $\beta^{i,j}$ and fiberwise Lyapunov exponents
$\lambda^{F}_{i,\mu}$ are linear functions on the common vector space
$A\simeq\mathbb{R}^{n-1}$. We say that a root $\beta^{i,j}$ is resonant with a
fiberwise Lyapunov exponent $\lambda^{F}_{i,\mu}$ of $\mu$ if they are
positively proportional; that is $\beta^{i,j}$ is resonant with
$\lambda^{F}_{i,\mu}$ if there is a $c>0$ with

$\beta^{i,j}=c\lambda^{F}_{i,\mu}.$

Otherwise we say that $\beta^{i,j}$ is not resonant with
$\lambda^{F}_{i,\mu}$. We say that a root $\beta^{i,j}$ of $G$ is nonresonant
if it is not resonant with any fiberwise Lyapunov exponent
$\lambda^{F}_{i,\mu}$ for the ergodic, $A$-invariant measure $\mu$.

The following is the key proposition from [AWBFRHZW-latticemeasure].

######  Proposition 15.5 ([AWBFRHZW-latticemeasure, Proposition 5.1]).

Suppose $\mu$ is an ergodic, $A$-invariant measure on $M^{\alpha}$ projecting
to the Haar measure on $\mathrm{SL}(n,\mathbb{R})/\Gamma$ under the projection
$\pi\colon M^{\alpha}\to\mathrm{SL}(n,\mathbb{R})/\Gamma$.

Then, for every nonresonant root $\beta^{i,j}$, the measure $\mu$ is
$U^{i,j}$-invariant.

######  Remark 15.6.

Since each root $\beta^{i,j}$ is a nonzero functional on $A$, if a fiberwise
exponent $\lambda^{F}_{i,\mu}$ is zero, then every root $\beta^{i,j}$ is not
resonant with $\lambda^{F}_{i,\mu}$. Since no roots of
$\mathrm{SL}(n,\mathbb{R})$ are positively proportional, if there are $p$
fiberwise Lyapunov exponents $\\{\lambda^{F}_{i,\mu},1\leq i\leq p\\}$ or,
more generally, $p^{\prime}\leq p$ coarse fiberwise Lyapunov exponents
$\\{\chi^{F}_{i,\mu},1\leq i\leq p^{\prime}\\}$ then Proposition 15.5 implies
that $\mu$ is invariant under all-but-$p^{\prime}$ root subgroups $U^{i,j}$.
Moreover, if every fiberwise Lyapunov exponent $\lambda^{F}_{i,\mu}$ is in
general position with respect to every root $\beta^{i,j}$ then from
Proposition 15.5, $\mu$ is automatically $G$-invariant.

$\bigast$15.3. Coarse-Lyapunov Abramov–Rokhlin Theorem and Proof of
Proposition 15.5. The proof of Proposition 15.5 follows from a version of the
Abramov–Rokhlin theorem (see equation (8.2), page 8.2) for entropies
subordinated to coarse-Lyapunov foliations. We outline these ideas and the
proof of Proposition 15.5 in this section.

Each root $\beta^{i,j}$ of $\mathrm{SL}(n,\mathbb{R})$ is a Lyapunov exponent
for the $A$-action on $(M^{\alpha},\mu)$ (corresponding to vectors tangent to
$U^{i,j}$ orbits in $M^{\alpha}$.) Let $\chi^{i,j}$ denote the coarse Lyapunov
exponent for the $A$-action on $(M^{\alpha},\mu)$ containing $\beta^{i,j}$;
that is, $\chi^{i,j}$ is the equivalence class of all Lyapunov exponents for
the $A$-action on $(M^{\alpha},\mu)$ that are positively proportional to
$\beta^{i,j}$. Let $\\{\lambda^{F}_{i,\mu},1\leq i\leq p\\}$ denote the
collection of fiberwise Lyapunov exponents. We have that

$\text{$\chi^{i,j}=\\{\beta^{i,j}\\}$ if $\beta^{i,j}$ is not resonant with
any $\lambda^{F}_{i,\mu}$}.$

Otherwise, $\chi^{i,j}$ contains $\beta^{i,j}$ and all fiberwise Lyapunov
exponents $\lambda^{F}_{i,\mu}\colon A\to\mathbb{R}$ that are positively
proportional to $\beta^{i,j}$.

For $\mu$-a.e. $x\in M^{\alpha}$ there is a coarse Lyapunov manifold
$W^{\chi^{i,j}}(x)$ through $x$ (see Section 7.4.3). If
$\chi^{i,j}=\\{\beta^{i,j}\\}$ then for $x\in M^{\alpha}$, $W^{\chi^{i,j}}(x)$
is simply the $U^{i,j}$-orbit of $x$. Otherwise, $W^{\chi^{i,j}}(x)$ is a
higher-dimensional manifold which intersects the fibers of $M^{\alpha}$
nontrivially. The partition of $(M^{\alpha},\mu)$ into
$W^{\chi^{i,j}}$-manifolds forms an $A$-invariant partition
$\mathcal{F}^{\chi^{i,j}}$ with $C^{1+\beta}$-leaves.

If $\beta^{i,j}$ is resonant with some fiberwise Lyapunov exponent, let
$\chi^{i,j,F}$ denote the corresponding coarse fiberwise Lyapunov exponent;
that is, $\chi^{i,j,F}$ is the equivalence class of fiberwise Lyapunov
exponents that are positively proportional to $\beta^{i,j}$. If $\beta^{i,j}$
is not resonant with any fiberwise Lyapunov exponent, let $\chi^{i,j,F}$
denote the zero functional. If $\chi^{i,j,F}$ is nonzero, for $\mu$-a.e. $x\in
M^{\alpha}$ there is a coarse fiberwise Lyapunov manifold
$W^{\chi^{i,j,F}}(x)$ through $x$. (To construct fiberwise coarse Lyapunov
manifolds $W^{\chi^{i,j,F}}(x)$, recall that the fibers of $M^{\alpha}$ are
permuted by the dynamics of $A$; all constructions in Section 7 may be carried
out fiberwise in the setting of a skew-product of diffeomorphisms over a
measurable base if the $C^{1+\beta}$ norms of the fibers are uniformly
bounded.) If $\chi^{i,j,F}$ is zero, we simply define
$W^{\chi^{i,j,F}}(x)=\\{x\\}$. We have that $W^{\chi^{i,j,F}}(x)$ is contained
in the fiber through $x$ and that $W^{\chi^{i,j}}(x)$ is the $U^{i,j}$-orbit
of $W^{\chi^{i,j,F}}(x)$.

For each $\chi^{i,j}$ and $a\in A$ with $\beta^{i,j}(a)>0$ we define a
conditional entropy of $a$ conditioned on $\chi^{i,j}$-manifolds, denoted by
$h_{\mu}(a\mid\chi^{i,j})$ as in Section 8.7. Similarly, we can define a
conditional entropy of $a$ conditioned on the fiberwise coarse Lyapunov
manifolds associated to $\chi^{i,j,F}$, denoted by
$h_{\mu}(a\mid\chi^{i,j,F})$. In this setting, we have the following “coarse-
Lyapunov Abramov–Rokhlin formula.”

######  Theorem 15.7.

Let $\mu$ be an ergodic, $A$-invariant measure on $M^{\alpha}$ that projects
to the Haar measure on $\mathrm{SL}(n,\mathbb{R})/\Gamma$. For any $a\in A$
with $\beta^{i,j}(a)>0$,

$h_{\mu}(a\mid\chi^{i,j})=h_{\mathrm{Haar}}(a\mid\beta^{i,j})+h_{{\mu}}(a\mid\chi^{i,j,F}).$
(15.2)

Above,

$h_{\mathrm{Haar}}(a\mid\beta^{i,j})$

denotes the conditional entropy of translation by $a$ in
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ conditioned along $U^{i,j}$-orbits in
$\mathrm{SL}(n,\mathbb{R})/\Gamma$.

###### Proof of Theorem 15.7.

We first show the upper bound

$h_{\mu}(a\mid\chi^{i,j})\leq
h_{\mathrm{Haar}}(a\mid\beta^{i,j})+h_{{\mu}}(a\mid\chi^{i,j,F}).$ (15.3)

This is a standard estimate in abstract ergodic theory whose proof we include
for completeness.

Fix $a\in A$ with $\beta^{i,j}(a)>0$. Let $\hat{\eta}$ be an increasing
measurable partition of $G/\Gamma$ subordinate to the partition into
$U^{i,j}$-orbits. Let $\pi\colon M^{\alpha}\to G/\Gamma$ be the natural
projection and let $\eta=\pi^{-1}\hat{\eta}$. Let $\xi\succ\eta$ be an
increasing measurable partition of $(M^{\alpha},\mu)$ subordinate to the
partition into $W^{\chi^{i,j}}$-manifolds. Let $\zeta$ be the partition of
$(M^{\alpha},\mu)$ into the level sets of $\pi\colon M^{\alpha}\to G/\Gamma$;
that is, $\zeta$ is the partition of $M^{\alpha}$ into fibers of the fibration
$\pi\colon M^{\alpha}\to G/\Gamma$. Let $\xi^{F}:=\xi\vee\zeta$ be the join of
$\xi$ and $\zeta$. The partitions $\hat{\eta}$, $\xi$, and $\xi^{F}$ satisfy

  1. (1)

$h_{\mu}(a,\eta)=h_{\mathrm{Haar}}(a,\hat{\eta})=h_{\mathrm{Haar}}(a\mid\beta^{i,j})$,

  2. (2)

$h_{\mu}(a,\xi)=h_{\mu}(a\mid\chi^{i,j}),$ and

  3. (3)

$h_{\mu}(a,\xi^{F})=h_{\mu}(a\mid\chi^{i,j,F}).$

We have the following computation (see for example [MR2729332, Lemma 6.1]):

$\displaystyle h_{\mu}(a\mid\chi^{i,j})$ $\displaystyle:=h_{\mu}(a,\xi)$
$\displaystyle=h_{\mu}(a,\eta\vee\xi)$ $\displaystyle\leq
h_{\mu}(a,\eta)+h_{\mu}\left(a,\xi\vee\bigvee_{n\in\mathbb{Z}}a^{n}(\eta)\right)$
$\displaystyle=h_{\mathrm{Haar}}(a,\hat{\eta})+h_{\mu}(a,\xi\vee\zeta)$
$\displaystyle=h_{\mathrm{Haar}}(a\mid\beta^{i,j})+h_{\mu}(a\mid\chi^{i,j,F})$

and (15.3) follows.

On the other hand, summing over all roots $\beta$ with $\beta(a)>0$ we have
from the classical Abramov–Rokhlin theorem (8.2), the product structure of
entropy in Theorem 8.8, and an analogous version of Theorem 8.8 for the
fiberwise entropy $h_{\mu}(a\mid\zeta)$ appearing in (8.2) that

$\displaystyle h_{\mu}(a)$ $\displaystyle=\sum_{\chi(a)>0}h_{\mu}(a\mid\chi)$
$\displaystyle=\sum_{\beta^{i,j}(a)>0}h_{\mu}(a\mid\chi^{i,j})+\sum_{\begin{subarray}{c}\text{$\chi^{F}$
nonres.}\\\ \chi^{F}(a)>0\end{subarray}}h_{\mu}(a\mid\chi^{F})$
$\displaystyle\leq\sum_{\beta^{i,j}(a)>0}\left(h_{\mathrm{Haar}}(a\mid\beta^{i,j})+h_{\mu}(a\mid\chi^{i,j,F})\right)+\sum_{\begin{subarray}{c}\text{$\chi^{F}$
nonres.}\\\ \chi^{F}(a)>0\end{subarray}}h_{\mu}(a\mid\chi^{F})$
$\displaystyle=\sum_{\beta^{i,j}(a)>0}h_{\mathrm{Haar}}(a\mid\beta^{i,j})+\sum_{\chi^{F}(a)>0}h_{\mu}(a\mid\chi^{F})$
$\displaystyle=h_{\mathrm{Haar}}(a)+h_{\mu}(a\mid\zeta)$
$\displaystyle=h_{\mu}(a).$

In the second and third lines, the second sum is over all fiberwise coarse
Lyapunov exponents that are not resonant with any root $\beta$ of $G$. Since
entropies are non-negative quantities, it follows that

$h_{\mu}(a\mid\chi^{i,j})=h_{\mathrm{Haar}}(a\mid\beta^{i,j})+h_{\mu}(a\mid\chi^{i,j,F})$

for all $\beta^{i,j}$ with $\beta^{i,j}(a)>0$. ∎

######  Remark 15.8.

A more general version of Theorem 15.7 appears in [AWB-GLY-P3, Theorem 13.6]
where the factor map $\pi$ is allowed to be measurable and the measure
$\pi_{*}(\mu)$ on the factor system is an arbitrary ergodic, $A$-invariant
measure.

The proof of Proposition 15.5 is a straightforward consequence of Theorem
15.7.

###### Proof of Proposition 15.5.

Given a root $\beta^{i,j}$ and $a\in A$ such that $\beta^{i,j}(a)>0$ we have
defined the conditional entropy $h_{\mu}(a\mid\beta^{i,j})$ for the
translation by $a$ conditioned on $U^{i,j}$-orbits in $M^{\alpha}$. From an
appropriate version of the Margulis–Ruelle inequality (see Theorem 8.6(1) and
(9.3)), for $a\in A$ with $\beta^{i,j}(a)>0$ we have that

$h_{\mu}(a\mid\beta^{i,j})\leq\beta^{i,j}(a).$ (15.4)

On the other hand, if $\beta^{i,j}$ is nonresonant then $\chi^{i,j,F}$ is the
zero functional whence the coarse Lyapunov manifold $W^{\chi^{i,j}}(x)$
associated to $\chi^{i,j}$ is simply the $U^{i,j}$-orbit of $x$ for every
$x\in M^{\alpha}$ and the term $h_{\mathrm{\mu}}(a\mid\chi^{i,j,F})$ in (15.2)
of Theorem 15.7 vanishes. Hence, by Theorem 15.7,

$h_{\mu}(a\mid\beta^{i,j})=h_{\mu}(a\mid\chi^{i,j})=h_{\mathrm{Haar}}(a\mid\beta^{i,j})+0=\beta^{i,j}(a).$
(15.5)

From (15.4) and (15.5), we have that the conditional entropy
$h_{\mu}(a\mid\beta^{i,j})$ attains its maximal possible value. In particular,
from the invariance principle in Theorem 9.5(3), it follows that $\mu$ is
$U^{i,j}$-invariant. ∎

$\bigast$15.4. Proof of Theorem 11.4 using the advanced invariance principle.
We outline another proof of Theorem 11.4 based on Proposition 15.5. This more
closely mimics the arguments in [1608.04995].

###### Proof of Theorem 11.4 using Proposition 15.5.

From Claim 15.2, it is sufficient to construct a $G$-invariant probability
measure on $M^{\alpha}$. Note that $A\simeq\mathbb{R}^{n-1}$ is abelian (and
in particular amenable, see Remark 18.2) and that the space of probability
measures on $M^{\alpha}$ projecting to the Haar measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ is nonempty, $A$-invariant, and weak-$*$
compact. The Krylov-Bogolyubov theorem thus gives an $A$-invariant probability
measure $\mu$ on $M^{\alpha}$ projecting to the Haar measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$. Moreover, since the Haar measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ is $A$-ergodic, we may assume $\mu$ is
$A$-ergodic.

Let $\dim(M)=d\leq n-2$. The fiberwise tangent bundle $F$ of $M^{\alpha}$ is
$d$-dimensional and therefore there are at most $d$ fiberwise Lyapunov
exponents

$\lambda^{F}_{1,\mu},\cdots,\lambda^{F}_{k,\mu},\quad k\leq d.$

As no pair of roots of $\mathrm{SL}(n,\mathbb{R})$ is positively proportional,
there are at most $d$ roots that are resonant with the fiberwise Lyapunov
exponent $\lambda^{F}_{j,\mu}$. All other roots $\beta^{i,j}$ are nonresonant.
By Proposition 15.5, if $\beta^{i,j}$ is not resonant with any
$\lambda^{F}_{j,\mu}$, then $\mu$ is $U^{i,j}$-invariant.

Let $H\subset\mathrm{SL}(n,\mathbb{R})$ be the subgroup that preserves $\mu$.
We claim $H=G$ completing the proof. As $d\leq n-2$, $\mu$ is invariant under
$A$ and all-but-at-most-$(n-2)$ root subgroups $U^{i,j}$. Then $H$ has
codimension at most $(n-2)$. From [1608.04995, Lemma 2.5], we have that $H$ is
parabolic; that is, $H$ is conjugate to a group of block-upper-triangular
matrices (see Remark 10.8). However, the proper closed parabolic subgroups of
$\mathrm{SL}(n,\mathbb{R})$ of maximal codimension are conjugate to the
codimension $(n-1)$ subgroup

$\left\\{\left(\begin{array}[]{cccc}*&*&\cdots&*\\\ 0&*&\cdots&*\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&*&\cdots&*\end{array}\right)\right\\}.$
(15.6)

(See Section VII.7, especially Proposition 7.76 of [MR1920389] for discussion
on the structure of parabolic subgroups.) As $H$ has codimension at most
$n-2$, it thus follows that $H=G$ as there are no proper parabolic subgroups
of $G$ with codimension less than $(n-1)$. ∎

######  Remark 15.9.

The above proof has the advantage that it generalizes to give invariance of
measures in the optimal critical dimension for actions by lattices in other
Lie groups including $\mathrm{Sp}(2n,\mathbb{R}),$ $\mathrm{SO}(n,n),$ or
$\mathrm{SO}(n,n+1)$ on manifolds of the optimal dimension. As discussed in
Remark 15.4 for a lattice $\Gamma$ in a group such as
$G=\mathrm{Sp}(2n,\mathbb{R})$, the proof in Section 15.1 yields that any
$C^{1+\beta}$ action of $\Gamma$ on a manifold of dimension at most
$\mathrm{rank}(G)-1$, any $A$-invariant measure on $M^{\alpha}$ that projects
to Haar on $G/\Gamma$ is $G$-invariant. However, the above proof establishes
this result for manifolds $M$ where the critical dimension is $r(G)$, the
number in the last column of Table 1 (page 1) defined in [AWBFRHZW-
latticemeasure, 1608.04995] (see also Footnote 4.) For $\mathbb{R}$-split
groups $G$ we have $r(G)=d_{0}(G)$. In particular, the above proof can be
adapted to show the following:

######  Theorem 15.10.

Let $G$ be a higher-rank simple Lie group $G$ with finite center, let $\Gamma$
be a lattice in $G$, let $M$ be a closed manifold, and let
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$ be an action. Then

  1. (1)

if $\dim(M)\leq r(G)-1$, every $A$-invariant probability measure on
$M^{\alpha}$ that projects to the Haar measure on $G/\Gamma$ is $G$-invariant;

  2. (2)

if $\dim(M)\leq r(G)$ and $\alpha$ is volume-preserving, every $A$-invariant
probability measure on $M^{\alpha}$ that projects to the Haar measure on
$G/\Gamma$ is $G$-invariant.

In particular, if $\dim(M)\leq r(G)-1$, every action
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$ preserves a Borel
probability measure.

###  16\. Proof outline of Theorem 11.6

We outline the proof of Theorem 11.6 for the case of $C^{\infty}$ actions of
cocompact lattices in $\mathrm{SL}(n,\mathbb{R})$. That is, for $n\geq 3$, we
consider a cocompact lattice $\Gamma$ in $\mathrm{SL}(n,\mathbb{R})$ and show
that every homomorphism $\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ has
finite image when

  1. (1)

$M$ is a compact manifold of dimension at most $(n-2)$, or

  2. (2)

$M$ is a compact manifold of dimension at most $(n-1)$ and $\alpha$ preserves
a volume form $\mathrm{vol}$.

The broad outline of the proof consists of 3 steps.

####  16.1. Step 1: Subexponential growth

In the case that $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ is cocompact, using
its action on $\mathrm{SL}(n,\mathbb{R})$ and that $\mathrm{SL}(n,\mathbb{R})$
is a proper length space one may show that $\Gamma$ is finitely generated (see
for example [MR2850125, Theorem 8.2]). More generally, it is a classical fact
that all lattices $\Gamma$ in semisimple Lie groups are finitely generated.

Fix a finite symmetric generating set $S$ for $\Gamma$. Given
$\gamma\in\Gamma$, let $|\gamma|=|\gamma|_{S}$ denote the word-length of
$\gamma$ relative to this generating set; that is,

$|\gamma|=\min\\{k:\gamma=s_{k}\cdots s_{1},s_{i}\in S\\}.$

Note that if we replace the finite generating set $S$ with another finite
generating set $S^{\prime}$, there is a uniform constant $C$ such that the
word-lengths are uniformly distorted:

$|\gamma|_{S^{\prime}}\leq C|\gamma|_{S}.$

Thus all definitions below will be independent of the choice of $S$.

Equip $TM$ with a Riemannian metric and corresponding norm.

######  Definition 16.1.

We say that an action $\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M)$ has uniform
subexponential growth of derivatives if for every $\epsilon>0$ there is a
$C=C_{\epsilon}$ such that for every $\gamma\in\Gamma$,

$\sup_{x\in M}\|D_{x}\alpha(\gamma)\|\leq Ce^{\epsilon|\gamma|}.$

Note that if $\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M)$ has uniform
subexponential growth of derivatives it follows for every $\epsilon>0$ that
there is a $C=C_{\epsilon}$ such that for every $\gamma\in\Gamma$,

$\sup_{x\in M}\|D_{x}\alpha(\gamma)\|\geq Ce^{-\epsilon|\gamma|}.$

The following is the main result of [1608.04995] in the case of cocompact
lattices in $\mathrm{SL}(n,\mathbb{R})$.

######  Theorem 16.2 ([1608.04995, Theorem 2.8]).

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a cocompact
lattice. Let $\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$ be an action. Suppose
that either

  1. (1)

$\dim(M)\leq n-2$, or

  2. (2)

$\dim(M)=n-1$ and $\alpha$ preserves a smooth volume.

Then $\alpha$ has uniform subexponential growth of derivatives.

######  Remark 16.3.

The proof of Theorem 16.2 is the only place in the proof of Theorem 11.6 where
cocompactness of $\Gamma$ is used. It is not required for Steps 2 or 3 below.
For $\Gamma=\mathrm{SL}(m,\mathbb{Z})$, the analogue of Theorem 16.2 is
established in [1710.02735] and has been announced for general lattices in
[BFHWM].

####  16.2. Step 2: Strong property (T) and averaging Riemannian metrics

Assume $\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ is an action by
$C^{\infty}$ diffeomorphisms.777For $C^{2}$ actions, one replaces the Hilbert
Sobolev spaces $W^{2,k}(S^{2}(T^{*}M)))$ below with appropriate Banach Sobolev
spaces $W^{p,1}(S^{2}(T^{*}M)))$ and verifies such spaces are of the type
$\mathcal{E}_{10}$ considered in [MR3407190]. The action $\alpha$ of $\Gamma$
on $M$ induces an action $\alpha_{\\#}$ of $\Gamma$ on tensor powers of the
cotangent bundle of $M$ by pull-back: Given $\omega\in(T^{*}M)^{\otimes k}$
write

$\alpha_{\\#}(\gamma)\omega=\alpha(\gamma^{-1})^{*}\omega;$

that is, if $v_{1},\dots v_{k}\in T_{x}M$ then

$\alpha_{\\#}(\gamma)\omega(x)(v_{1},\dots,v_{k})=\omega(x)(D_{x}\alpha(\gamma^{-1})v_{1},\dots,D_{x}\alpha(\gamma^{-1})v_{k}).$

In particular, we obtain an action of $\Gamma$ on the set of Riemannian
metrics which naturally sits as a half-cone inside $S^{2}(T^{*}M)$, the vector
space of all symmetric 2-forms on $M$. Note that $\alpha_{\\#}$ preserves
$C^{\ell}(S^{2}(T^{*}(M)))$, the subspace of all $C^{\ell}$ sections of
$S^{2}(T^{*}M)$ for any $\ell\in\mathbb{N}$.

Fix a volume form $\mathrm{vol}$ on $M$. The norm on $TM$ induced by the
background Riemannian metric induces a norm on each fiber of $S^{2}(T^{*}M)$.
We then obtain a natural notion of measurable and integrable sections of
$S^{2}(T^{*}M)$ with respect to $\mathrm{vol}$. Let
$\mathcal{H}^{k}=W^{2,k}(S^{2}(T^{*}M))$ be the Sobolev space of symmetric
2-forms whose weak derivatives of order $\ell$ are bounded with respect to the
$L^{2}(\mathrm{vol})$-norm for $0\leq\ell\leq k$. Then $\mathcal{H}^{k}$ is a
Hilbert space. Let $\|\cdot\|_{\mathcal{H}^{k}}$ denote the corresponding
Sobolev norm on $\mathcal{H}^{k}$ as well as the induced operator norm on the
space $B(\mathcal{H}^{k})$ of bounded operators on $\mathcal{H}^{k}$. Working
in local coordinates, the Sobolev embedding theorem implies that

$\mathcal{H}^{k}\subset C^{\ell}(S^{2}(T^{*}(M)))$

as long as

$\ell<k-\dim(M)/2.$

In particular, for $k$ sufficiently large, an element $\omega$ of
$\mathcal{H}^{k}$ is a $C^{\ell}$ section of $S^{2}(T^{*}M)$ which will be a
$C^{\ell}$ Riemannian metric on $M$ if it is positive definite.

The action $\alpha_{\\#}$ is a representation of $\Gamma$ by bounded operators
on $\mathcal{H}^{k}$. From Theorem 16.2, we obtain strong control on the norm
growth of the induced representation $\alpha_{\\#}$. In particular, we obtain
that the representation $\alpha_{\\#}\colon\Gamma\to B(\mathcal{H}^{k})$ has
subexponential norm growth:

######  Lemma 16.4.

Let $\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ have uniform
subexponential growth of derivatives. Then, for all $\epsilon^{\prime}>0$
there is $C>0$ such that

$\|\alpha_{\\#}(\gamma)\|_{\mathcal{H}^{k}}\leq
Ce^{\epsilon^{\prime}|\gamma|}$

for all $\gamma\in\Gamma$.

The proof of Lemma 16.4 follows from the chain rule, Leibniz rule, and
computations that bound the growth of higher-order derivatives by polynomial
functions in the growth of the first derivative. See [MR2198325, Lemma 6.4]
and discussion in [1608.04995, Section 6.3].

We use the main result from [MR3407190, MR2423763]: cocompact lattices
$\Gamma$ in higher-rank simple Lie groups (such as $\mathrm{SL}(n,\mathbb{R})$
for $n\geq 3$) satisfy Lafforgue’s strong Banach property (T) first introduced
in [MR2423763]. The result for $\mathrm{SL}(n,\mathbb{R})$ and its cocompact
lattices (as well as other higher-rank simple Lie groups containing a subgroup
isogenous to $\mathrm{SL}(3,\mathbb{R})$) is established by Lafforgue in
Corollary 4.1 and Proposition 4.3 of [MR2423763]; for cocompact lattices in
certain other higher-rank Lie groups (containing a subgroup isogenous to
$\mathrm{Sp}(4,\mathbb{R})$), the results of [MR3407190] are needed. See also
[delaSallenonuniform] for the case of nonuniform lattices. Strong Banach
property (T) considers representations $\pi$ of $\Gamma$ by bounded operators
on certain Banach spaces $E$ (of type $\mathcal{E}_{10}$). If such
representations have sufficiently slow exponential norm growth, then there
exists a sequence of averaging operators $p_{n}$ converging to a projection
$p_{\infty}$ such that for any vector $v\in E$, the limit $p_{\infty}(v)$ is
$\pi$-invariant. In the case that $E$ is a Hilbert space (which we may assume
when $\alpha$ is an action by $C^{\infty}$ diffeomorphisms) we have the
following formulation. Note that Lemma 16.4 (which follows from Theorem 16.2)
ensures our representation $\alpha_{\\#}$ satisfies the hypotheses of the
theorem.

######  Theorem 16.5 ([MR3407190, delaSallenonuniform, MR2423763]).

Let $\mathcal{H}$ be a Hilbert space and for $n\geq 3$, let $\Gamma$ be a
lattice in $\mathrm{SL}(n,\mathbb{R})$.

There exists $\epsilon>0$ such that for any representation $\pi\colon\Gamma\to
B(\mathcal{H})$, if there exists $C_{\epsilon}>0$ such that

$\displaystyle\|\pi(\gamma)\|\leq C_{\epsilon}e^{\epsilon|\gamma|}$

for all $\gamma\in\Gamma$ then there exists a sequence of averaging operators
$p_{n}=\sum w_{i}\pi(\gamma_{i})$ in $B(\mathcal{H})$—where $w_{i}\geq 0$,
$\sum w_{i}=1$, and $w_{i}=0$ for every $\gamma_{i}\in\Gamma$ of word-length
larger than $n$—such that for any vector $v\in\mathcal{H}$, the sequence
$v_{n}=p_{n}(v)\in\mathcal{H}$ converges to an invariant vector
$v^{*}=p_{\infty}(v)$.

Moreover the convergence is exponentially fast: there exist $0<\lambda<1$ and
$C=C_{\lambda}$ such that $\|v_{n}-v_{*}\|\leq C\lambda^{n}\|v\|$.

Theorem 16.5 as stated in [MR3407190, MR2423763] requires that $\Gamma$ be
cocompact. The extension to nonuniform lattices is announced in
[delaSallenonuniform]. The exponential convergence in Theorem 16.5 is often
not explicitly stated in the definition of strong property (T) or in
statements of theorems establishing that the property holds for lattices in
higher-rank simple Lie groups; however, the exponential convergence follows
from the proofs.

We complete Step 2 with the following computation.

######  Proposition 16.6.

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice and
let $\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ be an action with uniform
subexponential growth of derivatives. Then for any $\ell$, there is a
$C^{\ell}$ Riemannian metric $g$ on $M$ such that

$\alpha(\Gamma)\subset\mathrm{Isom}_{g}(M).$

###### Proof.

Consider an arbitrary $C^{\infty}$ Riemannian metric $g$. For any $k$, we have
$g\in\mathcal{H}^{k}$. We apply Theorem 16.5 and its notation to the
representation $\alpha_{\\#}\colon\Gamma\to B(\mathcal{H}^{k})$ with the $g$
the initial vector $v$. As averages of finitely many Riemannian metrics are
still Riemannian metrics we have that $g_{n}:=p_{n}(g)$ is positive definite
for every $n$. In particular, the limit $g_{\infty}=p_{\infty}(g)$ is in the
closed cone of positive (possibly indefinite) symmetric 2-tensors in
$\mathcal{H}^{k}$. Having taken $k$ sufficiently large we have that
$g_{\infty}$ is $C^{\ell}$; in particular, $g_{\infty}$ is continuous,
everywhere defined, and positive everywhere. We need only confirm that
$g_{\infty}$ is non-degenerate, i.e. is positive definite on $T_{x}M$ for
every $x\in M$.

Given any $x\in M$ and unit vector $\xi\in T_{x}M$, for any $\epsilon>0$ we
have from Definition 16.1 that there is a $C_{\epsilon}>0$ such that

$\displaystyle p_{n}(g)(\xi,\xi)$ $\displaystyle=\left(\sum
w_{i}\alpha_{\\#}(\gamma_{i})g\right)(\xi,\xi)$ $\displaystyle=\sum
w_{i}g(D_{x}\alpha(\gamma_{i}^{-1})\xi,D_{x}\alpha(\gamma_{i}^{-1})\xi)$
$\displaystyle\geq\frac{1}{C_{\epsilon}^{2}}e^{-2\epsilon n}$

where we use that $w_{i}>0$ only when $\gamma_{i}$ has word-length at most
$n$.

On the other hand, from the exponential convergence in Theorem 16.5 we have

$\displaystyle|p_{n}(g)(\xi,\xi)-p_{\infty}(g)(\xi,\xi)|$ $\displaystyle\leq
C_{\lambda}\lambda^{n}.$

Thus

$p_{\infty}(g)(\xi,\xi)\geq\frac{1}{C_{\epsilon}^{2}}e^{-2\epsilon
n}-C_{\lambda}\lambda^{n}$

for all $n\geq 0$. Taking $\epsilon>0$ sufficiently small we can ensure that

${C_{\epsilon}^{2}}e^{2\epsilon n}<\frac{1}{C_{\lambda}}\lambda^{-n}$

for all sufficiently large $n$. Then, for all sufficiently large $n$ we have

$\frac{1}{C_{\epsilon}^{2}}e^{-2\epsilon n}>C_{\lambda}\lambda^{n}$

and thus $p_{\infty}(g)(\xi,\xi)>0.$ ∎

####  16.3. Step 3: Margulis superrigidity with compact codomain

From Steps 1 and 2 we have that any action
$\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ as in Theorem 11.6 preserves
a $C^{\ell}$ Riemannian metric $g$. In the general case of $C^{2}$-actions (or
even $C^{1+\beta}$-actions), we have that any action
$\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$ preserves a continuous Riemannian
metric $g$. See [1608.04995, Theorem 2.7]. We thus have

$\alpha\colon\Gamma\to\mathrm{Isom}_{g}(M).$

The group $\mathrm{Isom}_{g}(M)$ is compact. When $g$ is at least $C^{1}$, it
follows from the classical result of Myers and Steenrod [MR1503467] that
$\mathrm{Isom}_{g}(M)$ is a Lie group; for continuous $g$ we use the solution
to the Hilbert–Smith conjecture for actions by bi-Lipschitz maps [MR1464908]
to conclude that $\mathrm{Isom}_{g}(M)$ is a Lie group. Let $\dim(M)=m$. Then

$\dim(\mathrm{Isom}_{g}(M))\leq\dfrac{m(m+1)}{2}.$ (16.1)

Indeed, the orbit of any point $p\in M$ under $\mathrm{Isom}_{g}(M)$ has
dimension at most $m$ and the dimension of the stabilizer of a point is at
most $\frac{m(m-1)}{2}$, the dimension of $\mathrm{SO}(m)$; thus

$\dim(\mathrm{Isom}_{g}(M))\leq m+\frac{m(m-1)}{2}.$

With $K=\mathrm{Isom}_{g}(M)$ we thus obtain a compact-valued representation
$\alpha\colon\Gamma\to K$. By equation (16.1), if
$m<\frac{1}{2}\sqrt{8n^{2}-7}-\frac{1}{2}$ then
$\dim(\mathfrak{su}(n))=n^{2}-1>\dim(K)$; by conclusion (2) of Theorem 12.4,
$\alpha(\Gamma)$ is thus contained in a 0-dimensional subgroup of $K$. This
holds in particular if $m\leq n-1$. We thus conclude that the image

$\alpha(\Gamma)\subset K=\mathrm{Isom}_{g}(M)$

is finite.

Summarizing the arguments from Steps 2 and 3, we obtain the following.

######  Theorem 16.7.

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice. Let
$\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$ be an action with uniform
subexponential growth of derivatives.

Then, if

$\dim(M)<\frac{1}{2}\sqrt{8n^{2}-7}-\frac{1}{2},$

the image $\alpha(\Gamma)$ is finite.

###  17\. Proof outline of Theorem 16.2

To establish Theorem 11.6, from the discussion in Section 16 it is enough to
establish Theorem 16.2: the action $\alpha$ has uniform subexponential growth
of derivatives. We outline the proof of Theorem 16.2.

####  17.1. Setup for proof

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a cocompact
lattice. Let $M$ be a compact manifold and let
$\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$ be an action. Assume either that
$\dim(M)\leq n-2$ or that $\dim(M)\leq n-1$ and that $\alpha$ preserves a
volume form. We recall the following constructions from the proof of Theorem
11.4:

  1. (1)

The manifold $M^{\alpha}=(\mathrm{SL}(n,\mathbb{R})\times M)/\Gamma$ is the
suspension space introduced in Section 14.1. $M^{\alpha}$ is a fiber bundle
over $\mathrm{SL}(n,\mathbb{R})/\Gamma$ with fibers diffeomorphic to $M$.
Moreover, $M^{\alpha}$ and $\mathrm{SL}(n,\mathbb{R})/\Gamma$ have natural
(left) $\mathrm{SL}(n,\mathbb{R})$-actions and the projection $\pi\colon
M^{\alpha}\to\mathrm{SL}(n,\mathbb{R})/\Gamma$ intertwines these $G$-actions.

  2. (2)

$A\subset\mathrm{SL}(n,\mathbb{R})$ denotes the subgroup of diagonal matrices
with positive entries. We have $A\simeq\mathbb{R}^{n-1}$ which is a higher-
rank, free abelian group if $n\geq 3$.

  3. (3)

Given an ergodic, $A$-invariant Borel probability measure $\mu$ on
$M^{\alpha}$ we have fiberwise Lyapunov exponents.

$\lambda^{F}_{1,\mu},\dots,\lambda^{F}_{p,\mu}\colon A\to\mathbb{R}$

for the restriction of the derivative of the $A$-action on $M^{\alpha}$ to the
fibers of $M^{\alpha}$ introduced in Section 14.2.

  4. (4)

$\beta^{i,j}\colon A\to\mathbb{R}$ are the roots of
$\mathrm{SL}(n,\mathbb{R})$ and $U^{i,j}$ are the corresponding root subgroups
introduced in Section 13.1.

####  17.2. Two key propositions

The proof of Theorem 16.2 is by contradiction and follows from the following
two propositions. Our first key proposition is an analogue of Proposition 7.2.

######  Proposition 17.1.

Suppose that $\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M)$ fails to have uniform
subexponential growth of derivatives. Then there exists a Borel probability
measure $\mu^{\prime}$ on $M^{\alpha}$ such that

  1. (1)

$\mu^{\prime}$ is $A$-invariant and ergodic;

  2. (2)

there exists a nonzero fiberwise Lyapunov exponent
$\lambda^{F}_{j,\mu^{\prime}}\colon A\to\mathbb{R}$.

The proof of Proposition 17.1 is very similar to the proof of Proposition 7.2
with some minor modifications and notational differences. We include an
outline of the proof in Section 18.3; see also [1608.04995, Section 4] for
complete details. Obtaining uniform control on the growth of a cocycle from
bounds on the top Lyapunov exponent over all invariant measures is a standard
technique in dynamical systems; see for instance [MR1986306, MR1643183,
Hurtado_Burnside, MR2776369].

The measure $\mu^{\prime}$ in Proposition 17.1 projects to an ergodic,
$A$-invariant measure on $\mathrm{SL}(n,\mathbb{R})/\Gamma$. If $\mu^{\prime}$
projected to the Haar measure on $\mathrm{SL}(n,\mathbb{R})/\Gamma$ then, from
Theorem 15.1 and the bounds on the dimension $M$, the measure $\mu^{\prime}$
would be $G$-invariant and, as explained below, the proof of Theorem 16.2
would be complete. However, there may exist ergodic $A$-invariant measures on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ that are not the Haar measure.888In fact,
for certain lattices $\Gamma$ there exist ergodic $A$-invariant measures on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ that have positive entropy for some element
of $A$ as shown by M. Rees; see [MR1989231, Section 8].

By carefully averaging the measure $\mu^{\prime}$ along root subgroups
$U^{i,j}$ and applying Ratner’s measure classification theorem [MR1262705] to
the projected measure on $\mathrm{SL}(n,\mathbb{R})/\Gamma$ we obtain the
following.

######  Proposition 17.2.

Let $\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M)$ be an action. Suppose there
exists an ergodic, $A$-invariant measure $\mu^{\prime}$ on the suspension
space $M^{\alpha}$ with a nonzero fiberwise Lyapunov exponent
$\lambda^{F}_{j^{\prime},\mu^{\prime}}\colon A\to\mathbb{R}$. Then there
exists a Borel probability measure $\mu$ on $M^{\alpha}$ such that

  1. (1)

$\mu$ is $A$-invariant and ergodic;

  2. (2)

there exists a nonzero fiberwise Lyapunov exponent $\lambda^{F}_{j,\mu}\colon
A\to\mathbb{R}$;

  3. (3)

$\mu$ projects to the Haar measure on $\mathrm{SL}(n,\mathbb{R})/\Gamma$.

######  Remark 17.3.

  1. (1)

Propositions 17.1 and 17.2 hold in full generality; they do not depend on the
comparison between the dimension of $M$ and the rank of
$\mathrm{SL}(n,\mathbb{R})$. The constraint on the dimension of $M$ is used to
obtain a contradiction in the proof of Theorem 16.2 by applying Theorem 15.1
and Zimmer’s cocycle superrigidity to the fiberwise derivative cocycle.

  2. (2)

Propositions 17.1 and 17.2 heavily use the fact that $\Gamma$ is cocompact in
$\mathrm{SL}(n,\mathbb{R})$ so that the manifold $M^{\alpha}$ is compact. For
instance, if $M^{\alpha}$ is not compact then the proof of Proposition 17.1
(compare with proof of Proposition 7.2) fails as there may be escape of mass
into the cusp of $G/\Gamma$. Thus, more subtle arguments are required to
establish the analogue of Theorem 16.2 in the case that $\Gamma$ is
nonuniform. In the case that $\Gamma=\mathrm{SL}(n,\mathbb{Z})$, such
arguments appear in [1710.02735].

  3. (3)

Both Proposition 17.1 and Proposition 17.2 hold for $C^{1}$ actions. The
$C^{1+\beta}$ hypotheses is later used (along with the dimension bounds) to
conclude that the $A$-invariant measure $\mu$ obtained in Proposition 17.2 is,
in fact, $G$-invariant by applying Theorem 15.1.

####  17.3. Proof of Theorem 16.2

We deduce Theorem 16.2 from Proposition 17.1, Proposition 17.2, Theorem 15.1,
and Theorem 12.2.

###### Proof of Theorem 16.2.

Let $\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$ be as in Theorem 16.2. For the
sake of contradiction, assume that

$\alpha\colon\Gamma\to\mathrm{Diff}^{2}(M)$

fails to have uniform subexponential growth of derivatives. Let $\mu^{\prime}$
be the measure guaranteed by Proposition 17.1. We then apply Proposition 17.2
to obtain an ergodic, $A$-invariant Borel probability measure $\mu$ on
$M^{\alpha}$ that projects to the Haar measure on $G/\Gamma$ and has a non-
zero fiberwise Lyapunov exponent. In either case considered in Theorem 16.2,
it follows from Theorem 15.1 that $\mu$ is $G$-invariant.

Recall that we write $\pi\colon M^{\alpha}\to\mathrm{SL}(n,\mathbb{R})/\Gamma$
for the natural projection and let $F$ be the fiberwise tangent bundle; that
is, $F$ is sub-vector-bundle of $TM^{\alpha}$ given by $F=\ker D\pi$. As $F$
is $G$-invariant, we may apply Zimmer’s cocycle superrigidity theorem, Theorem
12.2, to the fiberwise derivative cocycle
$\mathcal{A}(g,x)={D_{x}g}{\restriction_{{F(x)}}}$ of the $\mu$-preserving
$\mathrm{SL}(n,\mathbb{R})$-action on $M^{\alpha}$. Since the fibers have
dimension at most $n-1$ and since there are no non-trivial representations
$\rho\colon\mathrm{SL}(n,\mathbb{R})\to\mathrm{SL}(d,\mathbb{R})$ for $d<n$,
it follows from Theorem 12.2 that the fiberwise derivative cocycle
$\mathcal{A}(g,x)={D_{x}g}{\restriction_{{F(x)}}}$ is cohomologous to a
compact-valued cocycle: there is a compact group
$K\subset\mathrm{SL}(d,\mathbb{R})$ and a measurable map $\Phi\colon
M^{\alpha}\to\mathrm{GL}(d,\mathbb{R})$ such that

$\Phi(g\cdot x){D_{x}g}{\restriction_{{F(x)}}}\Phi(x)^{-1}\in K.$

By Poincaré recurrence to sets on which the norm and conorm of $\Phi$ are
bounded, it follows for any $g\in G$ and $\epsilon>0$ that the set of $x\in
M^{\alpha}$ such that

$\liminf_{n\to\infty}\frac{1}{n}\log\|{D_{x}g^{n}}{\restriction_{{F(x)}}}\|\geq{\epsilon}$

has $\mu$-measure zero. This contradicts the existence of nonzero fiberwise
Lyapunov exponent for $\mu$. This contradiction completes the proof of Theorem
16.2. ∎

###  18\. Discussion of the proof of Propositions 17.1 and 17.2

We outline the main steps in the proof of Propositions 17.1 and 17.2.

####  18.1. Averaging measures on $M^{\alpha}$

Let $H=\\{h^{t}:t\in\mathbb{R}\\}$ be a 1-parameter subgroup of
$\mathrm{SL}(n,\mathbb{R})$. Given a measure $\mu$ on $M^{\alpha}$ and $T\geq
0$ we define

$H^{T}\ast\mu:=\frac{1}{T}\int_{0}^{T}(h^{t})_{*}\mu\ dt$

to be the measure obtained by averaging the translates of $\mu$ over the
interval $[0,T]$.

Let $s\in A$. Given any $s$-invariant measure $\mu$ on $M^{\alpha}$, the
average top fiberwise Lyapunov exponent of $s$ with respect to $\mu$ is

$\lambda^{F}_{\mathrm{top}}(s,\mu)=\inf_{n\geq
1}\frac{1}{n}\int\log\|{D_{x}(s^{n})}{\restriction_{{F}}}\|\ d\mu(x).$ (18.1)

Note that if $\mu$ is moreover $A$-invariant and $A$-ergodic with fiberwise
Lyapunov exponents $\lambda^{F}_{1,\mu},\dots,\lambda^{F}_{p,\mu}\colon
A\to\mathbb{R}$ then

$\lambda^{F}_{\mathrm{top}}(s,\mu)=\max_{1\leq i\leq
p}\lambda^{F}_{i,\mu}(s).$

We have the following facts which we invoke throughout our averaging
procedures.

######  Claim 18.1.

Let $s\in A$ and let $\mu$ be an $s$-invariant measure on $M^{\alpha}$. Let
$H=\\{h^{t},t\in\mathbb{R}\\}$ be a one-parameter group contained in the
centralizer of $s$ in $\mathrm{SL}(n,\mathbb{R})$.

  1. (1)

The measure $H^{T}\ast\mu$ is $s$-invariant for every $T\geq 0$.

  2. (2)

Any weak-$*$ limit point of $\\{H^{T}\ast\mu\\}$ as $T\to\infty$ is
$s$-invariant.

  3. (3)

Any weak-$*$ limit point of $\\{H^{T}\ast\mu\\}$ as $T\to\infty$ is
$H$-invariant.

  4. (4)

$\lambda^{F}_{\mathrm{top}}(s,H^{T}\ast\mu)=\lambda^{F}_{\mathrm{top}}(s,\mu)$
for every $T\geq 0$.

  5. (5)

If $\mu^{\prime}$ is a weak-$*$ limit point of $\\{H^{T}\ast\mu\\}$ as
$T\to\infty$ then

$\lambda^{F}_{\mathrm{top}}(s,\mu^{\prime})\geq\lambda^{F}_{\mathrm{top}}(s,\mu).$

(1) is clear from definition and (2) follows since the set of $s$-invariant
measures is closed. (3) follows from (the proof of) the Krylov-Bogolyubov
theorem (see 7.3). (4) is a standard computation which follows from the
compactness of $M^{\alpha}$ and hence boundedness of the cocycle. Indeed we
have

$\displaystyle\lambda^{F}_{\mathrm{top}}$
$\displaystyle(s,H^{T}\ast\mu)=\inf_{n\to\infty}\frac{1}{n}\int\log\|{D_{x}(s^{n})}{\restriction_{{F}}}\|\
d(H^{T}\ast\mu)(x)$
$\displaystyle=\inf_{n\to\infty}\frac{1}{n}\frac{1}{T}\int\int_{t=0}^{T}\log\|{D_{h^{t}\cdot
x}(s^{n})}{\restriction_{{F}}}\|\ dt\,d\mu(x)$
$\displaystyle=\inf_{n\to\infty}\frac{1}{n}\frac{1}{T}\int\int_{t=0}^{T}\log\|{D_{h^{t}\cdot
x}(h^{t}s^{n}h^{-t})}{\restriction_{{F}}}\|\ dt\,d\mu(x)$
$\displaystyle\leq\inf_{n\to\infty}\frac{1}{n}\frac{1}{T}\int\int_{t=0}^{T}\log\|{D_{h^{t}\cdot
x}(h^{-t})}{\restriction_{{F}}}\|$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\log\|{D_{x}(s^{n})}{\restriction_{{F}}}\|+\log\|{D_{s\cdot
x}(h^{t})}{\restriction_{{F}}}\|\ dt\,d\mu(x)$
$\displaystyle\leq\inf_{n\to\infty}\frac{1}{n}\left(\int\log\|{D_{x}(s^{n})}{\restriction_{{F}}}\|\
d\mu(x)+2K\right)$

where

$K=\sup\left\\{\log\|{D_{x}(h^{t})}{\restriction_{{F}}}\|:x\in
M,t\in[-T,T]\right\\}.$

(5) follows from the well-known fact that the average top Lyapunov exponent is
upper-semicontinuous on the set of $s$-invariant measures (see for example
[MR3289050] or [1608.04995, Lemma 3.2(b)]). Indeed, in the weak-$*$ topology,
for each $n$ the function

$\mu\mapsto\frac{1}{n}\int\log\|{D_{x}(s^{n})}{\restriction_{{F}}}\|\ d\mu(x)$

is continuous. The pointwise infimum of a family of continuous functions is
upper-semicontinuous.

######  Remark 18.2.

Recall that a Følner sequence in a Lie group $H$ equipped with a left-Haar
measure $m_{H}$ is a sequence $\\{F_{n}\\}$ of Borel subsets $F_{n}\subset H$,
with $0<m_{H}(F_{n})<\infty$, such that for every compact subset $Q\subset H$
one has

$\lim_{n\to\infty}\sup_{h\in Q}\dfrac{m_{H}\big{(}(h\cdot F_{n})\triangle
F_{n}\big{)}}{m_{H}(F_{n})}=0.$

If $H$ admits a Følner sequence then $H$ is said to be amenable. When
$H=\mathbb{R}$, a Følner sequence is given by $F_{n}=[0,n]$. Examples of
amenable groups include abelian groups, nilpotent groups, solvable groups, and
compact groups. See [BdHV] for more details.

Consider $H$ to be an amenable Lie subgroup of $G=\mathrm{SL}(n,\mathbb{R})$.
Given a Borel probability measure $\mu$ on $M^{\alpha}$ and a Følner sequence
$\\{F_{n}\\}$ in $H$ we define

$F_{n}\ast\mu:=\dfrac{1}{m_{H}(F_{n})}\int_{F_{n}}h_{*}\mu\ dm_{H}(h).$

By a computation analogous to (7.3) in the proof of 7.3, any weak-$*$ limit
point $\hat{\mu}$ of the sequence $\\{F_{n}\ast\mu\\}$ as $n\to\infty$ is an
$H$-invariant measure on $M^{\alpha}$. Moreover, properties analogous to those
in Claim 18.1 hold when averaging an $s$-invariant measure $\mu$ against a
Følner sequence $\\{F_{n}\\}$ in an amenable subgroup $H$ contained in the
centralizer $C_{G}(s)$ of $s$. See [1608.04995, Lemma 3.2] for precise
formulations.

####  18.2. Averaging measures on $\mathrm{SL}(n,\mathbb{R})/\Gamma$

When averaging probability measures on $\mathrm{SL}(n,\mathbb{R})/\Gamma$
along 1-parameter unipotent subgroups we obtain additional properties of the
limiting measures. The results stated in the following proposition are
consequences of Ratner’s measure classification and equidistribution theorems
for unipotent flows [MR1262705, MR1054166, MR1075042]. See also [MR2158954].
We do not formulate Ratner’s theorems here but only the consequences we use in
the remainder.

######  Proposition 18.3.

Let $\hat{\mu}$ be a Borel probability measure on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$. For each 1-parameter root subgroup
$U^{i,j}$

  1. (1)

the weak-$*$ limit

$U^{i,j}\ast\hat{\mu}:=\lim_{T\to\infty}\\{(U^{i,j})^{T}\ast\hat{\mu}:T\geq
0\\}$

exists;

  2. (2)

if $\hat{\mu}$ is $A$-invariant, so is $U^{i,j}\ast\hat{\mu}$;

  3. (3)

if $\hat{\mu}$ is $A$-invariant and $A$-ergodic, the measure
$U^{i,j}\ast\hat{\mu}$ is $A$-ergodic;

  4. (4)

if $\hat{\mu}$ is $A$-invariant and $U^{i,j}$-invariant then $\hat{\mu}$ is
$U^{j,i}$-invariant.

Proposition 18.3(1) follows from Ratner’s measure classification and
equidistribution theorems for unipotent flows. When $U$ is higher-dimensional,
we use an analogue of Proposition 18.3(1) due to Shah [MR1291701, Corollary
1.3]. Proposition 18.3(2) follows from the fact that $A$ normalizes $U^{i,j}$
and that the limit in Proposition 18.3(1) exists and is hence unique.
Proposition 18.3(4) is a consequence of Theorem 9 in [MR1262705] or
Proposition 2.1 in [MR1054166].

Proposition 18.3(3) is a short argument that uses the $A$-invariance of
$\hat{\mu}$ and the pointwise ergodic theorem: Since there is $s\in A$ such
that $U^{i,j}$-orbits are contracted by $s$, by the pointwise ergodic theorem,
the measurable hull of the partition into $U^{i,j}$-orbits refines the ergodic
decomposition for $A$. (See Proposition 8.12 and Theorem C.4 in Appendix C.)
Let $\eta$ be the measurable hull of the partition into $U^{i,j}$-orbits and
let $\\{\hat{\mu}^{\eta}_{x}\\}$ be a family of conditional measures of
$\hat{\mu}$ for this partition. (Note that from Ratner’s equidistribution
theorem, we have that $\hat{\mu}^{\eta}_{x}$ is a homogeneous measure on a
closed homogeneous submanifold.) If $\phi$ is a bounded, $A$-invariant
measurable function then for $\hat{\mu}$-a.e. $x$, $\phi$ is constant
$\hat{\mu}^{\eta}_{x}$-almost surely; in particular,

$\phi(x)=\int\phi\ d\hat{\mu}^{\eta}_{x}$

for $\hat{\mu}$-a.e. $x$. Then $x\mapsto\int\phi\ d\hat{\mu}^{\eta}_{x}$ is a
$\mu$-almost everywhere defined, $A$-invariant function. In particular,
$x\mapsto\int\phi\ d\hat{\mu}^{\eta}_{x}$ is constant $\mu$ a.s. by ergodicity
of $\mu$. It follows that $\phi$ is constant $\hat{\mu}$-a.s. and ergodicity
follows.

####  18.3. Proof of Proposition 17.1

We outline the proof of Proposition 17.1. Recall the notation introduced in
Section 14.1. In particular, $\pi\colon M^{\alpha}\to G/\Gamma$ is the
canonical projection and $F=\ker(D\pi)$ is the fiberwise tangent bundle of
$M^{\alpha}$. We write the derivative of translation by $g$ in $M^{\alpha}$ as
$Dg$ and the restriction to the fiber of $F$ through $x\in M^{\alpha}$ by
${D_{x}g}{\restriction_{{F(x)}}}.$ Equip $M^{\alpha}$ with any Riemannian
metric and write

$\|{Dg}{\restriction_{{F}}}\|=\sup_{x\in
M^{\alpha}}\|{D_{x}g}{\restriction_{{F(x)}}}\|.$

Let $K=\mathrm{SO}(n)$. We equip $G$ with a right-invariant,
left-$K$-invariant Riemannian metric and induced distance function
$d(\cdot,\cdot)$. We note that relative to such a metric, all $A$-orbits are
geodesically embedded in $G$. We have the following elementary claim which
allows us to transfer exponential growth properties between the
$\Gamma$-action on $M$ and the $G$-action on the fibers of $M^{\alpha}$.

######  Claim 18.4.

If $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ is cocompact and if $M$ is
compact, then any action

$\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M)$

has uniform subexponential growth of derivatives if and only if for every
$\epsilon>0$ there is a $C$ such that for all $g\in\mathrm{SL}(n,\mathbb{R})$,

$\|{Dg}{\restriction_{{F}}}\|\leq Ce^{\epsilon d(e,g)}.$

With the above claim, we outline the main steps in the proof of Proposition
17.1.

###### Proof of Proposition 17.1.

We assume $\alpha\colon\Gamma\to\mathrm{Diff}^{1}(M)$ fails to have uniform
subexponential growth of derivatives. Then, by 18.4, there exist $\epsilon>0$,
integers $m_{n}\in\mathbb{N}$ with ${m_{n}}\to\infty$, elements $g_{m_{n}}\in
G$ with $d(g_{m_{n}},e)={m_{n}}$, points $x_{m_{n}}\in M_{\alpha},$ and unit
vectors $v_{m_{n}}\in T_{x_{m_{n}}}M_{\alpha}$ such that

$\|D_{x_{m_{n}}}g_{m_{n}}(v_{m_{n}})\|\geq e^{\epsilon{m_{n}}}.$

Let $UF$ denote the unit sphere bundle in $F$ and, given $g\in G$, let $UDg$
denote the induced action on $UF$: given $x\in M^{\alpha}$ and $v\in UF(x)$
write

$UD_{x}g(v)=\frac{D_{x}g(v)}{\|D_{x}g(v)\|}$

and

$UDg(x,v)=\left(g\cdot x,UD_{x}g(v)\right).$

By the singular value decomposition of matrices, the group
$G=\mathrm{SL}(n,\mathbb{R})$ can be written as $G=KAK$ where
$K=\mathrm{SO}(n)$. (For general simple Lie groups $G$ we use the Cartan
decomposition). We can thus write each $g_{m_{n}}\in G$ as

$g_{m_{n}}=k_{n}a_{n}k^{\prime}_{n}$

where $k_{n},k^{\prime}_{n}\in K$ and $a_{n}\in A$. Write

$x^{\prime}_{n}=k^{\prime}_{n}\cdot x_{m_{n}},\quad
x^{\prime\prime}_{n}=a_{n}k^{\prime}_{n}\cdot x_{m_{n}},$
$v^{\prime}_{n}=UD_{x_{m_{n}}}k^{\prime}_{n}(v_{m_{n}}),\quad
v^{\prime\prime}_{n}=UD_{x_{m_{n}}}(a_{n}k^{\prime}_{n})(v_{m_{n}}).$

Then

$\displaystyle\|$ $\displaystyle
D_{x_{m_{n}}}g_{m_{n}}(v_{m_{n}})\|=\|D_{x_{n}^{\prime\prime}}k_{n}(v_{n}^{\prime\prime})\|\cdot\|D_{x_{n}^{\prime}}a_{n}(v_{n}^{\prime})\|\cdot\|D_{x_{m_{n}}}k^{\prime}_{n}(v_{m_{n}})\|$

and so

$\epsilon\leq\lim_{n\to\infty}\frac{1}{{m_{n}}}\log\|D_{x_{m_{n}}}g_{m_{n}}(v_{m_{n}})\|=\lim_{n\to\infty}\frac{1}{{m_{n}}}\log\|D_{x^{\prime}_{n}}a_{n}(v^{\prime}_{n})\|$

as $\|{D_{x}k}{\restriction_{{F}}}\|$ is uniformly bounded over all $k\in K$
and $x\in M^{\alpha}$.

Note that

$|{m_{n}}-d(a_{n},e)|=|d(g_{m_{n}},e)-d(a_{n},e)|\leq
d(k_{n},e)+d(k^{\prime}_{n},e)$

is uniformly bounded in $n$. Thus ${m_{n}}^{-1}d(a_{n},e)\to 1$. As
$A\simeq\mathbb{R}^{n-1}$, for each $n$ there is a unique $\tilde{a}_{n}$ with
$a_{n}=(\tilde{a}_{n})^{m_{n}}$; moreover, as $A$ is geodesically embedded in
$G$, we have $d(\tilde{a}_{n},e)\to 1$.

For each $n$, let $\nu_{n}$ be the empirical measure on $UF$ given by

$\nu_{n}=\frac{1}{{m_{n}}}\sum_{j=0}^{{m_{n}}-1}(\tilde{a}_{n})^{j}_{*}\delta_{(x_{n}^{\prime},v_{n}^{\prime})}.$

Taking a subsequence $\\{n_{j}\\}$, we may assume that $\nu_{n_{j}}$ converges
to some $\nu_{\infty}$ and that $\tilde{a}_{n_{j}}$ converges to some $s\in
A$. Note that $d(s,e)=1$. Let $\overline{\mu}$ denote the image of
$\nu_{\infty}$ under the natural projection $UF\to M^{\alpha}$. Adapting the
proofs of 7.3 and Proposition 7.2 one can show that

  1. (1)

$\nu_{\infty}$ is $UD{s}$-invariant whence $\overline{\mu}$ is $s$-invariant;

  2. (2)

$\lambda^{F}_{\mathrm{top}}(s,\overline{\mu})\geq\epsilon>0$.

Take a Følner sequence $\\{F_{n}\\}$ in $A$ and let $\tilde{\mu}$ be any
weak-$*$ limit point of $\\{F_{n}\ast\overline{\mu}\\}$ as $n\to\infty$. Then,
from analogues of the properties in 18.1 for averaging over Følner sequences,
we have that

  1. (1)

$\tilde{\mu}$ is $A$-invariant;

  2. (2)

$\lambda^{F}_{\mathrm{top}}(s,\tilde{\mu})\geq\epsilon>0$.

We take $\mu^{\prime}$ to be an $A$-ergodic component of $\tilde{\mu}$ with
$\lambda^{F}_{\mathrm{top}}(s,\mu^{\prime})\geq\epsilon>0$. ∎

####  18.4. Proof of Proposition 17.2 for $\mathrm{SL}(3,\mathbb{R})$

To simplify ideas, we outline the proof of Proposition 17.2 assuming that
$\Gamma$ is a cocompact lattice in $\mathrm{SL}(3,\mathbb{R})$. We perform two
averaging procedures on the measure $\mu^{\prime}$ from the hypotheses of
Proposition 17.2 to obtain the measure $\mu$ in the conclusion of Proposition
17.2.

###### Proof of Proposition 17.2 for $\Gamma\subset\mathrm{SL}(3,\mathbb{R})$.

Take $\mu_{0}=\mu^{\prime}$ to be the ergodic, $A$-invariant probability
measure in the hypotheses of Proposition 17.2 with nonzero fiberwise exponent

$\lambda^{F}_{j,\mu_{0}}\colon A\to\mathbb{R}$, $\lambda^{F}_{j,\mu_{0}}\ \neq
0$.

First averaging. Consider the elements

$s=\mathrm{diag}(\tfrac{1}{4},2,2)$ and
$\overline{s}=\mathrm{diag}(2,2,\tfrac{1}{4})$

of $A\subset\mathrm{SL}(3,\mathbb{R})$. Note that $s$ and $\overline{s}$ are
linearly independent and hence form a basis for $A\simeq\mathbb{R}^{2}$. As
the linear functional $\lambda^{F}_{j,\mu_{0}}$ is nonzero, either

$\lambda^{F}_{j,\mu_{0}}(s)\neq 0$ or
$\lambda^{F}_{j,\mu_{0}}(\overline{s})\neq 0$.

Without loss of generality we may assume that

$\lambda^{F}_{j,\mu_{0}}(s)\neq 0.$

Take $s_{0}$ to be either $s$ or $s^{-1}$ so that
$\lambda^{F}_{j,\mu_{0}}(s_{0})>0$.

Consider the 1-parameter subgroup

$U^{2,3}=\left\\{\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&t\\\
0&0&1\end{array}\right):t\in\mathbb{R}\right\\}.$

Note that $U^{2,3}$ commutes with $s_{0}$. Let $\mu_{1}$ be any weak-$*$ limit
point of $\\{(U^{2,3})^{T}\ast\mu\\}$ as $T\to\infty$. From Claim 18.1,
$\mu_{1}$ is $s_{0}$-invariant and
$\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{1})\geq\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{0}).$

We now average $\mu_{1}$ over a Følner sequence in $A$: identifying $A$ with
$\mathbb{R}^{2}$ let $A^{T}=[0,T]\times[0,T]$ define a Følner sequence
$\\{A^{T}\\}$ in $A$. Then

$A^{T}\ast\mu_{1}:=\frac{1}{T^{2}}\int_{0}^{T}\int_{0}^{T}(t_{1},t_{2})_{*}\mu_{1}\
d(t_{1},t_{2}).$

Let $\mu_{2}$ be any weak-$*$ limit point of $\\{A^{T}\ast\mu_{1}\\}$ as
$T\to\infty$. Then, from facts analogous to those in Claim 18.1, $\mu_{2}$ is
$A$-invariant and

$\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2})\geq\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{1})>0.$

Note that $\mu_{2}$ might no longer be $U^{2,3}$-invariant.

We investigate properties of the projection of each measure $\mu_{0},\mu_{1},$
and $\mu_{2}$ to $\mathrm{SL}(3,\mathbb{R})/\Gamma$. For each $j$, we denote
by $\hat{\mu}_{j}=\pi_{*}(\mu_{j})$ the image of $\mu_{j}$ under the
projection $\pi\colon M^{\alpha}\to\mathrm{SL}(3,\mathbb{R})/\Gamma$.

Observe that $\hat{\mu}_{1}=U^{2,3}\ast\hat{\mu}_{0}$ is $U^{2,3}$-invariant.
Since $\hat{\mu}_{0}$ was $A$-invariant, from Proposition 18.3(2) we have that
$\hat{\mu}_{1}$ is $A$-invariant and it follows that
$\hat{\mu}_{1}=\hat{\mu}_{2}$ so $\hat{\mu}_{2}$ is $U^{2,3}$-invariant and
$A$-invariant. From Proposition 18.3(4), $\hat{\mu}_{2}$ is invariant under
the subgroup

$\left\\{\left(\begin{array}[]{ccc}\ast&0&0\\\ 0&\ast&\ast\\\
0&\ast&\ast\end{array}\right)\right\\}\subset\mathrm{SL}(3,\mathbb{R})$

generated by $A$, $U^{2,3}$ and $U^{3,2}$ in $\mathrm{SL}(3,\mathbb{R})$.
Moreover, since $\hat{\mu}_{0}$ was $A$-ergodic, from Proposition 18.3(3) the
measure $\hat{\mu}_{1}=\hat{\mu}_{2}$ is $A$-ergodic.

Returning to $M^{\alpha}$, as $\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2})>0$
and as $\hat{\mu}_{2}$ is $A$-ergodic, we may replace $\mu_{2}$ with an
$A$-ergodic component $\mu_{2}^{\prime}$ of $\mu_{2}$ such that

  1. (1)

$\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2}^{\prime})>0$, and

  2. (2)

the projection of $\mu_{2}^{\prime}$ to $\mathrm{SL}(3,\mathbb{R})/\Gamma$ is
$\hat{\mu}_{2}$.

Let
$\lambda^{F}_{1,\mu_{2}^{\prime}},\dots,\lambda^{F}_{p^{\prime},\mu_{2}^{\prime}}\colon
A\to\mathbb{R}$ denote the fiberwise Lyapunov exponents for the $A$-invariant,
$A$-ergodic measure $\mu_{2}^{\prime}$. Then
$0<\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s_{0})=\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2}^{\prime})$
for some $1\leq j^{\prime}\leq p^{\prime}$ whence some fiberwise Lyapunov
exponent $\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}\colon A\to\mathbb{R}$ is a
nonzero linear functional.

Second averaging. Consider now the elements $s=(2,2,\frac{1}{4})$ and
$\overline{s}=(2,\frac{1}{4},2)$ in $A$. Again, either

$\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s)\neq 0$ or
$\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(\overline{s})\neq 0$.

Case 1: $\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s)\neq 0$. Take $s_{1}=s$
or $s_{1}=s^{-1}$ so that
$\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s_{1})>0$. Consider the one-
parameter group $U^{1,2}$ which commutes with $s_{1}$. As above, any weak-$*$
limit point $\mu_{3}$ of $\\{(U^{1,2})^{T}\ast\mu_{2}^{\prime}\\}$ as
$T\to\infty$ is $s_{1}$-invariant, with

$\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{3})\geq\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{2}^{\prime})>0.$

Let $\mu_{4}$ be any weak-$*$ limit point of $\\{A^{T}\ast\mu_{3}\\}$ as
$T\to\infty$ (where $A^{T}\ast\mu_{3}$ is as in the first averaging). Then
$\mu_{4}$ is $A$-invariant and

$\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{4})\geq\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{3})>0.$

We claim that the projection $\hat{\mu}_{4}$ of $\mu_{4}$ to
$\mathrm{SL}(3,\mathbb{R})/\Gamma$ is the Haar measure. Since the groups
$U^{1,2}$ and $U^{3,2}$ commute and since $\hat{\mu}_{2}$ was
$U^{3,2}$-invariant, it follows that $\hat{\mu}_{3}=U^{1,2}\ast\hat{\mu}_{2}$
is $U^{3,2}$-invariant. Also, since $\hat{\mu}_{2}$ was $A$-invariant,
Proposition 18.3(2) shows that $\hat{\mu}_{3}$ is $A$-invariant. Thus
$\hat{\mu}_{3}=\hat{\mu}_{4}$ and $\hat{\mu}_{4}$ is also invariant under the
actions of $A$, $U^{1,2}$, and $U^{3,2}.$ By Proposition 18.3(4) it follows
that $\hat{\mu}_{4}$ is invariant under the groups $U^{2,1}$ and $U^{2,3}$; in
particular $\hat{\mu}_{4}$ is invariant under the following subgroups of
$\mathrm{SL}(3,\mathbb{R})$:

$\left\\{\left(\begin{array}[]{ccc}\ast&0&0\\\ 0&\ast&\ast\\\
0&\ast&\ast\end{array}\right)\right\\},\quad\quad\left\\{\left(\begin{array}[]{ccc}\ast&\ast&0\\\
\ast&\ast&0\\\ 0&0&\ast\end{array}\right)\right\\}.$

These two groups generate all of $\mathrm{SL}(3,\mathbb{R})$, and hence
$\hat{\mu}_{4}$ is the Haar measure.

Case 2: $\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(\overline{s})\neq 0$. Take
$s_{1}=\overline{s}$ or $s_{1}=\overline{s}^{-1}$ so that
$\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s_{1})>0$. Consider the one-
parameter group $U^{1,3}$ which commutes with $s_{1}$. As above, any weak-$*$
limit point $\mu_{3}$ of $\\{(U^{1,3})^{T}\ast\mu_{2}^{\prime}\\}$ as
$T\to\infty$ is $s_{1}$-invariant, with

$\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{3})\geq\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{2}^{\prime})>0.$

Let $\mu_{4}$ be any weak-$*$ limit point of $\\{A^{T}\ast\mu_{3}\\}$ as
$T\to\infty$. Then $\mu_{4}$ is $A$-invariant and

$\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{4})\geq\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{3})>0.$

Again, we claim that $\hat{\mu}_{4}=U^{1,3}\ast\hat{\mu}_{2}$ is the Haar
measure. Since the groups $U^{1,3}$ and $U^{2,3}$ commute, it follows that
$\hat{\mu}_{3}$ is $U^{2,3}$-invariant. Also, since $\hat{\mu}_{2}$ was
$A$-invariant, Proposition 18.3(2) shows that $\hat{\mu}_{3}$ is
$A$-invariant. Thus $\hat{\mu}_{3}=\hat{\mu}_{4}$ and $\hat{\mu}_{4}$ is also
invariant under the actions of $A$, $U^{1,3}$ and $U^{2,3}$. By Proposition
18.3(4) it follows that $\hat{\mu}_{4}$ is invariant under the following
subgroups of $\mathrm{SL}(3,\mathbb{R})$:

$\left\\{\left(\begin{array}[]{ccc}\ast&0&0\\\ 0&\ast&\ast\\\
0&\ast&\ast\end{array}\right)\right\\},\quad\quad\left\\{\left(\begin{array}[]{ccc}\ast&0&\ast\\\
0&\ast&0\\\ \ast&0&\ast\end{array}\right)\right\\}.$

Again, these two groups generate all of $\mathrm{SL}(3,\mathbb{R})$, and hence
$\hat{\mu}_{4}$ is the Haar measure.

Completion of proof. In either Case 1 or Case 2, since the Haar measure
$\hat{\mu}_{4}$ is $A$-ergodic, we may take an $A$-ergodic component
$\mu_{4}^{\prime}$ of $\mu_{4}$ projecting to the Haar measure with

$\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{4}^{\prime})>0.$

If
$\lambda^{F}_{1,\mu_{4}^{\prime}},\dots,\lambda^{F}_{p^{\prime\prime},\mu_{4}^{\prime}}\colon
A\to\mathbb{R}$ denote the fiberwise Lyapunov exponents for the $A$-invariant,
$A$-ergodic measure $\mu_{4}^{\prime}$ then
$0<\lambda^{F}_{j^{\prime\prime},\mu_{4}^{\prime}}(s_{1})=\lambda^{F}_{\mathrm{top}}(s_{1},\mu_{4}^{\prime})$
for some $1\leq j^{\prime\prime}\leq p^{\prime\prime}$ whence some fiberwise
Lyapunov exponent $\lambda^{F}_{j^{\prime\prime},\mu_{4}^{\prime}}\colon
A\to\mathbb{R}$ is a nonzero linear functional.

This completes the proof of Proposition 17.2. ∎

####  18.5. Modifications of the proof of Proposition 17.2 in
$\mathrm{SL}(n,\mathbb{R})$

When $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n,\mathbb{R})$ we replace
the first averaging step with a more complicated averaging.

First averaging. We again take $\mu_{0}=\mu^{\prime}$ to be the $A$-invariant
measure in Proposition 17.2 with nonzero fiberwise exponent

$\lambda^{F}_{j,\mu_{0}}\colon A\to\mathbb{R}$, $\lambda^{F}_{j,\mu_{0}}\ \neq
0$.

Without loss of generality (by conjugating by a permutation matrix) we may
assume that for the element

$s=\mathrm{diag}(\tfrac{1}{2^{n-1}},2,\dots,2)$

of $A\subset\mathrm{SL}(n,\mathbb{R})$, we have

$\lambda^{F}_{j,\mu_{0}}(s)\neq 0.$

Take $s_{0}$ to be either $s,$ or $s^{-1}$ so that
$\lambda^{F}_{j,\mu_{0}}(s_{0})>0$.

Consider the unipotent subgroup $U\subset\mathrm{SL}(n,\mathbb{R})$ of
matrices of the form

$U=\left\\{\left(\begin{array}[]{ccccc}1&0&0&\cdots&0\\\
0&1&\ast&\cdots&\ast\\\ \vdots&&\ddots&&\vdots\\\ 0&0&\cdots&1&\ast\\\
0&0&\cdots&0&1\\\ \end{array}\right)\right\\}.$

Note that $U$ commutes with $s_{0}$.

Let $\\{F_{n}\\}$ be a Følner sequence in $U$ and let $\mu_{1}$ be any
weak-$*$ limit point of $\\{F_{n}\ast\mu_{0}\\}$ as $n\to\infty$ where

$F_{n}\ast\mu_{0}=\frac{1}{m_{U}(F_{n})}\int_{F_{n}}u_{*}\mu_{0}\ du.$

From facts analogous to those in Claim 18.1, we have that $\mu_{1}$ is
$s_{0}$-invariant and
$\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{1})\geq\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{0})>0.$
Moreover, as $U$ is higher-dimensional, we use [MR1291701, Corollary 1.3]
rather than Proposition 18.3(1) to conclude (at least for certain Følner
sequences $\\{F_{n}\\}$ in $U$ with nice geometry) that the projection
$\hat{\mu}_{1}$ of $\mu_{1}$ to $G/\Gamma$ is the limit

$\hat{\mu}_{1}=\lim F_{n}\ast\hat{\mu}_{0}$

and is $A$-invariant, ergodic, and $U$-invariant.

We again average $\mu_{1}$ over a Følner sequence of the form

$A^{T}=[0,T]\times\dots\times[0,T]$

in $A$ (identified with $\mathbb{R}^{n-1}$) and let $\mu_{2}$ be any weak-$*$
limit point of $\\{A^{T}\ast\mu_{1}\\}$ as $T\to\infty$. Then $\mu_{2}$ is
$A$-invariant and

$\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2})\geq\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{1})>0.$

Again, we have equality of the projected measures
$\hat{\mu}_{1}=\hat{\mu}_{2}$ so $\hat{\mu}_{2}$ is $U$-invariant and
$A$-invariant. From Proposition 18.3(4), $\hat{\mu}_{2}$ is also invariant
under the subgroup

$H=\left\\{\left(\begin{array}[]{ccccc}1&0&0&\cdots&0\\\ 0&\ast&\ast&&\ast\\\
0&\ast&\ast&&\ast\\\ \vdots&&&\ddots&\vdots\\\ 0&\ast&\ast&\cdots&\ast\\\
\end{array}\right)\right\\}.$

As $\hat{\mu}_{2}$ is $A$-ergodic, we may replace $\mu_{2}$ with an
$A$-ergodic component $\mu_{2}^{\prime}$ of $\mu_{2}$ such that

  1. (1)

$\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2}^{\prime})>0$, and

  2. (2)

the projection of $\mu_{2}^{\prime}$ to $\mathrm{SL}(n,\mathbb{R})/\Gamma$ is
$\hat{\mu}_{2}$.

Then, if
$\lambda^{F}_{1,\mu_{2}^{\prime}},\dots,\lambda^{F}_{p^{\prime},\mu_{2}^{\prime}}\colon
A\to\mathbb{R}$ denote the fiberwise Lyapunov exponents for
$\mu_{2}^{\prime}$, we have
$0<\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s_{0})=\lambda^{F}_{\mathrm{top}}(s_{0},\mu_{2}^{\prime})$
for some $1\leq j^{\prime}\leq p^{\prime}$.

Second averaging. Consider now the roots $\beta^{1,2}$ and $\beta^{1,n}$ of
$G$. Since $\beta^{1,2}$ and $\beta^{1,n}$ are not proportional, at most one
of $\beta^{1,2}$ and $\beta^{1,n}$ is proportional to
$\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}$. In particular, we may find either
$s$ or $\overline{s}$ in $A$ such that

  1. (1)

$\beta^{1,2}(s)=0$ but $\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s)\neq 0$;
or

  2. (2)

$\beta^{1,n}(\overline{s})=0$ but
$\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(\overline{s})\neq 0$.

The two cases in the second averaging step of Section 18.4 are then identical
to the above, where we either average over the 1-parameter group $U^{1,2}$ in
the case $\lambda^{F}_{j^{\prime},\mu_{2}^{\prime}}(s)\neq 0$ or $U^{1,n}$ in
the case $\lambda^{F}_{j,\mu_{2}^{\prime}}(\overline{s})\neq 0$. The structure
theory of $\mathrm{SL}(n,\mathbb{R})$ will then imply that the measure
obtained after the second averaging projects to the Haar measure.

$\bigast$19\. Zimmer’s conjecture for actions by lattices in other Lie groups

Consider a connected, simple Lie group $G$ with finite center. Let
$\Gamma\subset G$ be a cocompact lattice. The proof of Theorem 11.6 discussed
above, particularly the use of Theorem 15.1 in Section 17.3 can be adapted
almost verbatim to show the following. See also [Cantat] where Theorem 19.1 is
stated and given a mostly self-contained proof.

######  Theorem 19.1.

Let $G$ be a connected, simple Lie group $G$ with finite center and rank at
least $2$. Let $\Gamma\subset G$ be a cocompact lattice. Let $M$ be a compact
manifold.

  1. (1)

If $\dim(M)<\mathrm{rank}(G)$ then any homomorphism
$\Gamma\rightarrow\mathrm{Diff}^{2}(M)$ has finite image.

  2. (2)

In addition, if $\mathrm{vol}$ is a volume form on $M$ and if
$\dim(M)\leq\mathrm{rank}(G)$ then any homomorphism
$\Gamma\rightarrow\mathrm{Diff}^{2}_{\mathrm{vol}}(M)$ has finite image.

As mentioned in Section 11.4, Theorem 19.1 holds for $C^{1}$ actions; see
Theorem 11.7.

Theorem 19.1 fails to give the optimal dimension bounds for the analogue of
Conjecture 11.2 given in Conjecture 11.3 for actions by lattices in Lie groups
other than $\mathrm{SL}(n,\mathbb{R})$. See Table 1 for various conjectured
critical dimensions arising in Zimmer’s conjecture for other Lie groups.

To state the most general (as of 2018) result towards solving Conjecture 11.3,
to any simple Lie group $G$, we associate a non-negative integer $r(G)$. See
[1608.04995, Section 2.2] or Footnote 4, page 4, for equivalent definitions of
$r(G)$ and Table 1 for values of $r(G)$ in various examples of $G$. For
actions of lattices in a general Lie group $G$, the main result of
[1608.04995], as well as the announced extension, gives finiteness of the
action up to the critical dimension $r(G)$.

######  Theorem 19.2 ([1608.04995] cocompact case; [BFHWM] nonuniform case).

Let $\Gamma\subset G$ be a lattice in a higher-rank simple Lie group $G$ with
finite center. Let $M$ be a compact manifold.

  1. (1)

If $\dim(M)<r(G)$ then any homomorphism
$\Gamma\rightarrow\mathrm{Diff}^{1+\beta}(M)$ has finite image.

  2. (2)

In addition, if $\mathrm{vol}$ is a volume form on $M$ and if $\dim(M)=r(G)$
then any homomorphism
$\Gamma\rightarrow\mathrm{Diff}^{1+\beta}_{\mathrm{vol}}(M)$ has finite image.

When $G$ is exceptional or not a split real form, our number $r(G)$ is lower
than the conjectured critical dimension in Conjecture 11.3(1) and (2).
However, for lattices in all Lie groups that are non-exceptional, split real
forms Theorem 19.2 confirms Conjecture 11.3(1) and (2). For instance, for
actions by lattices in symplectic groups we have the following.

######  Theorem 19.3 ([1608.04995, Theorem 1.3] cocompact case; [BFHWM]
nonuniform case).

For $n\geq 2$, if $M$ is a compact manifold with $\dim(M)<2n-1$ and if
$\Gamma\subset\mathrm{Sp}(2n,\mathbb{R})$ is a lattice then any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}^{2}(M)$ has finite image. In
addition, if $\dim(M)=2n-1$ then any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}_{\mathrm{vol}}^{2}(M)$ has finite
image.

Similarly, for actions by lattices in split orthogonal groups we have the
following.

######  Theorem 19.4 ([1608.04995, Theorem 1.4] cocompact case; [BFHWM]
nonuniform case).

Let $M$ be a compact manifold.

  1. (1)

For $n\geq 4$, if $\Gamma\subset\mathrm{SO}(n,n)$ is a lattice and if
$\dim(M)<2n-2$ then any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}^{2}(M)$ has finite image. If
$\dim(M)=2n-2$ then any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}^{2}_{\mathrm{vol}}(M)$ has finite
image.

  2. (2)

For $n\geq 3$, if $\Gamma\subset\mathrm{SO}(n,n+1)$ is a lattice and if
$\dim(M)<2n-1$ then any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}^{2}(M)$ has finite image. If
$\dim(M)=2n-1$ then any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}^{2}_{\mathrm{vol}}(M)$ has finite
image.

For actions by lattices $\Gamma$ in simple Lie groups that are not split real
forms such as $G=\mathrm{SL}(n,\mathbb{C})$, $\mathrm{SO}(n,m)$ for $m\geq
n+2$, or $\mathrm{SU}(n,m)$, Theorem 19.2 above (the main result of
[1608.04995] for cocompact case, [BFHWM] in general) gives finiteness of all
actions on manifolds whose dimension is below a certain critical dimension.
However, this critical dimension may be below the dimension conjectured by the
analogue of Conjecture 11.3 for these groups. See Table 1.

##  Part IV A selection of other measure rigidity results

###  20\. Nonuniformly hyperbolic $\mathbb{Z}^{k}$-actions

Instead of considering $\mathbb{Z}^{2}$-actions by automorphisms of
$\mathbb{T}^{3}$ as in Theorem 2.8, we might consider $\mathbb{Z}^{2}$-actions
on the torus $\mathbb{T}^{3}$ generated by two commuting diffeomorphisms
$f,g\colon\mathbb{T}^{3}\to\mathbb{T}^{3}$.

Recall that for any homeomorphism $f\colon\mathbb{T}^{3}\to\mathbb{T}^{3}$
there exists a unique $M\in\mathrm{GL}(3,\mathbb{Z})$ so that any lift
$\tilde{f}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ of $f$ is of the form

$\tilde{f}(x)=Mx+\psi(x)$

where $\psi\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ is
$\mathbb{Z}^{3}$-periodic. The linear map $M$ can also be seen as the induced
action of $f$ on first homology of $\mathbb{T}^{3}$. We call $M$ the linear
data of $f$. By a theorem of Franks [MR0271990], if $M$ has no eigenvalues of
modulus 1 then there is a continuous, surjective
$h\colon\mathbb{T}^{d}\to\mathbb{T}^{d}$, homotopic to the identity, such that

$h\circ f=L_{M}\circ h$ (20.1)

where $L_{M}\colon\mathbb{T}^{3}\to\mathbb{T}^{3}$ is the induced automorphism
of the torus; such a map $h$ is called a semiconjugacy.

If $f,g\colon\mathbb{T}^{3}\to\mathbb{T}^{3}$ are commuting homeomorphisms
with linear data $A$ and $B$, respectively, one can verify that $A$ and $B$
commute. Indeed if $\tilde{f}(x)=Ax+\psi(x)$ and $\tilde{g}(x)=Bx+\phi(x)$ are
lifts of $f$ and $g$, respectively, then

$\tilde{f}\circ\tilde{g}(x)=ABx+A\phi(x)+\psi(x)$

and

$\tilde{g}\circ\tilde{f}(x)=BAx+B\psi(x)+\phi(x)$

are both lifts of $f\circ g=g\circ f$ whence $AB=BA$.

If $A$ has no eigenvalues of modulus 1, we may take a map
$h\colon\mathbb{T}^{3}\to\mathbb{T}^{3}$ with

$h\circ f=L_{A}\circ h$

as in (20.1). Following [MR2643892, Lemma 1] (correcting [MR2261075, Lemma
1.2]) the map $h$ conjugates the $\mathbb{Z}^{2}$-action generated by $f$ and
$g$ to an affine action on $\mathbb{T}^{3}$ whose linear part is generated by
$L_{A}$ and $L_{B}$. That is, if
$\alpha\colon\mathbb{Z}^{2}\to\mathrm{Diff}(\mathbb{T}^{3})$ is the non-linear
action

$\alpha(n_{1},n_{2})=f^{n_{1}}g^{n_{2}}$

then there is an affine action
$\alpha_{0}\colon\mathbb{Z}^{2}\to\mathrm{Diff}(\mathbb{T}^{3})$ of the form

$\alpha_{0}(n_{1},n_{2})(x)=L_{A}^{n_{1}}L_{B}^{n_{2}}(x)+v_{(n_{1},n_{2})}$

for some $v_{(n_{1},n_{2})}\in\mathbb{T}^{3}$ such that for all
$(n_{1},n_{2})\in\mathbb{Z}^{2}$

$h\circ\alpha(n_{1},n_{2})=\alpha_{0}(n_{1},n_{2})\circ h.$ (20.2)

We note that the translation term $(n_{1},n_{2})\mapsto v_{(n_{1},n_{2})}$ is
a cocycle:

$v_{(n_{1},n_{2})+(m_{1},m_{2})}=L_{A}^{n_{1}}L_{B}^{n_{2}}v_{(m_{1},m_{2})}+v_{(n_{1},n_{2})}.$

Moreover, the action $\alpha_{0}$ has a fixed point if and only if
$v_{(n_{1},n_{2})}$ is a coboundary:

$v_{(n_{1},n_{2})}=L_{A}^{n_{1}}L_{B}^{n_{2}}\eta-\eta$

for some $\eta\in\mathbb{T}^{3}$. The presence of the translation term
$v_{(n_{1},n_{2})}$ is due to the non-uniqueness of the map $h$ satisfying
(20.1). However, all maps $h$ satisfying (20.1) differ by a translation by an
element of the finite set of fixed points for $L_{A}$. Thus, the translation
terms $v_{(n_{1},n_{2})}$ take only finitely many possible values. See
discussion in [MR2643892] for more details. We note that it is possible to
construct genuinely affine Anosov actions $\alpha_{0}$ without fixed points as
in Remark 10.6. See for example [MR1236179, Theorem 2]. In particular, it may
be that the action $\alpha$ is not semiconjugate to any action by
automorphisms. However, restricting to a subgroup
$\Sigma\subset\mathbb{Z}^{2}$ of finite index, one has that
${\alpha_{0}}{\restriction_{{\Sigma}}}\colon\Sigma\to\mathrm{Diff}(\mathbb{T}^{3})$
is an action by automorphisms:

$\alpha_{0}(n_{1},n_{2})(x)=L_{A}^{n_{1}}L_{B}^{n_{2}}(x)$

for all $(n_{1},n_{2})\in\Sigma$.

If $f$ is Anosov then its linear data $A$ is known to have no eigenvalues of
modulus 1 and the map $h$ in (20.1) is a homeomorphism. Suppose further that
$f$ and $g$ generate a “genuine” $\mathbb{Z}^{2}$-action so that the group of
matrices generated by their linear data $A$ and $B$ is not virtually cyclic.
This implies that the linear action generated by $L_{A}$ and $L_{B}$ satisfies
Theorem 2.8. Restricted to a finite-index subgroup, the map $h$ conjugates the
action $\alpha$ to a linear action of the type of action considered in Example
2.6. Since invertible maps preserve entropy, Theorem 2.8 classifies all
positive entropy measures that are jointly $f$\- and $g$-invariant. We remark
also that under the above assumptions, from [MR2318497], we know in this
setting that the conjugating map $h$ satisfying (20.2) is smooth. Note that
the assumption that the group of matrices generated by $A$ and $B$ is not
virtually cyclic is essential; for instance, if $g$ is a power of $f$ we
expect no rigidity of jointly invariant measures or smoothness of the
conjugacy $h$.

If neither $f$ nor $g$ is Anosov, the map $h$ in (20.2) may be non-invertible.
In particular, there may exist ergodic, $\alpha$-invariant measures $\mu$ on
$\mathbb{T}^{3}$ with $h_{\mu}(f)>0$ such that the push-forward measure
$h_{*}(\mu)$ has zero entropy for $\alpha_{0}(\mathbf{n})$ for every
$\mathbf{n}\in\mathbb{Z}^{2}.$ Note however that if

$h_{h_{*}(\mu)}(\alpha_{0}(\mathbf{n}))>0$

for some $\mathbf{n}\in\mathbb{Z}^{2}$ then $h_{*}(\mu)$ is necessarily Haar
by Theorem 2.8.

When the map $h$ in (20.2) is non-invertible, analysis of measures invariant
under the affine action $\alpha_{0}$ gives less information about measures
jointly invariant under $f$ and $g$. However, the method of proof of Theorem
2.8 can be adapted to study measures jointly invariant under $f$ and $g$; in
particular, one can show the following theorem which is a simplified version
of the main results of [MR2261075, MR2285730].

######  Theorem 20.1 ([MR2261075, MR2285730]).

Suppose $f,g\colon\mathbb{T}^{3}\to\mathbb{T}^{3}$ are commuting $C^{1+\beta}$
diffeomorphisms. Suppose the linear data of $f$ and $g$ are, respectively, the
matrices $A$ and $B$ in Example 2.6. Then any ergodic probability measure
$\mu$ that is invariant under both $f$ and $g$ and such that $h_{*}(\mu)$ is
Haar is absolutely continuous with respect to the Riemannian volume on
$\mathbb{T}^{3}$. Such a measure always exists and is, moreover, unique.

For actions on more general manifolds, there may be no a priori semiconjugacy
between the nonlinear action and an affine Anosov action. However, under
certain dynamical hypotheses on the action, the structure of the algebraic
toral action can be reconstructed. Consider a $\mathbb{Z}^{2}$-action $\alpha$
on a 3-manifold $M$ generated by two commuting diffeomorphisms $f,g\colon M\to
M$. Given an ergodic, $\mathbb{Z}^{2}$-invariant probability measure $\mu$,
one can define Lyapunov exponent functionals for the $\mathbb{Z}^{2}$-action
as in Theorem 7.4. These extend to linear functionals on $\mathbb{R}^{2}$.
Note that there are at most 3 (the dimension of $M$) Lyapunov exponent
functionals. Under some genericity assumptions on the Lyapunov exponent
functionals, an analogue of Theorem 2.8 and Theorem 20.1 was obtained in
[MR2811602].

######  Theorem 20.2 ([MR2811602]).

Let $\alpha$ be a $\mathbb{Z}^{2}$-action by $C^{1+\beta}$-diffeomorphisms of
a 3-manifold and let $\mu$ be an ergodic, $\alpha$-invariant measure. Assume
there are 3 nonzero, Lyapunov exponent functionals
$\lambda_{\mu}^{1},\lambda_{\mu}^{2},\lambda_{\mu}^{3}$ and that no pair of
exponents is proportional.

If some element $\alpha(n_{1},n_{2})$ has positive entropy with respect to
$\mu$, then $\mu$ is absolutely continuous with respect to the Riemannian
volume on $M$.

In [MR3503686], it is shown in the setting of Theorem 20.2 that one can
reconstruct an action by (infra-)toral automorphism and a measurable
semiconjugacy $h$ between the non-linear action $\alpha$ (restricted to a
finite-index subgroup of $\mathbb{Z}^{2}$) and the algebraic action. Moreover,
the semiconjugacy $h$ takes (an ergodic component of) $\mu$ to the Lebesgue
measure on the (infra-)torus, is differentiable along stable manifolds, and is
differentiable (in the Whitney sense) off sets of arbitrarily small measure.
This, in particular, implies that the exponents $\lambda^{i}_{\mu}(n)$ are
logarithms of algebraic numbers for every $n\in\mathbb{Z}^{2}$.

###  21\. Invariant measures for Cartan flows

In Section 13.2, we introduced an important example of a higher-rank,
continuous-time algebraic Anosov action, namely, the diagonal action (or
Cartan flow) on a higher-rank semisimple homogeneous space. We review its
properties, referring back to Section 13 for details.

######  Example 21.1.

Let $G=\mathrm{SL}(3,\mathbb{R})$ and let $\Gamma=\mathrm{SL}(3,\mathbb{Z})$
or any lattice in $G$. Let $X$ denote the coset space $X=G/\Gamma$. This is an
$8$ dimensional manifold (which is noncompact for
$\Gamma=\mathrm{SL}(3,\mathbb{Z})$.) $G$ acts on $X$ by left translation.

The group $A\subset G$ of diagonal matrices with positive entries is
isomorphic to $\mathbb{R}^{2}$. The action $\alpha\colon A\times X\to X$ of
$A$ on $X$ is given by $\alpha(s)(x)=sx.$ There are $6$ roots
$\beta^{i,j}\colon A\to\mathbb{R}$ given by
$\beta^{i,j}(\mathrm{diag}(e^{t_{1}},e^{t_{1}},e^{t_{3}}))=t_{i}-t_{j}$ each
with an associated root subgroup $U^{i,j}\subset G$. For $x\in X$,
$W^{i,j}(x)$ is the orbit of $x$ under the 1-parameter group $U^{i,j}$:

$W^{i,j}(x)=\\{U^{i,j}\cdot x:t\in\mathbb{R}\\}.$

For $s\in A$, the action $\alpha(s)$ dilates distances in $W^{i,j}(x)$ by
exactly $e^{\beta^{i,j}(s)}$.

One might ask whether an analogue of Theorem 2.8 holds in Example 21.1. That
is, if $\mu$ is an ergodic, $A$-invariant probability measure on $X$ such that
there is some $s\in A$ with $h_{\mu}(\alpha(s))>0$, is $\mu$ necessarily the
Haar measure on $X$ or on a homogeneous submanifold of $X$?

The answer is no. The extension of the proof of Theorem 2.8 breaks down in
this setting as the trick in Lemma 6.6 fails. Indeed, for every root
$\beta^{i,j}$, we have that $\beta^{i,j}$ and $\beta^{j,i}=-\beta^{i,j}$ are
negatively proportional. Moreover, explicit examples of diagonally invariant
measures with positive entropy (for some element of the diagonal) on spaces of
the form $\mathrm{SL}(3,\mathbb{R})/\Gamma$ for certain (cocompact) lattices
$\Gamma$ were constructed by Mary Rees in an unpublished manuscript. See
[MR1989231] for detailed constructions of such measures.

A related problem is the classification of orbit closures for the $A$-action
on $X=\mathrm{SL}(n,\mathbb{R})/\Gamma$. Rees’s construction yields $A$-orbit
closures in $\mathrm{SL}(n,\mathbb{R})/\Gamma$ that are non-homogeneous and of
intermediate Hausdorff dimensions for certain lattices $\Gamma$. It was shown
in [MR2630049] that there exist non-homogeneous $A^{\prime}$-orbit closures in
$\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ (for $n\geq 6$) for
certain higher-rank subgroups $A^{\prime}$ of the full diagonal group $A$.

Returning to the classification of invariant measures, in the case that
$\Gamma=\mathrm{SL}(3,\mathbb{Z})$, the Margulis conjecture asserts that all
ergodic $A$-invariant measures $\mu$ on $X$ should be algebraic. See
[MR2247967, Conjecture 1.1] and discussion in [MR1754775, §1.2]. For measures
with positive entropy, this conjecture was solved in [MR2247967] (see Theorem
21.5 below). We outline the main results used in [MR2247967], namely the high
and low entropy methods.

To discuss the high and low entropy methods, first note that there are some
key differences in the structure of the foliations in this setting versus the
setting of Example 2.6. First note that any two transverse foliations
$W^{i},W^{j}$ of the torus $\mathbb{T}^{3}$ by lines are jointly integrable;
that is there is a foliation of $\mathbb{T}^{3}$ by planes $W^{i,j}$ with
$W^{i}(x)\subset W^{i,j}(x)$ and $W^{j}(x)\subset W^{i,j}(x)$ for all $x$.
This follows as $\mathbb{T}^{3}$ has an abelian group structure. In
$X=\mathrm{SL}(3,\mathbb{R})/\Gamma$, Lyapunov foliations do not jointly
integrate as the corresponding subgroups may not commute. For instance, the
subgroups $U^{1,2}$ and $U^{2,3}$ do not commute and thus the foliations
$W^{1,2}(x)$ and $W^{2,3}(x)$ do not jointly integrate. This is the key idea
behind the high entropy method. Moreover, translations along Lyapunov
directions $E^{i}$ are isometries in the torus $\mathbb{T}^{3}$. In
$X=G/\Gamma$, translation by an element of a 1-parameter subgroup $U^{i,j}$ is
not isometric; there is some polynomial shearing. This is a key step in the
proof of Ratner’s measure classification theorem for unipotent flows (see
[MR1262705]) and is also a key idea in the low entropy method.

We state the versions of the high entropy and low entropy methods for Example
21.1. Given a measure $\mu$ on $X$, for $i\neq j$ let $\mu^{i,j}_{x}$ denote
the locally finite leaf-wise measures obtained by conditioning $\mu$ along
$W^{i,j}$-manifolds.

######  Theorem 21.2 (High entropy method [MR1989231]).

Let $\mu$ be an ergodic, $A$-invariant measure on
$X=\mathrm{SL}(3,\mathbb{R})/\Gamma$. Let $i,j,$ and $k$ be distinct elements
of $\\{1,2,3\\}$. If $\mu^{i,j}_{x}$ and $\mu^{j,k}_{x}$ are nonatomic for a
positive measure set of $x$ then $\mu$ is $U^{i,k}$-invariant.

Note that the subgroups $U^{i,j}$ and $U^{j,k}$ do not commute; precisely, we
have $[U^{i,j},U^{j,k}]=U^{i,k}$. Theorem 21.2 states that if both Lyapunov
exponents $\beta^{i,j}$ and $\beta^{j,k}$ contribute entropy (so that
$\mu^{i,j}_{x}$ and $\mu^{j,k}_{x}$ are nonatomic) then the measure $\mu$ is
invariant under their bracket $U^{i,k}=[U^{i,j},U^{j,k}]$. The
noncommutativity of $U^{i,j}$ and $U^{j,k}$ is essential in the proof of the
theorem.

From Theorem 21.2 one can derive the following corollary.

######  Theorem 21.3 ([MR1989231, Theorem 4.1]).

Let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice and let $\mu$ be an
ergodic, $A$-invariant measure on $X=\mathrm{SL}(n,\mathbb{R})/\Gamma$ such
that for every nontrivial $s\in A$,

$h_{\mu}(\alpha(s))>0.$

Then $\mu$ is the Haar measure on $X$.

We note that the statements of Theorems 21.2 and 21.3 are specific for the
group $\mathrm{SL}(n,\mathbb{R})$. More general high entropy methods appear in
[MR2191228].

The low entropy method is a bit more difficult to state. We state a version
for $X=\mathrm{SL}(3,\mathbb{R})/\Gamma$. For each $i\neq j$, let
$A^{\prime}_{i,j}$ denote the kernel of $\beta^{i,j}$ and let
$C(A^{\prime}_{i,j})$ denote the centralizer of $A^{\prime}_{i,j}$ in
$G=\mathrm{SL}(3,\mathbb{R}).$

######  Theorem 21.4 (Low entropy method, [MR2247967, Theorem 2.3.]).

Let $\mu$ be an ergodic, $A$-invariant measure on
$X=\mathrm{SL}(3,\mathbb{R})/\Gamma$. If $\mu^{i,j}_{x}$ and $\mu^{j,i}_{x}$
are nonatomic for some $i,j$ and $\mu^{i^{\prime},j^{\prime}}_{x}$ is atomic
for all other pairs $i^{\prime},j^{\prime}$ then either

  1. (1)

$\mu$ is $U^{i,j}$-invariant, or

  2. (2)

there is $x_{0}\in X$ and $s\in A^{\prime}_{i,j}$ with
$\alpha(s)(x_{0})=x_{0}$ such that $\mu$ is supported on the orbit
$C(A^{\prime}_{i,j})x_{0}$.

Conclusion (2) is specific for the case that $X$ is of the form
$X=\mathrm{SL}(n,\mathbb{R})/\Gamma$ for $n=3$. For $n>3$, the appropriate
version of (2) is slightly more complicated.

To show that Theorem 21.2 and Theorem 21.4 cover all cases it is shown
[MR2247967, Corollary 3.4] for any ergodic, $A$-invariant measure $\mu$ and
every pair $i,j$, the measure $\mu^{i,j}_{x}$ is nonatomic if and only if
$\mu^{j,i}_{x}$ is nonatomic. Thus every $A$-invariant measure $\mu$ with
positive entropy is considered in either Theorem 21.2 and Theorem 21.4.

To apply the low entropy method, one typically does additional work to rule
out conclusion (2) in Theorem 21.4. Note that conclusion (2) of Theorem 21.4
occurs in Rees’s examples so it can not be ruled out for all lattices
$\Gamma$. However, for certain lattices, it can be shown that (2) of Theorem
21.4 does not happen. In particular, this is verified for
$\Gamma=\mathrm{SL}(n,\mathbb{Z})$ in [MR2247967]. The high and low entropy
method combine to give the following.

######  Theorem 21.5 ([MR2247967, Theorem 1.3, Corollary 1.4]).

Let $\mu$ be an ergodic, $A$-invariant measure on
$X=\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$. Assume $\mu$ has
positive entropy for some nontrivial element of $A$. Then $\mu$ is the Haar
measure on $X$.

Note the conclusion that $\mu$ is the Haar measure above follows as we assume
$n=3$ which is prime. For the general result on
$\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$, the conclusion is that
$\mu$ is algebraic.

The study of invariant measures and orbit closures for various subgroups
$H\subset\mathrm{SL}(n,\mathbb{R})$ acting on
$X=\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ is related to several
important problems in number theory. See for instance the proof of Margulis’s
proof [MR882782, MR993328] of the Oppenheim conjecture which reduces to the
study of $H=\mathrm{SO}(2,1)$-orbit closures in
$\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$. See [MR2158954, Section
1.2] for further discussion. One motivation for studying $A$-invariant
measures on $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ is its
relationship to Littlewood’s conjecture. An important consequence of Theorem
21.5 is that the set of values for which Littlewood’s conjecture fails has
Hausdorff dimension zero. See [MR2247967] as well as [MR1754775, §1.2] and
[MR2358379] for details.

## References

AbramovL. M.RokhlinV. A.Entropy of a skew product of mappings with invariant
measure1962ISSN 0146-924xVestnik Leningrad. Univ.175–13@article{MR0140660,
author = {Abramov, L.~M.}, author = {Rokhlin, V.~A.}, title = {Entropy of a
skew product of mappings with invariant measure}, date = {1962}, issn =
{0146-924x}, journal = {Vestnik Leningrad. Univ.}, volume = {17}, pages =
{5\ndash 13}} AlvesJosé FerreiraSRB measures for non-hyperbolic systems with
multidimensional expansion2000ISSN 0012-9593Ann. Sci. École Norm. Sup.
(4)331–32Link@article{MR1743717, author = {Alves, Jos{\'e}~Ferreira}, title =
{S{RB} measures for non-hyperbolic systems with multidimensional expansion},
date = {2000}, issn = {0012-9593}, journal = {Ann. Sci. \'Ecole Norm. Sup.
(4)}, volume = {33}, pages = {1\ndash 32}, url =
{http://dx.doi.org/10.1016/S0012-9593(00)00101-4}} AlvesJosé
F.BonattiChristianVianaMarceloSRB measures for partially hyperbolic systems
whose central direction is mostly expanding2000ISSN 0020-9910Invent.
Math.140351–398Link@article{MR1757000, author = {Alves, Jos{\'e}~F.}, author =
{Bonatti, Christian}, author = {Viana, Marcelo}, title = {S{RB} measures for
partially hyperbolic systems whose central direction is mostly expanding},
date = {2000}, issn = {0020-9910}, journal = {Invent. Math.}, volume = {140},
pages = {351\ndash 398}, url = {http://dx.doi.org/10.1007/s002220000057}}
AlvesJosé F.DiasCarla L.LuzzattoStefanoPinheiroViltonSRB measures for
partially hyperbolic systems whose central direction is weakly
expanding2017ISSN 1435-9855J. Eur. Math. Soc.
(JEMS)192911–2946@article{MR3712997, author = {Alves, Jos\'e~F.}, author =
{Dias, Carla~L.}, author = {Luzzatto, Stefano}, author = {Pinheiro, Vilton},
title = {S{RB} measures for partially hyperbolic systems whose central
direction is weakly expanding}, date = {2017}, issn = {1435-9855}, journal =
{J. Eur. Math. Soc. (JEMS)}, volume = {19}, pages = {2911\ndash 2946}}
AnosovD. V.Geodesic flows on closed Riemannian manifolds of negative
curvature1967ISSN 0371-9685Proc. Steklov Inst. Math.901–235@article{MR0224110,
author = {Anosov, D.~V.}, title = {Geodesic flows on closed {R}iemannian
manifolds of negative curvature}, date = {1967}, issn = {0371-9685}, journal =
{Proc. Steklov Inst. Math.}, volume = {90}, pages = {1\ndash 235}}
AvilaArturVianaMarceloExtremal Lyapunov exponents: an invariance principle and
applications2010ISSN 0020-9910Invent. Math.181115–189Link@article{MR2651382,
author = {Avila, Artur}, author = {Viana, Marcelo}, title = {Extremal
{L}yapunov exponents: an invariance principle and applications}, date =
{2010}, issn = {0020-9910}, journal = {Invent. Math.}, volume = {181}, pages =
{115\ndash 189}, url = {http://dx.doi.org/10.1007/s00222-010-0243-1}}
BarreiraLuisPesinYakovSchmelingJörgDimension and product structure of
hyperbolic measures1999ISSN 0003-486XAnn. of Math.
(2)149755–783@article{MR1709302, author = {Barreira, Luis}, author = {Pesin,
Yakov}, author = {Schmeling, J{\"o}rg}, title = {Dimension and product
structure of hyperbolic measures}, date = {1999}, issn = {0003-486X}, journal
= {Ann. of Math. (2)}, volume = {149}, pages = {755\ndash 783}} BekkaBachirde
la HarpePierreValetteAlainKazhdan’s property (t)New Mathematical
Monographs11Cambridge University Press, Cambridge2008xiv+472ISBN
978-0-521-88720-5Document@book{BdHV, author = {Bekka, Bachir}, author = {de la
Harpe, Pierre}, author = {Valette, Alain}, title = {Kazhdan's property (T)},
series = {New Mathematical Monographs}, volume = {11}, publisher = {Cambridge
University Press, Cambridge}, date = {2008}, pages = {xiv+472}, isbn =
{978-0-521-88720-5}, doi = {10.1017/CBO9780511542749}}
BenedicksMichaelCarlesonLennartOn iterations of $1-ax^{2}$ on $(-1,1)$1985ISSN
0003-486XAnn. of Math. (2)1221–25Link@article{MR799250, author = {Benedicks,
Michael}, author = {Carleson, Lennart}, title = {On iterations of {$1-ax^2$}
on {$(-1,1)$}}, date = {1985}, issn = {0003-486X}, journal = {Ann. of Math.
(2)}, volume = {122}, pages = {1\ndash 25}, url =
{https://doi.org/10.2307/1971367}} BenedicksMichaelCarlesonLennartThe dynamics
of the Hénon map1991ISSN 0003-486XAnn. of Math.
(2)13373–169Link@article{MR1087346, author = {Benedicks, Michael}, author =
{Carleson, Lennart}, title = {The dynamics of the {H}\'enon map}, date =
{1991}, issn = {0003-486X}, journal = {Ann. of Math. (2)}, volume = {133},
pages = {73\ndash 169}, url = {https://doi.org/10.2307/2944326}}
BenedicksMichaelVianaMarceloSolution of the basin problem for Hénon-like
attractors2001ISSN 0020-9910Invent. Math.143375–434Link@article{MR1835392,
author = {Benedicks, Michael}, author = {Viana, Marcelo}, title = {Solution of
the basin problem for {H}\'enon-like attractors}, date = {2001}, issn =
{0020-9910}, journal = {Invent. Math.}, volume = {143}, pages = {375\ndash
434}, url = {http://dx.doi.org/10.1007/s002220000109}}
BenedicksMichaelYoungLai-SangSinaĭ-Bowen-Ruelle measures for certain Hénon
maps1993ISSN 0020-9910Invent. Math.112541–576Link@article{MR1218323, author =
{Benedicks, Michael}, author = {Young, Lai-Sang}, title =
{Sina\u\i-{B}owen-{R}uelle measures for certain {H}\'enon maps}, date =
{1993}, issn = {0020-9910}, journal = {Invent. Math.}, volume = {112}, pages =
{541\ndash 576}, url = {http://dx.doi.org/10.1007/BF01232446}} BenoistYvesFive
lectures on lattices in semisimple Lie groups2009Géométries à courbure
négative ou nulle, groupes discrets et rigiditésSémin. Congr.18Soc. Math.
France, Paris117–176@incollection{MR2655311, author = {Benoist, Yves}, title =
{Five lectures on lattices in semisimple {L}ie groups}, date = {2009},
booktitle = {G\'eom\'etries \\`a courbure n\'egative ou nulle, groupes
discrets et rigidit\'es}, series = {S\'emin. Congr.}, volume = {18}, publisher
= {Soc. Math. France, Paris}, pages = {117\ndash 176}} BenoistYvesQuintJean-
FrançoisMesures stationnaires et fermés invariants des espaces
homogènes2011ISSN 0003-486XAnn. of Math.
(2)1741111–1162Link@article{MR2831114, author = {Benoist, Yves}, author =
{Quint, Jean-Fran{\c{c}}ois}, title = {Mesures stationnaires et ferm\'es
invariants des espaces homog\\`enes}, date = {2011}, issn = {0003-486X},
journal = {Ann. of Math. (2)}, volume = {174}, pages = {1111\ndash 1162}, url
= {http://dx.doi.org/10.4007/annals.2011.174.2.8}} BenoistYvesQuintJean-
FrançoisStationary measures and invariant subsets of homogeneous spaces
(II)2013ISSN 0894-0347J. Amer. Math. Soc.26659–734Link@article{MR3037785,
author = {Benoist, Yves}, author = {Quint, Jean-Fran{\c{c}}ois}, title =
{Stationary measures and invariant subsets of homogeneous spaces ({II})}, date
= {2013}, issn = {0894-0347}, journal = {J. Amer. Math. Soc.}, volume = {26},
pages = {659\ndash 734}, url =
{http://dx.doi.org/10.1090/S0894-0347-2013-00760-2}} BenoistYvesQuintJean-
FrançoisStationary measures and invariant subsets of homogeneous spaces
(iii)2013Ann. of Math. (2)1781017–1059@article{BQIII, author = {Benoist,
Yves}, author = {Quint, Jean-Fran{\c c}ois}, title = {Stationary measures and
invariant subsets of homogeneous spaces (iii)}, date = {2013}, journal = {Ann.
of Math. (2)}, volume = {178}, pages = {1017\ndash 1059}} BenvenisteElie
JeromeRigidity and deformations of lattice actions preserving geometric
structuresProQuest LLC, Ann Arbor, MI1996LinkThesis (Ph.D.)–The University of
Chicago@book{MR2716615, author = {Benveniste, Elie~Jerome}, title = {Rigidity
and deformations of lattice actions preserving geometric structures},
publisher = {ProQuest LLC, Ann Arbor, MI}, date = {1996}, url =
{http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9629245},
note = {Thesis (Ph.D.)–The University of Chicago}} BenvenisteE. J.Rigidity of
isometric lattice actions on compact Riemannian manifolds2000ISSN
1016-443XGeom. Funct. Anal.10516–542Link@article{MR1779610, author =
{Benveniste, E.~J.}, title = {Rigidity of isometric lattice actions on compact
{R}iemannian manifolds}, date = {2000}, issn = {1016-443X}, journal = {Geom.
Funct. Anal.}, volume = {10}, pages = {516\ndash 542}, url =
{http://dx.doi.org/10.1007/PL00001627}} BenvenisteE.
JeromeFisherDavidNonexistence of invariant rigid structures and invariant
almost rigid structures2005ISSN 1019-8385Comm. Anal.
Geom.1389–111Link@article{MR2154667, author = {Benveniste, E.~Jerome}, author
= {Fisher, David}, title = {Nonexistence of invariant rigid structures and
invariant almost rigid structures}, date = {2005}, issn = {1019-8385}, journal
= {Comm. Anal. Geom.}, volume = {13}, pages = {89\ndash 111}, url =
{http://projecteuclid.org/euclid.cag/1117649081}} BerendDanielMulti-invariant
sets on tori1983ISSN 0002-9947Trans. Amer. Math.
Soc.280509–532Link@article{MR716835, author = {Berend, Daniel}, title =
{Multi-invariant sets on tori}, date = {1983}, issn = {0002-9947}, journal =
{Trans. Amer. Math. Soc.}, volume = {280}, pages = {509\ndash 532}, url =
{http://dx.doi.org/10.2307/1999631}} BogenschützThomasCrauelHansThe Abramov-
Rokhlin formula1992Ergodic theory and related topics, III (Güstrow,
1990)Lecture Notes in Math.1514SpringerBerlin32–35Link@incollection{MR1179170,
author = {Bogensch{\"u}tz, Thomas}, author = {Crauel, Hans}, title = {The
{A}bramov-{R}okhlin formula}, date = {1992}, booktitle = {Ergodic theory and
related topics, {III} ({G}\"ustrow, 1990)}, series = {Lecture Notes in Math.},
volume = {1514}, publisher = {Springer}, address = {Berlin}, pages = {32\ndash
35}, url = {http://dx.doi.org/10.1007/BFb0097526}}
BonattiChristianVianaMarceloSRB measures for partially hyperbolic systems
whose central direction is mostly contracting2000ISSN 0021-2172Israel J.
Math.115157–193Link@article{MR1749677, author = {Bonatti, Christian}, author =
{Viana, Marcelo}, title = {S{RB} measures for partially hyperbolic systems
whose central direction is mostly contracting}, date = {2000}, issn =
{0021-2172}, journal = {Israel J. Math.}, volume = {115}, pages = {157\ndash
193}, url = {http://dx.doi.org/10.1007/BF02810585}} BorelArmandHarish-
ChandraArithmetic subgroups of algebraic groups1962ISSN 0003-486XAnn. of Math.
(2)75485–535Link@article{MR0147566, author = {Borel, Armand}, author =
{Harish-Chandra}, title = {Arithmetic subgroups of algebraic groups}, date =
{1962}, issn = {0003-486X}, journal = {Ann. of Math. (2)}, volume = {75},
pages = {485\ndash 535}, url = {https://doi.org/10.2307/1970210}}
BorelArmandJiLizhenCompactifications of symmetric and locally symmetric
spaces2005Lie theoryProgr. Math.229Birkhäuser Boston, Boston,
MA69–137Link@incollection{MR2126641, author = {Borel, Armand}, author = {Ji,
Lizhen}, title = {Compactifications of symmetric and locally symmetric
spaces}, date = {2005}, booktitle = {Lie theory}, series = {Progr. Math.},
volume = {229}, publisher = {Birkh\"auser Boston, Boston, MA}, pages =
{69\ndash 137}, url = {http://dx.doi.org/10.1007/0-8176-4430-X_2}}
BourgainJeanFurmanAlexLindenstraussElonMozesShaharStationary measures and
equidistribution for orbits of nonabelian semigroups on the torus2011ISSN
0894-0347J. Amer. Math. Soc.24231–280Link@article{MR2726604, author =
{Bourgain, Jean}, author = {Furman, Alex}, author = {Lindenstrauss, Elon},
author = {Mozes, Shahar}, title = {Stationary measures and equidistribution
for orbits of nonabelian semigroups on the torus}, date = {2011}, issn =
{0894-0347}, journal = {J. Amer. Math. Soc.}, volume = {24}, pages =
{231\ndash 280}, url = {http://dx.doi.org/10.1090/S0894-0347-2010-00674-1}}
BowenRufusEquilibrium states and the ergodic theory of Anosov
diffeomorphismsLecture Notes in Mathematics, Vol. 470Springer-
VerlagBerlin1975@book{MR0442989, author = {Bowen, Rufus}, title = {Equilibrium
states and the ergodic theory of {A}nosov diffeomorphisms}, series = {Lecture
Notes in Mathematics, Vol. 470}, publisher = {Springer-Verlag}, address =
{Berlin}, date = {1975}} BowenRufusWeak mixing and unique ergodicity on
homogeneous spaces1976ISSN 0021-2172Israel J.
Math.23267–273Link@article{MR0407233, author = {Bowen, Rufus}, title = {Weak
mixing and unique ergodicity on homogeneous spaces}, date = {1976}, issn =
{0021-2172}, journal = {Israel J. Math.}, volume = {23}, pages = {267\ndash
273}, url = {http://dx.doi.org/10.1007/BF02761804}} BowenRufusRuelleDavidThe
ergodic theory of Axiom A flows1975ISSN 0020-9910Invent.
Math.29181–202@article{MR0380889, author = {Bowen, Rufus}, author = {Ruelle,
David}, title = {The ergodic theory of {A}xiom {A} flows}, date = {1975}, issn
= {0020-9910}, journal = {Invent. Math.}, volume = {29}, pages = {181\ndash
202}} BridsonMartin R.GrunewaldFritzVogtmannKarenActions of arithmetic groups
on homology spheres and acyclic homology manifolds2014ISSN 0025-5874Math.
Z.276387–395Link@article{MR3150210, author = {Bridson, Martin~R.}, author =
{Grunewald, Fritz}, author = {Vogtmann, Karen}, title = {Actions of arithmetic
groups on homology spheres and acyclic homology manifolds}, date = {2014},
issn = {0025-5874}, journal = {Math. Z.}, volume = {276}, pages = {387\ndash
395}, url = {http://dx.doi.org/10.1007/s00209-013-1205-2}} BridsonMartin
R.VogtmannKarenActions of automorphism groups of free groups on homology
spheres and acyclic manifolds2011ISSN 0010-2571Comment. Math.
Helv.8673–90Link@article{MR2745276, author = {Bridson, Martin~R.}, author =
{Vogtmann, Karen}, title = {Actions of automorphism groups of free groups on
homology spheres and acyclic manifolds}, date = {2011}, issn = {0010-2571},
journal = {Comment. Math. Helv.}, volume = {86}, pages = {73\ndash 90}, url =
{http://dx.doi.org/10.4171/CMH/218}} BrownAaronSmoothness of stable holonomies
inside center-stable manifolds 2016Preprint<EMAIL_ADDRESS>author = {Brown, Aaron}, title = {Smoothness of stable holonomies inside
center-stable manifolds}, journal = { }, date = {2016}, note = {Preprint,
arXiv:1608.05886}} BrownAaronSmooth ergodic theory of $\mathbb{Z}^{d}$-actions
part 2: entropy formulas for rank-1 systems2016Preprint,
arXiv:1610.09997@misc{AWB-GLY-P2, author = {Brown, Aaron}, title = {Smooth
ergodic theory of {$\Bbb Z^d$}-actions part 2: Entropy formulas for rank-1
systems}, date = {2016}, note = {Preprint, arXiv:1610.09997}}
BrownAaronDamjanovicDanijelatitle=$C^{1}$ actions on manifolds by lattices in
Lie Groups date=2018Zhang, ZhiyuanIn preparation@misc{BDZ, author = {Brown,
Aaron}, author = {Damjanovic, Danijela}, author = {{Zhang, Zhiyuan}
title={$C^1$ actions on manifolds by lattices in Lie Groups} date={2018}},
note = {In preparation}} BrownAaronFisherDavidHurtadoSebastianZimmer’s
conjecture: subexponential growth, measure rigidity, and strong property
(t)2016Preprint<EMAIL_ADDRESS>author = {Brown, Aaron},
author = {Fisher, David}, author = {Hurtado, Sebastian}, title = {Zimmer's
conjecture: Subexponential growth, measure rigidity, and strong property (T)},
date = {2016}, note = {Preprint, arXiv:1608.04995}}
BrownAaronFisherDavidHurtadoSebastianZimmer’s conjecture for actions of
$\mathrm{SL}(m,\mathbb{Z})$2017Preprint<EMAIL_ADDRESS>author = {Brown, Aaron}, author = {Fisher, David}, author = {Hurtado,
Sebastian}, title = {Zimmer's conjecture for actions of
$\mathrm{SL}(m,\mathbb{Z})$}, date = {2017}, note = {Preprint,
arXiv:1710.02735}} BrownAaronFisherDavidHurtadoSebastianZimmer’s conjecture
for non-uniform latticesdate=2018In preparation@misc{BFHWM, author = {Brown,
Aaron}, author = {Fisher, David}, author = {Hurtado, Sebastian}, title =
{{Zimmer's conjecture for non-uniform lattices}date={2018}}, note = {In
preparation}} BrownAaronRodriguez HertzFedericoSmooth ergodic theory of
$\mathbb{Z}^{d}$-actions part 1: lyapunov exponents, dynamical charts, and
coarse lyapunov manifolds2016Preprint, arXiv:1610.09997@misc{AWB-GLY-P1,
author = {Brown, Aaron}, author = {Rodriguez~Hertz, Federico}, title = {Smooth
ergodic theory of {$\Bbb Z^d$}-actions part 1: Lyapunov exponents, dynamical
charts, and coarse Lyapunov manifolds}, date = {2016}, note = {Preprint,
arXiv:1610.09997}} BrownAaronRodriguez HertzFedericoMeasure rigidity for
random dynamics on surfaces and related skew products2017J. Amer. Math.
<EMAIL_ADDRESS>author = {Brown, Aaron}, author =
{Rodriguez~Hertz, Federico}, title = {Measure rigidity for random dynamics on
surfaces and related skew products}, date = {2017}, journal = {J. Amer. Math.
Soc.}, volume = {30}, pages = {1055\ndash 1132}} BrownAaronRodriguez
HertzFedericoWangZhirenInvariant measures and measurable projective factors
for actions of higher-rank lattices on manifolds2016Preprint,
arXiv:1609.05565@misc{AWBFRHZW-latticemeasure, author = {Brown, Aaron}, author
= {Rodriguez~Hertz, Federico}, author = {Wang, Zhiren}, title = {Invariant
measures and measurable projective factors for actions of higher-rank lattices
on manifolds}, date = {2016}, note = {Preprint, arXiv:1609.05565}}
BrownAaronRodriguez HertzFedericoWangZhirenSmooth ergodic theory of
$\mathbb{Z}^{d}$-actions part 3: product structure of entropy2016Preprint,
arXiv:1610.09997@misc{AWB-GLY-P3, author = {Brown, Aaron}, author =
{Rodriguez~Hertz, Federico}, author = {Wang, Zhiren}, title = {Smooth ergodic
theory of {$\Bbb Z^d$}-actions part 3: Product structure of entropy}, date =
{2016}, note = {Preprint, arXiv:1610.09997}} BrownAaronRodriguez
HertzFedericoWangZhirenGlobal smooth and topological rigidity of hyperbolic
lattice actions2017Ann. of Math. (2)186913–972@article{BRHW1, author = {Brown,
Aaron}, author = {Rodriguez~Hertz, Federico}, author = {Wang, Zhiren}, title =
{Global smooth and topological rigidity of hyperbolic lattice actions}, date =
{2017}, journal = {Ann. of Math. (2)}, volume = {186}, pages = {913\ndash
972}} BurgerM.MonodN.Continuous bounded cohomology and applications to
rigidity theory2002ISSN 1016-443XGeom. Funct.
Anal.12219–280Link@article{MR1911660, author = {Burger, M.}, author = {Monod,
N.}, title = {Continuous bounded cohomology and applications to rigidity
theory}, date = {2002}, issn = {1016-443X}, journal = {Geom. Funct. Anal.},
volume = {12}, pages = {219\ndash 280}, url =
{http://dx.doi.org/10.1007/s00039-002-8245-9}} BurnsKeithWilkinsonAmieOn the
ergodicity of partially hyperbolic systems2010ISSN 0003-486XAnn. of Math.
(2)171451–489@article{MR2630044, author = {Burns, Keith}, author = {Wilkinson,
Amie}, title = {On the ergodicity of partially hyperbolic systems}, date =
{2010}, issn = {0003-486X}, journal = {Ann. of Math. (2)}, volume = {171},
pages = {451\ndash 489}} CantatSergeVersion kählérienne d’une conjecture de
Robert J. Zimmer2004ISSN 0012-9593Ann. Sci. École Norm. Sup.
(4)37759–768Link@article{MR2103473, author = {Cantat, Serge}, title = {Version
k\"ahl\'erienne d'une conjecture de {R}obert {J}. {Z}immer}, date = {2004},
issn = {0012-9593}, journal = {Ann. Sci. \'Ecole Norm. Sup. (4)}, volume =
{37}, pages = {759\ndash 768}, url =
{https://doi.org/10.1016/j.ansens.2004.04.003}} CantatSergeProgrès récents
concernant le programme de zimmer2017Séminaire Bourbaki. Vol. 2017–2018.
Exposés 1136@article{Cantat, author = {Cantat, Serge}, title = {Progr\\`es
r\'ecents concernant le programme de zimmer}, date = {2017}, note =
{S\'eminaire Bourbaki. Vol. 2017–2018. Expos\'es 1136}}
CantatSergeXieJunyiAlgebraic actions of discrete groups: the $p$-adic
method2018ISSN 0001-5962Acta Math.2202239–295Link@article{MR3849285, author =
{Cantat, Serge}, author = {Xie, Junyi}, title = {Algebraic actions of discrete
groups: the {$p$}-adic method}, date = {2018}, issn = {0001-5962}, journal =
{Acta Math.}, volume = {220}, number = {2}, pages = {239\ndash 295}, url =
{https://doi.org/10.4310/ACTA.2018.v220.n2.a2}}
CantatSergeZeghibAbdelghaniHolomorphic actions, Kummer examples, and Zimmer
program2012ISSN 0012-9593Ann. Sci. Éc. Norm. Supér.
(4)45447–489@article{MR3014483, author = {Cantat, Serge}, author = {Zeghib,
Abdelghani}, title = {Holomorphic actions, {K}ummer examples, and {Z}immer
program}, date = {2012}, issn = {0012-9593}, journal = {Ann. Sci. \'Ec. Norm.
Sup\'er. (4)}, volume = {45}, pages = {447\ndash 489}} CaoYongluoNon-zero
Lyapunov exponents and uniform hyperbolicity2003ISSN
0951-7715Nonlinearity1641473–1479Link@article{MR1986306, author = {Cao,
Yongluo}, title = {Non-zero {L}yapunov exponents and uniform hyperbolicity},
date = {2003}, issn = {0951-7715}, journal = {Nonlinearity}, volume = {16},
number = {4}, pages = {1473\ndash 1479}, url =
{https://doi.org/10.1088/0951-7715/16/4/316}}
ClimenhagaVaughnKatokAnatoleMeasure theory through dynamical eyes2012Preprint,
<EMAIL_ADDRESS>author = {Climenhaga, Vaughn}, author =
{Katok, Anatole}, title = {Measure theory through dynamical eyes}, date =
{2012}, note = {Preprint, arXiv:1208.4550}} CorletteKevinArchimedean
superrigidity and hyperbolic geometry1992ISSN 0003-486XAnn. of Math.
(2)135165–182Link@article{MR1147961, author = {Corlette, Kevin}, title =
{Archimedean superrigidity and hyperbolic geometry}, date = {1992}, issn =
{0003-486X}, journal = {Ann. of Math. (2)}, volume = {135}, pages = {165\ndash
182}, url = {https://doi.org/10.2307/2946567}}
DamjanovićDanijelaKatokAnatoleLocal rigidity of partially hyperbolic actions
I. KAM method and $\mathbb{Z}^{k}$ actions on the torus2010ISSN 0003-486XAnn.
of Math. (2)1721805–1858Link@article{MR2726100, author = {Damjanovi{\'c},
Danijela}, author = {Katok, Anatole}, title = {Local rigidity of partially
hyperbolic actions {I}. {KAM} method and {$\Bbb Z^k$} actions on the torus},
date = {2010}, issn = {0003-486X}, journal = {Ann. of Math. (2)}, volume =
{172}, pages = {1805\ndash 1858}, url =
{http://dx.doi.org/10.4007/annals.2010.172.1805}}
DamjanovićDanijelaKatokAnatoleLocal rigidity of partially hyperbolic actions.
II: The geometric method and restrictions of Weyl chamber flows on
$SL(n,\mathbb{R})/\Gamma$2011ISSN 1073-7928Int. Math. Res. Not.
IMRN4405–4430Link@article{MR2838045, author = {Damjanovi{\'c}, Danijela},
author = {Katok, Anatole}, title = {Local rigidity of partially hyperbolic
actions. {II}: {T}he geometric method and restrictions of {W}eyl chamber flows
on {$SL(n,\Bbb R)/\Gamma$}}, date = {2011}, issn = {1073-7928}, journal =
{Int. Math. Res. Not. IMRN}, pages = {4405\ndash 4430}, url =
{http://dx.doi.org/10.1093/imrn/rnq252}} DamjanovicDanijelaXuDishengOn
classification of higher rank anosov actions on compact manifold2018Preprint,
<EMAIL_ADDRESS>author = {Damjanovic, Danijela}, author =
{Xu, Disheng}, title = {On classification of higher rank Anosov actions on
compact manifold}, date = {2018}, note = {Preprint, arXiv:1801.04904}} DaniS.
G.Invariant measures of horospherical flows on noncompact homogeneous
spaces1978ISSN 0020-9910Invent. Math.47101–138Link@article{MR0578655, author =
{Dani, S.~G.}, title = {Invariant measures of horospherical flows on
noncompact homogeneous spaces}, date = {1978}, issn = {0020-9910}, journal =
{Invent. Math.}, volume = {47}, pages = {101\ndash 138}, url =
{http://dx.doi.org/10.1007/BF01578067}} DaniS. G.Invariant measures and
minimal sets of horospherical flows1981ISSN 0020-9910Invent.
Math.64357–385Link@article{MR629475, author = {Dani, S.~G.}, title =
{Invariant measures and minimal sets of horospherical flows}, date = {1981},
issn = {0020-9910}, journal = {Invent. Math.}, volume = {64}, pages =
{357\ndash 385}, url = {http://dx.doi.org/10.1007/BF01389173}} DaniS. G.Orbits
of horospherical flows1986ISSN 0012-7094Duke Math.
J.53177–188Link@article{MR835804, author = {Dani, S.~G.}, title = {Orbits of
horospherical flows}, date = {1986}, issn = {0012-7094}, journal = {Duke Math.
J.}, volume = {53}, pages = {177\ndash 188}, url =
{http://dx.doi.org/10.1215/S0012-7094-86-05312-3}} DaniS. G.MargulisG.
A.Values of quadratic forms at primitive integral points1989ISSN
0020-9910Invent. Math.98405–424Link@article{MR1016271, author = {Dani, S.~G.},
author = {Margulis, G.~A.}, title = {Values of quadratic forms at primitive
integral points}, date = {1989}, issn = {0020-9910}, journal = {Invent.
Math.}, volume = {98}, pages = {405\ndash 424}, url =
{http://dx.doi.org/10.1007/BF01388860}} DaniS. G.MargulisG. A.Orbit closures
of generic unipotent flows on homogeneous spaces of ${\rm SL}(3,{\bf
R})$1990ISSN 0025-5831Math. Ann.286101–128Link@article{MR1032925, author =
{Dani, S.~G.}, author = {Margulis, G.~A.}, title = {Orbit closures of generic
unipotent flows on homogeneous spaces of {${\rm SL}(3,{\bf R})$}}, date =
{1990}, issn = {0025-5831}, journal = {Math. Ann.}, volume = {286}, pages =
{101\ndash 128}, url = {http://dx.doi.org/10.1007/BF01453567}} DaniS.
G.SmillieJohnUniform distribution of horocycle orbits for Fuchsian
groups1984ISSN 0012-7094Duke Math. J.51185–194Link@article{MR744294, author =
{Dani, S.~G.}, author = {Smillie, John}, title = {Uniform distribution of
horocycle orbits for {F}uchsian groups}, date = {1984}, issn = {0012-7094},
journal = {Duke Math. J.}, volume = {51}, pages = {185\ndash 194}, url =
{http://dx.doi.org/10.1215/S0012-7094-84-05110-X}} de la SalleMikaelStrong (t)
for higher rank lattices2017Preprint,
arXiv:1711.01900@misc{delaSallenonuniform, author = {de~la Salle, Mikael},
title = {Strong (T) for higher rank lattices}, date = {2017}, note =
{Preprint, arXiv:1711.01900}} de LaatTimde la SalleMikaelStrong property (T)
for higher-rank simple Lie groups2015ISSN 0024-6115Proc. Lond. Math. Soc.
(3)111936–966Link@article{MR3407190, author = {de~Laat, Tim}, author = {de~la
Salle, Mikael}, title = {Strong property ({T}) for higher-rank simple {L}ie
groups}, date = {2015}, issn = {0024-6115}, journal = {Proc. Lond. Math. Soc.
(3)}, volume = {111}, pages = {936\ndash 966}, url =
{http://dx.doi.org/10.1112/plms/pdv040}}
EinsiedlerManfredFisherTravisDifferentiable rigidity for hyperbolic toral
actions2007ISSN 0021-2172Israel J. Math.157347–377Link@article{MR2342454,
author = {Einsiedler, Manfred}, author = {Fisher, Travis}, title =
{Differentiable rigidity for hyperbolic toral actions}, date = {2007}, issn =
{0021-2172}, journal = {Israel J. Math.}, volume = {157}, pages = {347\ndash
377}, url = {https://doi.org/10.1007/s11856-006-0016-0}}
EinsiedlerManfredKatokAnatoleInvariant measures on $G/\Gamma$ for split simple
Lie groups $G$2003ISSN 0010-3640Comm. Pure Appl.
Math.561184–1221Link@article{MR1989231, author = {Einsiedler, Manfred}, author
= {Katok, Anatole}, title = {Invariant measures on {$G/\Gamma$} for split
simple {L}ie groups {$G$}}, date = {2003}, issn = {0010-3640}, journal =
{Comm. Pure Appl. Math.}, volume = {56}, pages = {1184\ndash 1221}, url =
{http://dx.doi.org/10.1002/cpa.10092}} EinsiedlerManfredKatokAnatoleRigidity
of measures—the high entropy case and non-commuting foliations2005ISSN
0021-2172Israel J. Math.148169–238Link@article{MR2191228, author =
{Einsiedler, Manfred}, author = {Katok, Anatole}, title = {Rigidity of
measures—the high entropy case and non-commuting foliations}, date = {2005},
issn = {0021-2172}, journal = {Israel J. Math.}, volume = {148}, pages =
{169\ndash 238}, url = {http://dx.doi.org/10.1007/BF02775436}}
EinsiedlerManfredKatokAnatoleLindenstraussElonInvariant measures and the set
of exceptions to Littlewood’s conjecture2006ISSN 0003-486XAnn. of Math.
(2)164513–560Link@article{MR2247967, author = {Einsiedler, Manfred}, author =
{Katok, Anatole}, author = {Lindenstrauss, Elon}, title = {Invariant measures
and the set of exceptions to {L}ittlewood's conjecture}, date = {2006}, issn =
{0003-486X}, journal = {Ann. of Math. (2)}, volume = {164}, pages = {513\ndash
560}, url = {http://dx.doi.org/10.4007/annals.2006.164.513}}
EinsiedlerManfredLindenstraussElonRigidity properties of
$\mathbb{Z}^{d}$-actions on tori and solenoids2003ISSN 1079-6762Electron. Res.
Announc. Amer. Math. Soc.999–110 (electronic)Link@article{MR2029471, author =
{Einsiedler, Manfred}, author = {Lindenstrauss, Elon}, title = {Rigidity
properties of {$\Bbb Z^d$}-actions on tori and solenoids}, date = {2003}, issn
= {1079-6762}, journal = {Electron. Res. Announc. Amer. Math. Soc.}, volume =
{9}, pages = {99\ndash 110 (electronic)}, url =
{http://dx.doi.org/10.1090/S1079-6762-03-00117-3}}
EinsiedlerManfredLindenstraussElonOn measures invariant under diagonalizable
actions: the rank-one case and the general low-entropy method2008ISSN
1930-5311J. Mod. Dyn.283–128@article{MR2366231, author = {Einsiedler,
Manfred}, author = {Lindenstrauss, Elon}, title = {On measures invariant under
diagonalizable actions: the rank-one case and the general low-entropy method},
date = {2008}, issn = {1930-5311}, journal = {J. Mod. Dyn.}, volume = {2},
pages = {83\ndash 128}} EinsiedlerM.LindenstraussE.Diagonal actions on locally
homogeneous spaces2010Homogeneous flows, moduli spaces and arithmeticClay
Math. Proc.10Amer. Math. Soc., Providence, RI155–241@incollection{MR2648695,
author = {Einsiedler, M.}, author = {Lindenstrauss, E.}, title = {Diagonal
actions on locally homogeneous spaces}, date = {2010}, booktitle =
{Homogeneous flows, moduli spaces and arithmetic}, series = {Clay Math.
Proc.}, volume = {10}, publisher = {Amer. Math. Soc., Providence, RI}, pages =
{155\ndash 241}} EinsiedlerManfredWardThomasErgodic theory with a view towards
number theoryGraduate Texts in MathematicsSpringer-Verlag London, Ltd.,
London2011259ISBN 978-0-85729-020-5Link@book{MR2723325, author = {Einsiedler,
Manfred}, author = {Ward, Thomas}, title = {Ergodic theory with a view towards
number theory}, series = {Graduate Texts in Mathematics}, publisher =
{Springer-Verlag London, Ltd., London}, date = {2011}, volume = {259}, isbn =
{978-0-85729-020-5}, url = {https://doi.org/10.1007/978-0-85729-021-2}}
EskinAlexLindenstraussE.Random walks on locally homogeneous
spaces2018Preprint.@misc{ELlong, author = {Eskin, Alex}, author =
{Lindenstrauss, E.}, title = {Random walks on locally homogeneous spaces},
date = {2018}, note = {Preprint.}} EskinAlexMirzakhaniMaryamInvariant and
stationary measures for the ${\rm SL}(2,\mathbb{R})$ action on moduli
space2018ISSN 0073-8301Publ. Math. Inst. Hautes Études
Sci.12795–324Link@article{MR3814652, author = {Eskin, Alex}, author =
{Mirzakhani, Maryam}, title = {Invariant and stationary measures for the
{${\rm SL}(2,\Bbb R)$} action on moduli space}, date = {2018}, issn =
{0073-8301}, journal = {Publ. Math. Inst. Hautes \'Etudes Sci.}, volume =
{127}, pages = {95\ndash 324}, url =
{https://doi.org/10.1007/s10240-018-0099-2}}
EskinAlexMirzakhaniMaryamMohammadiAmirIsolation, equidistribution, and orbit
closures for the ${\rm SL}(2,\mathbb{R})$ action on moduli space2015ISSN
0003-486XAnn. of Math. (2)182673–721Link@article{MR3418528, author = {Eskin,
Alex}, author = {Mirzakhani, Maryam}, author = {Mohammadi, Amir}, title =
{Isolation, equidistribution, and orbit closures for the {${\rm SL}(2,\Bbb
R)$} action on moduli space}, date = {2015}, issn = {0003-486X}, journal =
{Ann. of Math. (2)}, volume = {182}, pages = {673\ndash 721}, url =
{http://dx.doi.org/10.4007/annals.2015.182.2.7}} FarbBensonMargalitDanA primer
on mapping class groupsPrinceton Mathematical SeriesPrinceton University
Press, Princeton, NJ201249@book{MR2850125, author = {Farb, Benson}, author =
{Margalit, Dan}, title = {A primer on mapping class groups}, series =
{Princeton Mathematical Series}, publisher = {Princeton University Press,
Princeton, NJ}, date = {2012}, volume = {49}} FarbBensonShalenPeterReal-
analytic actions of lattices1999ISSN 0020-9910Invent.
Math.135273–296Link@article{MR1666834, author = {Farb, Benson}, author =
{Shalen, Peter}, title = {Real-analytic actions of lattices}, date = {1999},
issn = {0020-9910}, journal = {Invent. Math.}, volume = {135}, pages =
{273\ndash 296}, url = {http://dx.doi.org/10.1007/s002220050286}}
FeresR.LabourieF.Topological superrigidity and Anosov actions of
lattices1998ISSN 0012-9593Ann. Sci. École Norm. Sup.
(4)31599–629Link@article{MR1643954, author = {Feres, R.}, author = {Labourie,
F.}, title = {Topological superrigidity and {A}nosov actions of lattices},
date = {1998}, issn = {0012-9593}, journal = {Ann. Sci. \'Ecole Norm. Sup.
(4)}, volume = {31}, pages = {599\ndash 629}, url =
{http://dx.doi.org/10.1016/S0012-9593(98)80001-3}} FisherDavidLocal rigidity
of group actions: past, present, future2007Dynamics, ergodic theory, and
geometryMath. Sci. Res. Inst. Publ.54Cambridge Univ. Press,
Cambridge45–97Link@incollection{MR2369442, author = {Fisher, David}, title =
{Local rigidity of group actions: past, present, future}, date = {2007},
booktitle = {Dynamics, ergodic theory, and geometry}, series = {Math. Sci.
Res. Inst. Publ.}, volume = {54}, publisher = {Cambridge Univ. Press,
Cambridge}, pages = {45\ndash 97}, url =
{https://doi.org/10.1017/CBO9780511755187.003}} FisherDavidDeformations of
group actions2008ISSN 0002-9947Trans. Amer. Math.
Soc.360491–505Link@article{MR2342012, author = {Fisher, David}, title =
{Deformations of group actions}, date = {2008}, issn = {0002-9947}, journal =
{Trans. Amer. Math. Soc.}, volume = {360}, pages = {491\ndash 505}, url =
{https://doi.org/10.1090/S0002-9947-07-04372-3}} FisherDavidGroups acting on
manifolds: around the Zimmer program2011Geometry, rigidity, and group
actionsChicago Lectures in Math.Univ. Chicago Press, Chicago,
IL72–157Link@incollection{MR2807830, author = {Fisher, David}, title = {Groups