# Non-left-orderability of lattices
in higher-rank semisimple Lie groups
(after Deroin and Hurtado)

Dave Witte Morris  Department of Mathematics and Computer Science, University
of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada
<EMAIL_ADDRESS>https://deductivepress.ca/dmorris

###### Abstract.

Let $G$ be a connected, semisimple, real Lie group with finite centre, with
$\operatorname{rank}_{\mathbb{R}}G\geq 2$. B. Deroin and S. Hurtado recently
proved the 30-year-old conjecture that no irreducible lattice in $G$ has a
left-invariant total order. (Equivalently, they proved that no such lattice
has a nontrivial, orientation-preserving action on the real line.) We will
explain many of the main ideas of the proof, by using them to prove the
analogous result for lattices in $p$-adic semisimple groups. The $p$-adic case
is easier, because some of the technical issues do not arise.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2A Measure theory
    2. 2B Topological groups
    3. 2C Group actions
    4. 2D Dynamical systems
  3. 3 Main ideas of the Deroin-Hurtado proof
    1. 3A Assumptions and notation
    2. 3B The almost-periodic space $Z$ and the induced $G$-space $X$
    3. 3C The probability measures $\mu_{Z}$, $\mu_{X}$, and $\mu_{X}^{P}$
    4. 3D Contraction on $\mathbb{R}$-orbits and the Key Proposition
    5. 3E Invariance under the centralizer
    6. 3F Completion of the proof by propagating invariance
    7. 3G Proof of proposition 3.33
  4. 4 Some additional explanations
    1. 4A Construction of the almost-periodic space $Z$
    2. 4B The measures $\mu_{Z}$ and $\mu_{X}$ are stationary
    3. 4C Contraction on the $\mathbb{R}$-orbits
    4. 4D $\mu_{X}^{P}$ is $G$-invariant
    5. 4E More proof of proposition 3.33
  5. 5 $K\Gamma$ might not be all of $G$

##  1\. Introduction

######  Definition 1.1 ([16, p. 7]).

A group $\Gamma$ is _left-orderable_ if it has a left-invariant total order.
This means there is a transitive binary relation $\prec$, such that, for all
$a,b,c\in\Gamma$,

  * •

exactly one of the following is true: $a\prec b$, or $a=b$, or $a\succ b$, and

  * •

if $a\prec b$, then $ca\prec cb$.

Understanding which groups are left-orderable is a longstanding problem in
algebra [25, 30] and is also important in topology [11]. The following result
proves a conjecture that was suggested informally (at least for the case where
$G$ is simple) by the current author more than 30 years ago, and became
official when it was proposed independently by É. Ghys [22, p. 200] in 1999.

######  Theorem 1.2 (Deroin-Hurtado [14, Thm. 1.3]).

If $\Gamma$ is an irreducible lattice in a connected, semisimple, real Lie
group $G$ with finite centre, and $\operatorname{rank}_{\mathbb{R}}G\geq 2$,
then $\Gamma$ is not left-orderable.

Theorem 1.2 can be restated as follows:

######  Corollary 1.3 (Deroin-Hurtado [14, Thm. 1.3]).

Assume $\Gamma$ is an irreducible lattice in a connected, semisimple, real Lie
group $G$ with finite centre, and $\operatorname{rank}_{\mathbb{R}}G\geq 2$.
Then $\Gamma$ has no nontrivial action on the real line by orientation-
preserving homeomorphisms.

###### Proof.

Suppose $\Gamma$ has a nontrivial, orientation-preserving action on
$\mathbb{R}$, and let $N$ be the kernel of the action. Then $\Gamma/N$ has a
faithful, orientation-preserving action on $\mathbb{R}$, so it is not
difficult to see that $\Gamma/N$ is left-orderable [23, Thm. 6.8].

On the other hand, since no finite groups are left-orderable [16, §1.4.1, p.
27], we know that $N$ is a normal subgroup of infinite index, so the Margulis
Normal Subgroup Theorem [29, Thm. 17.1.1, p. 347] tells us that $N$ is
contained in the centre of $G$. So $N$ is normal in $G$. Then $\Gamma/N$ is a
left-orderable, irreducible lattice in the semisimple Lie group $G/N$. This
contradicts theorem 1.2. ∎

######  Remark 1.4.

If we do not require the homeomorphisms to be orientation-preserving, then
there could be a nontrivial action: the map $x\mapsto-x$ is a homeomorphism of
order $2$, and it is possible that $\Gamma$ has a nontrivial homomorphism to
$\mathbb{Z}/2\mathbb{Z}$. However, it follows from theorem 1.2 that every
action of $\Gamma$ on $\mathbb{R}$ (whether orientation-preserving or not)
must be _virtually trivial_. This means that some finite-index subgroup of
$\Gamma$ acts trivially. (In fact, the subgroup will have index $1$ or $2$.)

By combining theorem 1.2 with the following result, Deroin and Hurtado also
obtained a strong result for actions on the circle $S^{1}$, not only actions
on $\mathbb{R}$.

######  Theorem 1.5 (Ghys [22], cf. Burger-Monod [9, Cor. 1.5], [8, Cor.
1.7]).

Assume $\Gamma$ is an irreducible lattice in a connected, semisimple, real Lie
group $G$, and $\operatorname{rank}_{\mathbb{R}}G\geq 2$. If no almost-simple
factor of $G$ is locally isomorphic to $\operatorname{SL}(2,\mathbb{R})$, then
every action of $\Gamma$ on the circle $S^{1}$ has a finite orbit.

######  Corollary 1.6 (Deroin-Hurtado [14]).

Assume $\Gamma$ is an irreducible lattice in a connected, semisimple, real Lie
group $G$, and $\operatorname{rank}_{\mathbb{R}}G\geq 2$. If no almost-simple
factor of $G$ is locally isomorphic to $\operatorname{SL}(2,\mathbb{R})$, then
every action of $\Gamma$ on the circle $S^{1}$ is virtually trivial.

###### Proof.

By theorem 1.5, we may let $x$ be a point whose orbit is finite. Then its
stabilizer $\Gamma_{\\!x}$ is a finite-index subgroup that acts on
$S^{1}\smallsetminus\\{x\\}$, which is homeomorphic to $\mathbb{R}$. Then we
see from 1.4 that a finite-index subgroup of $\Gamma_{\\!x}$ must act
trivially. ∎

######  Remark 1.7.

The group $\operatorname{PSL}(2,\mathbb{R})$ acts faithfully on
$\mathbb{R}\cup\\{\infty\\}\simeq S^{1}$ (by linear-fractional
transformations). Therefore, if some almost-simple factor of $G$ is locally
isomorphic to $\operatorname{SL}(2,\mathbb{R})$, then every lattice in $G$ has
a highly nontrivial action on $S^{1}$ (because there is a surjective
homomorphism $G\to\operatorname{PSL}(2,\mathbb{R})$). This shows that
corollary 1.6 would not be valid without the hypothesis that no factor is
locally isomorphic to $\operatorname{SL}(2,\mathbb{R})$.

For simplicity, the statements of the above results assume that the centre of
$G$ is finite. There are counterexamples without this assumption, but Deroin
and Hurtado proved that all of them arise from the fact that the action of
$\operatorname{PSL}(2,\mathbb{R})$ on $S^{1}$ lifts to an orientation-
preserving action of $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ (the
universal cover of $\operatorname{SL}(2,\mathbb{R})$) on $\mathbb{R}$. In
particular:

######  Corollary 1.8 (Deroin-Hurtado [14, Thm. 1.1 and §2]).

Assume $G$ is a connected, semisimple, real Lie group, and
$\operatorname{rank}_{\mathbb{R}}G\geq 2$. An irreducible lattice $\Gamma$ in
$G$ is left-orderable if and only if it is torsion-free and there is a
surjective homomorphism $G\to\widetilde{\operatorname{SL}}(2,\mathbb{R})$.

######  Remark 1.9.

Theorem 1.2 is a huge advance over what was previously known. In particular:

  1. (1)

There was not a single example of a cocompact, torsion-free lattice $\Gamma$
(in a connected, semisimple Lie group), for which it was known that no finite-
index subgroup of $\Gamma$ is left-orderable.

  2. (2)

For the noncocompact case, the theorem had been proved under the additional
assumption that either $\operatorname{rank}_{\mathbb{Q}}\Gamma\geq 2$ [32] or
$G$ has more than one noncompact simple factor [26]. However, the theorem was
not known to be true even for the special case where
$G=\operatorname{SL}(3,\mathbb{R})$.

See [23, 28] for expository discussions of actions of lattices (and related
groups) on $\mathbb{R}$ or $S^{1}$, and see [16, Chap. 4] for an introduction
to the viewpoint on (non-)left-orderability that is used in the proof of
theorem 1.2.

######  Remark 1.10.

The circle is a $1$-dimensional manifold, so the Deroin-Hurtado Theorem is a
part of the “Zimmer program,” which includes the study of actions of lattices
on low-dimensional manifolds. (However, unlike in theorem 1.2, the actions are
usually assumed to be differentiable, and are often $C^{\infty}$.) The recent
work of A. Brown, D. Fisher, and S. Hurtado [5, 6, 7] is a huge advance in the
area (and seems to have been part of the inspiration for Deroin and Hurtado).
See [18] for a survey of the Zimmer program, and see [10, 19] for expositions
of the first Brown-Fisher-Hurtado paper.

In this paper, we will explain some of the main ideas in the proof of theorem
1.2 (or, equivalently, corollary 1.3). However, $p$-adic groups are easier
than real ones, so we will actually prove the following:

######  Theorem 1.11 (Deroin-Hurtado).

If $\Gamma$ is an irreducible lattice in a $p$-adic semisimple Lie group $G$,
and $\operatorname{rank}_{\mathbb{Q}_{p}}G\geq 2$, then $\Gamma$ has no
nontrivial action on the real line by orientation-preserving homeomorphisms.

Here is an outline of the paper.

  * •

Section 2 recalls some standard definitions and notation, and a few basic
results. (The reader is encouraged to skip this Section, and refer back when
necessary.)

  * •

Section 3 sketches the proof of theorem 1.11.

  * •

Further explanation of several arguments can be found in Section 4. Some
details are omitted, but specific references to the original work of Deroin
and Hurtado [14] are provided.

  * •

Our exposition of the proof of theorem 1.11 relies on the simplifying
assumption that $G=K\Gamma$ (see 3.6), but Section 5 provides a brief
introduction to the main tool that allows this assumption to be eliminated
(namely, the theory of harmonic functions).

######  Note 1.12.

This is an expository paper. None of the nontrivial ideas or results are new,
so the author does not claim credit for any of them, even if no reference is
given. In particular, although theorem 1.11 is not stated in [14], it follows
directly from the methods there, so it can be considered to be implicit in the
paper and should be attributed to Deroin and Hurtado.

On the other hand, the author takes responsibility for any errors and other
deficiencies in the manuscript (of course).

######  Acknowledgments.

I thank B. Deroin, S. Hurtado, and J. Lécureux for informative discussions
related to the proof of theorem 1.2 and possible generalizations. I also
acknowledge support of the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne
Université) and LabEx CARMIN (ANR-10-LABX-59-01).

##  2\. Preliminaries

###  2A. Measure theory

######  Reminder 2.1.

Let $\mu$ be a measure on a separable, metrizable topological space $Y$.

  1. (1)

We say that $\mu$ is a _probability measure_ if $\mu(Y)=1$.

  2. (2)

The _support_ of $\mu$ is the (unique) smallest closed set whose complement
has measure $0$.

  3. (3)

$\mu$ is _atomic_ if $\mu(B)=\sum_{b\in B}\mu\bigl{(}\\{b\\}\bigr{)}$ for
every Borel set $B$. Informally, this means that $\mu$ is a sum of point
masses.

  4. (4)

A property is true _for $\mu$-a.e. $y\in Y$_ if the set of points where it is
false has measure $0$. In this case, we may also say that the property is true
_$\mu$ -a.e._, or that is is true for $\mu$-a.e. $y\in Y$.

  5. (5)

A measure $\mu^{\prime}$ is _absolutely continuous_ with respect to $\mu$ if,
for every Borel set $B$, such that $\mu(B)=0$, we also have
$\mu^{\prime}(B)=0$.

  6. (6)

If $\mu^{\prime}$ is absolutely continuous with respect to $\mu$, then there
is a non-negative measurable function $d\mu^{\prime}/d\mu$ (called the _Radon-
Nikodym derivative_ of $\mu^{\prime}$ with respect to $\mu$), such that
$\mu^{\prime}=(d\mu^{\prime}/d\mu)\,\mu$, in the sense that, for every real-
valued continuous function with compact support, we have

$\int_{Y}f\,d\mu^{\prime}=\int_{Y}f(y)\,\frac{d\mu^{\prime}}{d\mu}(y)\,d\mu(y).$

The Radon-Nikodym derivative is unique up to a set of $\mu$-measure $0$.

  7. (7)

If two measures are absolutely continuous with respect to each other, then
they are in the same _measure class_. (This is an equivalence relation.)

  8. (8)

If $f\colon Y\to Z$ is a continuous function to a topological space $Z$, then
we define a measure $f_{*}\mu$ on $Z$ by

$f_{*}\mu(B)=\mu\bigl{(}f^{-1}(B)\bigr{)}.$

###  2B. Topological groups

######  Reminder 2.2.

Assume $H$ is a locally compact topological group.

  * •

$H$ is said to be _locally compact_ if it has a nonempty, open subset whose
closure is compact. All groups in this paper (including $G$, $K$, $\Gamma$,
and $\mathbb{R}$) are locally compact (and second countable).

  * •

It is a basic fact in the theory of topological groups that every locally
compact group has a _Haar measure_. This is a measure $m_{H}$ on $H$, such
that:

    * $\circ$

$m_{H}$ is left-invariant: $m_{H}(hB)=m_{H}(B)$ for all $h\in H$ and every
Borel set $B$,

    * $\circ$

$m_{H}$ is locally finite: $m_{H}(C)<\infty$ for every compact set $C$, and

    * $\circ$

$m_{H}$ has full support: $m_{H}(U)>0$ for every open set $U$.

Furthermore, the Haar measure is unique, up to a scalar multiple: if $\mu$ is
another Haar measure on $H$, then there is a constant $c>0$, such that
$\mu(B)=c\,m_{H}(B)$ for every Borel set $B$.

  * •

$H$ is _unimodular_ if its Haar measure is bi-invariant. This means it is both
left-invariant and right-invariant: $\mu_{H}(hB)=\mu_{H}(B)=\mu_{H}(Bh)$. Note
that this implies that $\mu_{H}$ is invariant under conjugation:
$\mu_{H}(hBh^{-1})=\mu_{H}(B)$.

  * •

If $H$ is either a semisimple Lie group or a discrete group or a compact
group, then $H$ is unimodular.

  * •

If $H$ is compact, then $m_{H}(H)<\infty$ (because the measure of every
compact set is finite), and we usually choose the normalizing scalar to make
$m_{H}$ a probability measure.

  * •

If $H$ is not compact, then $m_{H}(H)=\infty$, so it is not possible for the
Haar measure on a noncompact group to be a probability measure.

######  Definition 2.3.

Assume $G$ is a unimodular topological group.

  * •

If $\Gamma$ is any discrete subgroup of $G$, then any Haar measure $m_{G}$
induces a well-defined $G$-invariant measure $m_{G/\Gamma}$ on $G/\Gamma$ by
setting:

$m_{G/\Gamma}(B)=m_{G}\bigl{(}\pi^{-1}(B)\cap\mathcal{F}\bigr{)},$

where $\pi\colon G\to G/\Gamma$ is the natural quotient map
($\pi(g)=g\Gamma$), and $\mathcal{F}$ is a measurable set of coset
representatives for $\Gamma$ in $G$ (or, in other words, $\mathcal{F}$ is a
_fundamental domain_ for $\Gamma$ in $G$).

  * •

A subgroup $\Gamma$ of a locally compact topological group $G$ is a _lattice_
in $G$ if $\Gamma$ is discrete, and the quotient space $G/\Gamma$ has finite
measure (i.e., $m_{G/\Gamma}(G/\Gamma)<\infty$).

  * •

A lattice $\Gamma$ is _irreducible_ if $\Gamma N$ is dense in $G$, for every
closed, noncompact, normal subgroup $N$ of $G$.

  * •

A closed subgroup $\Gamma$ of $G$ is _cocompact_ if $G/\Gamma$ is compact.

Every discrete, cocompact subgroup is a lattice (because the Haar measure of
every compact set is finite). The converse is not true in general, but the
first part of the following result is a special case of the fact that it is
true in $p$-adic groups:

######  Theorem 2.4.

Assume $\Gamma$ is an irreducible lattice in a $p$-adic semisimple Lie group
$G$, such that $\operatorname{rank}_{\mathbb{Q}_{p}}G\geq 2$. Then:

  1. (1)

$\Gamma$ is cocompact [27, Prop. IX.3.7, p. 313],

  2. (2)

the commutator subgroup $[\Gamma,\Gamma]$ has finite index in $\Gamma$ [27,
Thm. IX.5.4, p. 325], and

  3. (3)

if $\Gamma$ acts on the circle $S^{1}$ by homeomorphisms, then the action has
a finite orbit [33, Cor. 6.10].

###  2C. Group actions

######  Definition 2.5.

Assume

  * •

a topological group $H$ acts by homeomorphisms on a metrizable topological
space $Y$,

  * •

$\mu_{H}$ is a probability measure on $H$, and

  * •

$\mu_{Y}$ is a probability measure on $Y$.

Then:

  1. (1)

The action of $H$ is _free_ if $hy\neq y$ for all nonidentity $h\in H$ and all
$y\in Y$.

  2. (2)

The measure $\mu_{Y}$ is _$H$ -invariant_ if $h_{*}\mu_{Y}=\mu_{Y}$ for all
$h\in H$ (where $h_{*}\mu_{Y}$ was defined in 2.1(8)). In other words,
$\mu_{Y}(hB)=\mu_{Y}(B)$, for every $h\in H$ and every Borel set $B$.

  3. (3)

An $H$-invariant measure $\mu_{Y}$ is _ergodic_ if, for every $H$-invariant
Borel set $B$, either $\mu_{Y}(B)=0$, or $\mu_{Y}(Y\setminus B)=0$.

  4. (4)

The convolution $\mu_{H}*\mu_{Y}$ is the measure on $Y$ defined by

$\text{$\mu_{H}*\mu_{Y}=\int_{H}h_{*}\mu_{Y}\,d\mu_{H}(H)$, \quad so
\quad$(\mu_{H}*\mu_{Y})(B)=\int_{H}\mu_{Y}(h^{-1}B)\,d\mu_{H}(h)$}.$

  5. (5)

We say that $\mu_{Y}$ is _$\mu_{H}$ -stationary_ if $\mu_{H}*\mu_{Y}=\mu_{Y}$.
More concretely, for every Borel set $B$, this means that we have

$\mu_{Y}(B)=\int_{H}\mu_{Y}(h^{-1}B)\,d\mu_{H}(h).$

In other words, the measure of every set is equal to the average of the
measures of its translates.

  6. (6)

If $\mu_{Y}$ is $H$-invariant, then there is a unique probability measure
$\eta$ on the space $\mathcal{E}$ of ergodic $H$-invariant measures on $Y$,
such that

$\mu_{Y}=\int_{\mathcal{E}}\xi\,d\eta(\xi).$

(See [31, Thm., p. 77], for example.) Measures in the support of $\eta$ are
called _ergodic components_ of $\mu_{Y}$.

  7. (7)

A function $f\colon\mathbb{R}\to\mathbb{R}$ is _Lipschitz_ if there is a
constant $C\in\mathbb{R}$, such that

$\text{$|f(x)-f(y)|\leq C\,|x-y|$ \quad for all $x,y\in\mathbb{R}$}.$

###  2D. Dynamical systems

######  Proposition 2.6 (cf. [17, Cor. 4.2 and Thm. 4.4, pp. 98–99]).

Assume $\mathbb{R}$ acts continuously on a compact, metrizable space $Y$. Then
there is at least one $\mathbb{R}$-invariant probability measure on $Y$ that
is ergodic.

######  Theorem 2.7 (Pointwise Ergodic Theorem [17, Thm. 2.30, p. 44]).

Assume a cyclic group $\langle g\rangle$ acts by homeomorphisms on a
metrizable space $Y$, and $\mu$ is an ergodic $\langle g\rangle$-invariant
probability measure on $Y$. Then, for any continuous function $f$ on $Y$ with
compact support, and $\mu$-a.e. $y\in Y$, we have

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}f(g^{i}y)=\int_{Y}f\,d\mu.$

######  Proposition 2.8.

Let $\Gamma$ be a cocompact, irreducible lattice in a (real or $p$-adic)
semisimple Lie group $G$, and let $P$ be a parabolic subgroup of $G$. Then:

  1. (1)

The $P$-invariant probability measure on $G/\Gamma$ is unique (namely, the
$G$-invariant measure $m_{G/\Gamma}$ is the only $P$-invariant probability
measure).

  2. (2)

$G/\Gamma$ has no closed, nonempty, proper, $P$-invariant subsets. In other
words, for all $y\in G/\Gamma$, the orbit $Py$ is dense in $G/\Gamma$.

###### Idea of proof.

Since $P$ is parabolic, we know there is an $\mathbb{R}$-split semisimple
element $a\in P$, such that

$P=\bigl{\\{}\,p\in G\mid\text{$\\{\,a^{n}pa^{-n}\mid n\leq 0\,\\}$ is
bounded}\,\bigr{\\}}.$

By the Moore Ergodicity Theorem [29, Exer. 11.2#11, p. 218], we also know that
$a$ is ergodic on $G/\Gamma$ (with respect to the $G$-invariant probability
measure $m_{G/\Gamma}$).

(2) Since $a$ is ergodic, almost every $a$-orbit is dense. Hence, we can
choose some $y^{\prime}\approx y$, such that $\\{\,a^{n}y^{\prime}\mid n\geq
0\,\\}$ is dense in $G/\Gamma$. Let

$U_{a}^{-}=\bigl{\\{}\,u\in G\mid a^{n}ua^{-n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\mathbf{1}\,\\}.$

Then $U_{a}^{-}P$ contains a neighbourhood of $\mathbf{1}$ in $G$, so we may
write $y^{\prime}=upy$ with $u\in U_{a}^{-}$ and $p\in P$.

For any $x\in G/\Gamma$, there is some large $n>0$, such that
$a^{n}y^{\prime}\approx x$. Then

$x\approx a^{n}y^{\prime}=a^{n}upy=a^{n}u^{-}a^{-n}\cdot
a^{n}py\approx\mathbf{1}\cdot a^{n}py\in Py.$

(1) Let $\mu$ be an ergodic $P$-invariant probability measure on $G/\Gamma$,
let $f$ be a continuous function on $G/\Gamma$, and let

$Y=\left\\{\,y\in G/\Gamma\mathrel{\left|\vphantom{\left\\{y\in
G/\Gamma\mid\frac{1}{n}\sum_{i=1}^{n}f(a^{i}y)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{G/\Gamma}f\,d\mu\right\\}}\right.}\frac{1}{n}\sum_{i=1}^{n}f(a^{i}y)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{G/\Gamma}f\,d\mu\,\right\\}.$

We have $\mu(Y)=1$ by the Pointwise Ergodic Theorem (2.7). Since $\mu$ is
$P$-invariant, this implies that for $\mu$-a.e. $u\in Y$ and $m_{P}$-a.e.
$p\in P$, we have $py\in Y$. Also, it is not difficult to see that $uy\in Y$
for every $u\in U_{a}^{-}$. This easily implies that $m_{G/\Gamma}(Y)>0$.
Hence, by the Pointwise Ergodic Theorem (2.7), there exists $y\in Y$, such
that

$\frac{1}{n}\sum_{i=1}^{n}f(a^{i}y)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{G/\Gamma}f\,dm_{G/\Gamma}.$

So we must have $\int_{G/\Gamma}f\,d\mu=\int_{G/\Gamma}f\,dm_{G/\Gamma}$. ∎

##  3\. Main ideas of the Deroin-Hurtado proof

In this Section, we will present many of the main ideas in the Deroin-Hurtado
paper [14]. (For readers interested in more details, references to specific
statements in [14] will usually be provided. In many cases, additional details
are also available in Section 4.) Our goal is to establish theorem 1.11. For
the reader’s convenience, we reproduce the statement of this result:

######  Theorem 1.11 (Deroin-Hurtado).

If $\Gamma$ is an irreducible lattice in a $p$-adic semisimple Lie group $G$,
and $\operatorname{rank}_{\mathbb{Q}_{p}}G\geq 2$, then $\Gamma$ has no
nontrivial action on the real line by orientation-preserving homeomorphisms.

###  3A. Assumptions and notation

######  Notation 3.2.

Throughout the remainder of this paper:

  * •

$p$ is a prime number,

  * •

$G$ is a $p$-adic semisimple Lie group, such that
$\operatorname{rank}_{\mathbb{Q}_{p}}G\geq 2$,

  * •

$\Gamma$ is an irreducible lattice in $G$,

  * •

$K$ is a compact, open subgroup of $G$,

  * •

$P$ is a minimal parabolic subgroup of $G$, and

  * •

$A$ is a maximal $\mathbb{Q}_{p}$-split torus in $P$.

The proof of theorem 1.11 is by contradiction, so we assume:

######  Assumption 3.3.

$\Gamma$ has a faithful action by orientation-preserving homeomorphisms of
$\mathbb{R}$.

######  Remark 3.4.

  1. (1)

The proof of corollary 1.3 shows that any nontrivial, orientation-preserving
action of $\Gamma$ on $\mathbb{R}$ will become faithful after modding out a
finite, normal subgroup of $G$. Therefore, even though theorem 1.11 refers to
a _nontrivial_ action, no loss of generality results from 3.3’s requirement
that the action is _faithful_.

  2. (2)

The reader will not miss out on any of the main ideas if they assume that
$G=\operatorname{SL}(3,\mathbb{Q}_{p})$ or that
$G=\operatorname{SL}(2,\mathbb{Q}_{p})\times\operatorname{SL}(2,\mathbb{Q}_{p})$.
Even these two special cases were not known before the work of Deroin and
Hurtado.

  3. (3)

If $G=\operatorname{SL}(3,\mathbb{Q}_{p})$, we may let

$K=\operatorname{SL}(3,\mathbb{Z}_{p}),\qquad P=\left[\begin{matrix}*&*&*\\\
0&*&*\\\ 0&0&*\end{matrix}\right],\qquad A=\left[\begin{matrix}*&0&0\\\
0&*&0\\\ 0&0&*\end{matrix}\right].$

  4. (4)

The same proof applies when $G$ is a finite product of $p$-adic semisimple Lie
groups, for various primes $p$.

######  Note 3.5.

By theorem 2.4(1), we know that $G/\Gamma$ is compact. On the other hand,
since $K$ is open, we also know that $K\backslash G$ is discrete. Therefore,
the double-coset space $K\backslash G/\Gamma$ is both compact and discrete. So
$K\backslash G/\Gamma$ must be finite.

######  Assumption 3.6.

To simplify the proof, we will assume that the finite set $K\backslash
G/\Gamma$ has only one element. This means:

(3.7) $\displaystyle G=K\,\Gamma.$

(See Section 5 for some comments on the more complicated situation where
$K\backslash G/\Gamma$ is a larger finite set.)

######  Remark 3.8.

The conclusion of Note 3.5 is a key reason why the $p$-adic case is much
easier than the real case. If $G$ is a (noncompact) real semisimple Lie group
and $K$ is a compact subgroup of $G$, then $K\backslash G/\Gamma$ is a highly
nontrivial manifold that plays a role in the proof of the Deroin-Hurtado
Theorem (1.2).

######  Note 3.9.

It is not difficult to see that the real line has no nontrivial, orientation-
preserving homeomorphisms of finite order. Hence, 3.3 implies that $\Gamma$
has no nontrivial elements of finite order [16, §1.4.1, p. 27]. Since every
subgroup of $\Gamma$ is discrete, and every discrete subgroup of a compact
group is finite, this implies $\Gamma\cap K$ is trivial. By combining this
with (3.7), we see that

(3.10) $\displaystyle\text{the natural map $(k,\gamma)\mapsto k\gamma$ is a
$K$-equivariant homeomorphism $K\times\Gamma\simeq G$}.$

Also, we can identify $G/\Gamma$ with $K$.

Recall that a Borel measure $\mu$ on a topological space $Y$ is a _probability
measure_ if $\mu(Y)=1$.

######  Notation 3.11.

Let

  1. (1)

$m_{K}$ be the Haar measure on $K$ (normalized to be a probability measure),
and

  2. (2)

$\mu_{G}$ be a nice probability measure on $G$ whose support is compact and
generates $G$. We assume [14, Defn. 3.6]:

    1. (a)

$\mu_{G}$ is absolutely continuous with respect to the Haar measure $m_{G}$,

    2. (b)

$\mu_{G}$ is _bi- $K$-invariant_, which means
$\mu_{G}(B)=\mu_{G}(kB)=\mu_{G}(Bk)$ for every Borel set $B$ and every $k\in
K$, and

    3. (c)

$\mu_{G}$ is _symmetric_ , which means $\mu_{G}(B^{-1})=\mu_{G}(B)$ for every
Borel set $B$, where $B^{-1}=\\{\,b^{-1}\mid b\in B\,\\}$.

######  Warning 3.12.

The measure $\mu_{G}$ is definitely _not_ the Haar measure on $G$ (because the
Haar measure on a noncompact group can never be a probability measure and
cannot have compact support).

######  Definition 3.13.

Define an atomic probability measure $\mu_{\Gamma}$ on $\Gamma$ by
$\mu_{\Gamma}\bigl{(}\\{\gamma\\}\bigr{)}=\mu_{G}(K\gamma)$. (Note that the
support of $\mu_{\Gamma}$ is finite, since the support of $\mu_{G}$ is
compact.) Since $\mu_{G}$ is $K$-invariant on the left (and the Haar measure
on $K$ is unique), this implies

$\mu_{G}=m_{K}\times\mu_{\Gamma},$

where we are using the identification $G\simeq K\times\Gamma$ (see Note 3.9).
We assume $\mu_{G}$ is chosen so that the support of $\mu_{\Gamma}$ generates
$\Gamma$.

######  Remark 3.14.

It is more difficult to construct the appropriate measure $\mu_{\Gamma}$ when
3.6 does not hold (see 5.4(5)), and it is usually not true that $\mu_{\Gamma}$
has finite support.

###  3B. The almost-periodic space $Z$ and the induced $G$-space $X$

The action of $\Gamma$ on $\mathbb{R}$ obviously extends to an action of
$\Gamma$ on the one-point compactification $\mathbb{R}\cup\\{\infty\\}$.
However, $\Gamma$ fixes the point $\infty$ in this action. The following
theorem shows that $\Gamma$ acts with no fixed points on a much more
sophisticated compactification of $\mathbb{R}$. (This is a general result
about finitely generated groups that act on $\mathbb{R}$; it does not use our
assumption that $\Gamma$ is a lattice.) Deroin and Hurtado [14, §1.3.5] call
this compactification the _almost-periodic space_.

######  Theorem 3.15 (see theorem 4.1, [13, Thm. 1.1], [14, Thm. 5.4], [16,
Cor. 4.2.11]).

There is a (nonempty) compact metrizable space $Z$, such that:

  1. (1)

$\Gamma$ acts on $Z$ (by homeomorphisms) with no global fixed point,

  2. (2)

$\mathbb{R}$ has a continuous action on $Z$ that is free,

  3. (3)

each $\mathbb{R}$-orbit is $\Gamma$-invariant, and

  4. (4)

the action of $\Gamma$ on each $\mathbb{R}$-orbit is by orientation-preserving
Lipschitz maps.

######  Notation 3.16.

We will use additive notation for the action of $\mathbb{R}$ on $Z$:

$\text{the image of a point $z\in Z$ by $t\in\mathbb{R}$ is denoted $z+t$}.$

This additive notation allows us to also define subtraction for points that
are in the same $\mathbb{R}$-orbit. Namely, for $z_{1},z_{2}\in Z$ and
$t\in\mathbb{R}$:

$\text{if $z_{1}=z_{2}+t$, then we let $z_{1}-z_{2}\coloneqq t$}.$

Since the $\mathbb{R}$-action is assumed to be free, the difference
$z_{1}-z_{2}$ is well-defined (when it exists, i.e., whenever $z_{1}$ and
$z_{2}$ are in the same $\mathbb{R}$-orbit).

We “induce” (or “suspend”) the $\Gamma$-action to obtain a $G$-action (on a
different space).

######  Definition 3.17 ([14, §5.3], [34, p. 75]).

Let

$\text{$X=\mathop{\mathrm{Ind}}\nolimits_{\Gamma}^{G}Z\coloneqq(G\times
Z)/\Gamma$, \quad where \quad$(h,z)*\gamma=(h\gamma,\gamma^{-1}z)$}.$

Then $G$ acts on this quotient space by

$g[(h,z)]={[(gh,z)]},$

where we use $[(h,z)]$ to denote the image of $(h,z)$ under the natural
quotient map from $G\times Z$ to $(G\times Z)/\Gamma$.

######  Note 3.18.

  1. (1)

For $[(h,z)]\in X$ and $\gamma\in\Gamma$, we have

$[(h\gamma,z)]=[(h\gamma\cdot\gamma^{-1},\gamma z)]=[(h,\gamma z)].$

  2. (2)

The natural map $(k,z)\mapsto[(k,z)]$ is a $K$-equivariant homeomorphism
$K\times Z\simeq X$ (by (3.10)). This implies that $X$ is compact (because $K$
and $Z$ are compact).

  3. (3)

The function $[(h,z)]\mapsto h\Gamma$ is a well-defined $G$-equivariant map
$X\to G/\Gamma$. It gives $X$ the structure of a fibre bundle over $G/\Gamma$
with fibres homeomorphic to $Z$.

  4. (4)

For $g\in G$ and $k\in K$, there exist $k^{\prime}\in K$ and
$\gamma=\gamma(g,k)\in\Gamma$, such that $gk=k^{\prime}\gamma$ (by (3.7)).
Then, for all $z\in Z$, we have

$g[(k,z)]=[(gk,z)]=[(k^{\prime}\gamma,z)]=[(k^{\prime},\gamma z)].$

Hence, $g$ acts on the entire fibre $[\\{k\\}\times Z]$ via the element
$\gamma(g,k)$.

We have defined $z_{1}-z_{2}$ when $z_{1}$ and $z_{2}$ are points in $Z$ that
are in the same $\mathbb{R}$-orbit. We extend this to points in $X$:

######  Notation 3.19.

The action of $\mathbb{R}$ on $Z$ can be extended to a free action of
$\mathbb{R}$ on $X$ via the identification $X\simeq K\times Z$:
$[(k,z)]+t=[(k,z+t)]$. This yields a definition of $x-y$ for all $x,y\in X$,
such that $x$ and $y$ are in the same $\mathbb{R}$-orbit.

######  Warning 3.20.

For $g\in G$, it is usually _not_ true that $[(g,z)]+t=[(g,z+t)]$. To see
this, write $g=k\gamma$, with $k\in K$ and $\gamma\in\Gamma$ (by using (3.7)).
Then:

$[(g,z)]+t=[(k\gamma,z)]+t=[(k,\gamma z)]+t=[(k,\gamma
z+t)]=\bigl{[}\bigl{(}k\gamma,\gamma^{-1}(\gamma
z+t)\bigr{)}\bigr{]}=\bigl{[}\bigl{(}g,\gamma^{-1}(\gamma
z+t)\bigr{)}\bigr{]}.$

This will rarely be equal to $[(g,z+t)]$, because $\Gamma$ is not assumed to
commute with the $\mathbb{R}$-action.

However, we do have the following weaker property, which follows from Note
3.18(4) (and theorem 3.15(4)):

######  Lemma 3.21 (cf. [14, §5.3.1]).

The action of $G$ on $X$ respects the $\mathbb{R}$-orbits, and the restriction
of each element of $G$ to an $\mathbb{R}$-orbit is Lipschitz (with a Lipschitz
constant that does not depend on the choice of the orbit). More precisely, for
all $g\in G$, there exists $C=C(g)\in\mathbb{R}^{+}$, such that, for all $x\in
X$ and $t\in\mathbb{R}$, we have $g(x+t)\in gx+\mathbb{R}$ and

$|g(x+t)-gx|\leq C\,|t|.$

###  3C. The probability measures $\mu_{Z}$, $\mu_{X}$, and $\mu_{X}^{P}$

######  Notation 3.22.

Let

  * •

$\mu_{Z}$ be an ergodic $\mathbb{R}$-invariant probability measure on $Z$ (see
proposition 2.6 for the existence of such a measure), and

  * •

$\mu_{X}=m_{K}\times\mu_{Z}$ (under the identification $X\simeq K\times Z$),
so $\mu_{X}$ is a $K$-invariant probability measure on $X$.

######  Remark 3.23.

See 5.7 for a definition of $\mu_{X}$ when $X\not\simeq K\times Z$.

Lipschitz functions on $\mathbb{R}$ are differentiable almost everywhere (and
the derivative is bounded). Thus, we see from lemma 3.21 that each element of
$G$ has a derivative along the $\mathbb{R}$-orbits (a.e.):

######  Definition 3.24 ([14, §5.2.2]).

For $g\in G$ and $\mu_{X}$-a.e. $x\in X$, we let

$D^{\text{leaf}}_{g}(x)=\lim_{t\to 0}\frac{g(x+t)-x}{t}.$

We can similarly define $D^{\text{leaf}}_{\gamma}(z)$ for $\gamma\in\Gamma$
and $\mu_{Z}$-a.e. $z\in Z$.

By definition, the measure $\mu_{Z}$ is $\mathbb{R}$-invariant, so moving
along the $\mathbb{R}$-orbits preserves the measure. Since the $\Gamma$-action
preserves these $\mathbb{R}$-orbits, any non-invariance of $\mu_{Z}$ under
this action comes from distortion within the $\mathbb{R}$-orbits. Also, for
any $\gamma\in\Gamma$, the action on each $\mathbb{R}$-orbit is Lipschitz, so
the measure class of Lebesgue measure is preserved. This implies that
$\gamma_{*}\mu_{Z}$ is in the same measure class as $\mu_{Z}$. Furthermore:

######  Lemma 3.25 ([14, Lems. 5.5 and 5.10(4)]).

For every $\gamma\in\Gamma$, the Radon-Nikodym derivative
$d\gamma_{*}\mu_{Z}/d\mu_{Z}$ is equal to $D^{\text{leaf}}_{\gamma^{-1}}$
(a.e.). Similarly, for all $g\in G$, we have

$\frac{dg_{*}\mu_{X}}{d\mu_{X}}=D^{\text{leaf}}_{g^{-1}}\quad\text{{(}a.e.{)}}.$

The almost-periodic space $Z$ can be constructed so that the following holds.
(See 2.5(5) for the definition of “stationary.”)

######  Proposition 3.26 (see proposition 4.7, [14, Lem. 5.6]).

The measure $\mu_{Z}$ is $\mu_{\Gamma}$-stationary.

This has the following crucial consequence.

######  Corollary 3.27 (see corollary 4.8,[14, Prop. 5.11]).

$\mu_{X}$ is $\mu_{G}$-stationary.

Recall that $P$ is a minimal parabolic subgroup (see 3.2 and perhaps also
3.4(3)). It is known that all $\mu_{G}$-stationary measures come from
$P$-invariant measures, by averaging the $K$-translates:

######  Theorem 3.28 (Furstenburg [21, Thm. 2.1]).

There is a unique $P$-invariant probability measure $\mu_{X}^{P}$ on $X$, such
that

$\mu_{X}=\int_{K}k_{*}\,\mu_{X}^{P}\,dm_{K}(k).$

The remainder of the proof of the Deroin-Hurtado Theorem will show that the
stationary measure $\mu_{X}$ is $G$-invariant (or, equivalently, that
$\mu_{X}=\mu_{X}^{P}$). This will contradict the following result, and thereby
complete the proof of theorem 1.11, by establishing that 3.3 must be false,
because it leads to a contradiction.

######  Proposition 3.29 ([14, Prop. 5.12]).

We have:

  1. (1)

the stationary measure $\mu_{Z}$ is not $\Gamma$-invariant,

  2. (2)

the stationary measure $\mu_{X}$ is not $G$-invariant,

  3. (3)

the $P$-invariant measure $\mu_{X}^{P}$ is not $K$-invariant (so it is not
$G$-invariant), and

  4. (4)

$\mu_{X}\neq\mu_{X}^{P}$.

###### Proof.

(1) Suppose $\mu_{Z}$ is $\Gamma$-invariant. Then for all $\gamma\in\Gamma$ we
have

$\frac{d\gamma_{*}\mu_{Z}}{d\mu_{Z}}=\frac{d\mu_{Z}}{d\mu_{Z}}=1\qquad\text{a.e.},$

so lemma 3.25 tells us $D^{\text{leaf}}_{\gamma}=1$ a.e. This implies that
$\Gamma$ acts by translations on each $\mathbb{R}$-orbit. However, the group
of translations is abelian, and the abelianization of $\Gamma$ is finite (see
theorem 2.4(2)). This contradicts 3.3, which states that the action of
$\Gamma$ is faithful.

(2) It can be shown that if $\mu_{X}$ is $G$-invariant, then $\mu_{Z}$ is
$\Gamma$-invariant, which contradicts (1). Indeed, it is a general property of
induced actions that $\mu_{Z}$ is $\Gamma$-invariant if and only if $\mu_{X}$
is $G$-invariant, but see lemma 4.10 for a proof of the direction we need,
that is specialized to our situation.

(4) Suppose $\mu_{X}=\mu_{X}^{P}$. Since $\mu_{X}^{P}$ is $P$-invariant, this
assumption implies that $\mu_{X}$ is $P$-invariant. However, $\mu_{X}$ is also
known to be $K$-invariant (see 3.11(2)). These two subgroups generate all of
$G$, so we conclude that $\mu_{X}$ is $G$-invariant. This contradicts (2).

(3) Suppose $\mu_{X}^{P}$ is $K$-invariant. This means
$k_{*}\mu_{X}^{P}=\mu_{X}^{P}$ for all $k\in K$, so

$\mu_{X}=\int_{K}k_{*}\mu_{X}^{P}\,dm_{K}(k)=\int_{K}\mu_{X}^{P}\,dm_{K}(k)=\mu_{X}^{P}.$

This contradicts (4). ∎

###  3D. Contraction on $\mathbb{R}$-orbits and the Key Proposition

######  Definition 3.30 ([14, §7.1, p. 43]).

We define $\chi_{P}\colon A\to\mathbb{R}$ by

$\chi_{P}(a)=\int_{X}\log\,D^{\text{leaf}}_{\\!a}(x)\,d\mu_{X}^{P}(x).$

(There is a technical issue here: we know that $D^{\text{leaf}}_{\\!a}(x)$
exists for $\mu_{X}$-a.e. $x\in X$, but the definition assumes that it exists
for $\mu_{X}^{P}$-a.e. $x\in X$. See lemma 4.11 for a proof that the
derivative exists $\mu_{X}^{P}$-a.e.)

######  Lemma 3.31 (see lemma 4.12, [14, §7.1, p. 43]).

It follows from the Chain Rule that $\chi_{P}$ is a homomorphism.

Notice that if $\chi_{P}(a)<0$, then there is constant $C>1$, such that if $x$
and $y$ are two nearby points in the same $\mathbb{R}$-orbit, then, on
average,

$|ax-ay|<\frac{|x-y|}{C}.$

Applying $a$ repeatedly should decrease the distance by an exponential factor,
so one would expect the distance to tend to $0$ in the limit. This is indeed
true (if we start with two points that are not too far apart):

######  Lemma 3.32 (local contraction on $\mathbb{R}$-orbits, see lemma 4.14,
[14, Lem. 7.1]).

If $a\in A$, such that $\chi_{P}(a)<0$, then for $\mu_{X}^{P}$-a.e. $x\in X$,
there exists $\epsilon=\epsilon(x)>0$, such that

$\text{$a^{n}(x+\epsilon)\ -\ a^{n}(x-\epsilon)\ \to\ 0$ \quad as
$n\to\infty$}.$

With additional work, it can be shown that $a$ contracts entire
$\mathbb{R}$-orbits, not just $\epsilon$-intervals. The resulting contraction
property will be the key to the proof of theorem 1.11.

######  Key Proposition 3.33 (global contraction on $\mathbb{R}$-orbits, cf.
[14, Prop. 7.7]).

For all $a\in A$, if $\chi_{P}(a)<0$, then

$\text{for $\mu_{X}^{P}$-a.e.\ $x\in X$, \ for all $y\in x+\mathbb{R}$,
\quad$a^{n}\,x-a^{n}\,y\ \stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\ 0$}.$

###### Proof.

See Section 3G for an outline. More details can be found in Section 4E. ∎

###  3E. Invariance under the centralizer

######  Notation 3.34.

For $a\in A$, let

$U_{a}^{+}=\left\\{\,u\in G\mathrel{\left|\vphantom{\left\\{u\in
G\mid\begin{matrix}a^{n}ua^{-n}\to 1\\\\[-2.0pt] \text{as
$n\to-\infty$}\end{matrix}\right\\}}\right.}\begin{matrix}a^{n}ua^{-n}\to
1\\\\[-2.0pt] \text{as
$n\to-\infty$}\end{matrix}\,\right\\}\qquad\begin{pmatrix}\text{This is the
\emph{expanding horospherical}}\\\ \text{\emph{subgroup} corresponding
to~{}$a$.}\end{pmatrix}$

Proposition 3.33 has the following consequence, which will be used to show (by
a bootstrapping argument) that $\mu_{X}^{P}$ is $G$-invariant. This
contradicts proposition 3.29(3).

######  Corollary 3.35 (cf. [14, Prop. 8.3]).

If $a\in A$, such that $\chi_{P}(a)<0$ and $U_{a}^{+}\subseteq P$, then
$\mu_{X}^{P}$ is $C_{G}(a)$-invariant.

###### Idea of proof.

Let $c\in C_{G}(A)$. We wish to show $c_{*}\mu_{X}^{P}=\mu_{X}^{P}$.

The measure $\mu_{X}^{P}$ might not be ergodic for the action of $a$, so we
should work with ergodic components of the measure, but, for simplicity, we
ignore this technical issue. (See 4.16 for a comment on this.) Then, by the
Pointwise Ergodic Theorem (2.7), there exists $x\in X$, such that, for all
$f\in C(X)$, we have

(3.36)
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}f(a^{i}\,x)=\int_{X}f\,d\mu_{X}^{P}.$

We say that $x$ is a “Birkhoff-generic point” for $\mu_{X}^{P}$ (w.r.t. $a$),
and we call the average on the left-hand side a “Birkhoff average.”
Furthermore, since (3.36) holds for $\mu_{X}^{P}$-a.e. $x\in X$, we see from
proposition 3.33 that we may assume

(3.37) $\displaystyle\text{$a^{n}\,x\ -\ a^{n}\,y\quad\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\quad 0$ \quad for all $y\in x+\mathbb{R}$}.$

Since $\mu_{X}^{P}$ is $P$-invariant, the measure $c_{*}\mu_{X}^{P}$ is
${}^{c}\\!P$-invariant, where ${}^{c}\\!P=cPc^{-1}$ (a conjugate of $P$).
Define $\pi\colon X\to G/\Gamma$ by $\pi\bigl{(}[(h,z)]\bigr{)}=h\Gamma$, so
$\pi$ is a well-defined $G$-equivariant map (see Note 3.18(3)). Then
$\pi_{*}(c_{*}\mu_{X}^{P})$ is a ${}^{c}\\!P$-invariant probability measure on
$G/\Gamma$, so we see from proposition 2.8(2) that its support is all of
$G/\Gamma$. Since $\pi(cx)$ is Birkhoff generic for this measure (w.r.t. $a$),
we conclude that the $\langle a\rangle$-orbit of $\pi(cx)$ is dense in
$G/\Gamma$. Hence, we may choose $i\in\mathbb{Z}$, such that $\pi(a^{i}cx)$ is
very close to $\pi(x)$.

Now, if we let

$P_{a}^{-}=\left\\{\,g\in G\mathrel{\left|\vphantom{\left\\{g\in
G\mid\begin{matrix}\text{$a^{n}ga^{-n}$ is bounded}\\\\[-2.0pt] \text{for all
$n\geq 0$}\end{matrix}\right\\}}\right.}\begin{matrix}\text{$a^{n}ga^{-n}$ is
bounded}\\\\[-2.0pt] \text{for all $n\geq 0$}\end{matrix}\,\right\\},$

then the Lie algebra $\mathfrak{g}$ is the direct sum of the Lie algebra of
$P_{a}^{-}$ and the Lie algebra of $U_{a}^{+}$. Hence, the conclusion of the
preceding paragraph implies there exist small $g_{-}\in P_{a}^{-}$ and
$u^{+}\in U_{a}^{+}$, such that

$\displaystyle\pi(a^{i}\,cx)=g^{-}u^{+}\,\pi(x)=\pi(g^{-}u^{+}x).$

Writing $x=[(h,z)]$ with $h\in G$ and $z\in Z$, this means there exists
$\gamma\in\Gamma$, such that

$\displaystyle a^{i}\,ch\gamma=g^{-}u^{+}h,$

so (recalling that the $\mathbb{R}$-orbits in $Z$ are $\Gamma$-invariant) we
have

(3.38) $\displaystyle x$
$\displaystyle=c^{-1}a^{-i}a^{i}cx=c^{-1}a^{-i}[(a^{i}ch,z)]=c^{-1}a^{-i}[(a^{i}ch\gamma,\gamma^{-1}z)]$
$\displaystyle=c^{-1}a^{-i}[(g^{-}u^{+}h,\gamma^{-1}z)]\in
c^{-1}a^{-i}[(g^{-}u^{+}h,z)]+\mathbb{R}=c^{-1}a^{-i}g^{-}u^{+}x+\mathbb{R}.$

Let $x_{c}=a^{i}cx$ and $x_{0}=u^{+}x$. We see from (3.38) and (3.37) that

$a^{n}x-a^{n}(c^{-1}a^{-i}g^{-}x_{0})\to 0.$

Then, since $a^{i}c$ commutes with $a^{n}$, and is uniformly Lipschitz on the
$\mathbb{R}$-orbits (see lemma 3.21), we can multiply by $a^{i}c$ to conclude
that

(3.39) $\displaystyle a^{n}x_{c}-a^{n}(g^{-}x_{0})\to 0.$

However:

  * •

$d(a^{n}x_{0},a^{n}x_{c})\leq
d(a^{n}x_{0},a^{n}g^{-}x_{0})+d(a^{n}x_{c},a^{n}g^{-}x_{0})$.

  * •

Since $g_{-}\in P^{-}_{a}$, we know that $d(a^{n}x_{0},a^{n}g^{-}x_{0})$ stays
bounded at about the size of $g^{-}$, so it is uniformly small.

  * •

$d(a^{n}x_{c},a^{n}g^{-}x_{0})\leq
d_{\text{leaf}}(a^{n}x_{c},a^{n}g^{-}x_{0})\to 0$ (by (3.39)).

Therefore $a^{n}x_{0}$ is close to $a^{n}x_{c}$ for all $n$, so

$x_{0}$ and $x_{c}$ have almost the same Birkhoff averages.

Now, we have another technical issue. Since $x_{0}=u^{+}\,x$ and $x$ is
Birkhoff generic for $\mu_{X}^{P}$, we would like to be able to say that
$x_{0}$ is Birkhoff generic for $u^{+}_{*}\mu_{X}^{P}=\mu_{X}^{P}$. However,
$u^{+}$ does not centralize $a$ (far from it!), so this is not obvious. It is
true for most values of $u^{+}$ though: since $\mu_{X}^{P}$ is
$U_{a}^{+}$-invariant, one can show (for a.e. $x$) that the statement is true
for $m_{U_{a}^{+}}$-a.e. choice of $u^{+}$ in $U_{a}^{+}$. By perturbing the
generic point $x$, we can change the value of $u^{+}$ to get it into the good
set (see [14, p. 50]). Therefore, we can assume that $x_{0}$ is indeed
Birkhoff generic for $\mu_{X}^{P}$.

On the other hand, $x_{c}$ is Birkhoff generic for $c_{*}\mu_{X}^{P}$.
Combining this with the conclusions of the preceding two paragraphs, we
conclude that $\mu_{X}^{P}$ is $\epsilon$-close to $c_{*}\mu_{X}^{P}$ (for
every $\epsilon>0$). By letting $\epsilon$ tend to $0$, we conclude that
$\mu_{X}^{P}=c_{*}\mu_{X}^{P}$. ∎

######  Remark 3.40.

The assumption in corollary 3.35 that $U_{a}^{+}\subseteq P$ simply means that
$a$ is in a particular closed Weyl chamber in $A$.

######  Note 3.41.

Corollary 3.35 is not vacuous: there is an element $a$ of $A$, such that
$\chi_{P}(a)<0$ and $U_{a}^{+}\subseteq P$ (see proposition 4.23).

###  3F. Completion of the proof by propagating invariance

We can now complete the proof of theorem 1.11, by obtaining the following
contradiction to proposition 3.29(3). (This is where we use the assumption
that $\operatorname{rank}_{\mathbb{Q}_{p}}G\geq 2$. If $G$ has rank one, then
$C_{G}(a)$ is equal to $A$, so corollary 3.35 cannot show that $\mu_{X}^{P}$
is invariant under anything more than $A$.) The contradiction implies that our
assumption (3.3) that there is a nontrivial, orientation-preserving action of
$\Gamma$ on $\mathbb{R}$ must be false. So it does indeed complete the proof
of theorem 1.11.

######  Corollary 3.42 (“Propagating Invariance” [14, Thm. 8.1 (and §9)]).

The measure $\mu_{X}^{P}$ is $G$-invariant.

###### Idea of proof.

For concreteness, let us assume $G=\operatorname{SL}(3,\mathbb{Q}_{p})$. (See
corollary 4.17 for an explanation of the general case.) Let $\mathcal{W}_{P}$
be the closed Weyl chamber corresponding to $P$, so $U^{+}_{a}\subseteq P$ for
all $a\in\mathcal{W}_{P}$. We may assume $P$ is upper triangular, as suggested
in 3.4(3). Let

$\mathcal{H}=\\{\,a\in A\mid\chi_{P}(a)<0\,\\}.$

This is an open halfspace in $A$.

Case 1. Assume $\chi_{P}(a)<0$ for all nontrivial $a\in\mathcal{W}_{P}$.  Let
$a_{1}$ and $a_{2}$ be on the two walls of $\mathcal{W}_{P}$, as pictured in
Fig. 1(1).

$\begin{matrix}\includegraphics{pdf/propagate-easy.pdf}&\hskip
14.22636pt&\includegraphics{pdf/propagate-interior.pdf}&\hskip
14.22636pt&\includegraphics{pdf/propagate-bdry.pdf}\\\
\hbox{(a)}&&\hbox{(b)}&&\hbox{(c)}\end{matrix}$
Figure 1. The halfspace $\mathcal{H}$ is unshaded.

Specifically, we may let

$a_{1}=\left[\begin{matrix}1/p&&\\\ &1/p&\\\
&&p^{2}\end{matrix}\right]\qquad\text{and}\qquad
a_{2}=\left[\begin{matrix}1/p^{2}\\\ &p\\\ &&p\end{matrix}\right].$

By the assumption of this Section, we have $\chi_{P}(a_{1})<0$ and
$\chi_{P}(a_{2})<0$. Also, we know $U^{+}_{a_{1}}\subseteq P$ and
$U^{+}_{a_{2}}\subseteq P$, because $a_{1},a_{2}\in\mathcal{W}_{P}$. Indeed,

$U^{+}_{a_{1}}=\left[\begin{smallmatrix}1&&*\\\ &1&*\\\
&&1\end{smallmatrix}\right]\subset P\qquad\text{and}\qquad
U^{+}_{a_{1}}=\left[\begin{smallmatrix}1&*&*\\\ &1&\\\
&&1\end{smallmatrix}\right]\subset P.$

So we see from corollary 3.35 that $\mu_{X}^{P}$ is invariant under

$C_{G}(a_{1})=\left[\begin{smallmatrix}*&*\\\ *&*\\\
&&1\end{smallmatrix}\right]\qquad\text{and}\qquad
C_{G}(a_{2})=\left[\begin{smallmatrix}1&\\\ &*&*\\\
&*&*\end{smallmatrix}\right].$

These two centralizers generate $G$, so we conclude that $\mu_{X}^{P}$ is
invariant under all of $G$.

Case 2. The remaining case.  We know that:

  * •

$\mathcal{H}$ contains some element of $\mathcal{W}_{P}$ (by Note 3.41), and

  * •

$\mathcal{H}$ does not contain every nontrivial element of $\mathcal{W}_{P}$
(by the assumption of this case).

So either the boundary of $\mathcal{H}$ passes through the interior of
$\mathcal{W}_{P}$ (as in Fig. 1(1)), or the boundary of $\mathcal{H}$ contains
one the boundary rays of $\mathcal{W}_{P}$, and the rest of $\mathcal{W}_{P}$
is in $\mathcal{H}$ (as in Fig. 1(1)). In either case, one of the two boundary
rays of $\mathcal{W}_{P}$ is in $\mathcal{H}$.

Let $\mathcal{W}_{Q}$ be the Weyl chamber on the other side of this boundary
ray, with corresponding minimal parabolic subgroup $Q$, and let $a$ be a
nontrivial point on the ray. Then $a\in\mathcal{H}$, which means
$\chi_{P}(a)<0$. (Also, $a\in\mathcal{W}_{P}$, so $U_{a}^{+}\subseteq P$.) So
we see from corollary 3.35 that $\mu_{X}^{P}$ is $C_{G}(a)$-invariant.
However, we also know by the definition of $\mu_{X}^{P}$ that it is
$P$-invariant. So $\mu_{X}^{P}$ is invariant under the subgroup $\langle
C_{G}(a),P\rangle$. This is a parabolic subgroup of $G$ (because every
subgroup that contains a parabolic subgroup is also a parabolic subgroup). In
fact, since $a$ is on the boundary between $\mathcal{W}_{P}$ and
$\mathcal{W}_{Q}$, this subgroup is equal to $\langle P,Q\rangle$, and
therefore contains $Q$. So $\mu_{X}^{P}$ is $Q$-invariant.

By the definition of $\mu_{X}^{P}$, we also have
$\mu_{X}=\int_{K}k_{*}\mu_{X}^{P}\,dm_{K}(k)$ (see theorem 3.28). Hence, the
uniqueness in theorem 3.28 implies that $\mu_{X}^{Q}=\mu_{X}^{P}$. Then we see
from 3.30 that $\chi_{Q}=\chi_{P}$. Since it is clear from Fig. 1(1,1) that
every nontrivial element of $\mathcal{W}_{Q}$ is in $\mathcal{H}$, we have
$\chi_{P}(a)<0$ for all nontrivial $a\in\mathcal{W}_{Q}$. Then, since
$\chi_{Q}=\chi_{P}$, we conclude that the same inequality holds with
$\chi_{Q}$ in the place of $\chi_{P}$. Hence, Section 3F applies with the
parabolic subgroup $Q$ in the place of $P$. ∎

###  3G. Proof of proposition 3.33

Let $a\in A$, such that $\chi_{P}(a)<0$. We know from lemma 3.32 that the
action of $a$ contracts small intervals in $\mathbb{R}$-orbits (a.e.). In
order to establish proposition 3.33, we need to show that the contraction
happens on the entire $\mathbb{R}$-orbit. To this end, we will employ
contraction by generic elements of $G$, rather than only by elements of $A$.

For $g\in G$, we have $\mu_{X}(gX)=\mu_{X}(X)=1$, so (by lemma 3.25)
$\int_{X}D^{\text{leaf}}_{g}\,d\mu_{X}=1$. As $\mu_{X}$ is not $G$-invariant
(see proposition 3.29(2)), we also know that $D^{\text{leaf}}_{g}$ is not
identically equal to $1$ for all $g$, so, since $\log$ is a concave function,
we conclude from Jensen’s Inequality that

(3.43) $\displaystyle\int_{G}\int_{X}\log
D^{\text{leaf}}_{g}(x)\,d\mu_{X}(x)\,d\mu_{G}(g)<0.$

This implies that (on average) applying a random element of $G$ contracts
distances that are small. Much like what we saw in lemma 3.32 for $a^{n}$,
this implies that repeated application of random elements of $G$ will contract
small intervals in the $\mathbb{R}$-orbits. However, we also have the
following global contraction property of the $\Gamma$-action on each
$\mathbb{R}$-orbit:

######  Proposition 3.44 (see proposition 4.19, [14, Prop. 6.3]).

For every action of $\Gamma$ on $\mathbb{R}$ without fixed points, there is a
compact subset $I$ of $\mathbb{R}$, such that for every $\epsilon>0$ and all
$s,t\in\mathbb{R}$, there exists $\gamma\in\Gamma$, such that

$\text{$|\gamma s-\gamma t|<\epsilon$ \ and \ $\\{\gamma s,\gamma t\\}\subset
I$}.$

By using this, it is possible to prove that repeated application of random
elements of $G$ yields contraction on entire $\mathbb{R}$-orbits, rather than
only locally, and that the contraction is exponentially fast:

######  Proposition 3.45 (see proposition 4.20, [14, Props. 6.4 and 6.8]).

There is a constant $\delta<1$, such that, for $\mu_{X}$-a.e. $x\in X$, for
all $y\in\mathbb{R}x$, and for $\mu_{G}^{\infty}$-a.e. sequence
$(g_{i})_{i=1}^{\infty}$ of elements of $G$, we have

$\text{$|g_{n}g_{n-1}\cdots g_{1}x\ -\ g_{n}g_{n-1}\cdots g_{1}y|<\delta^{n}$
\quad for all sufficiently large~{}$n$}.$

We have been proving the $p$-adic case in this paper, but the next part of the
argument is easier to explain for real groups, so we will switch to that
setting for the remainder of this Section:

Assume, contrary to 3.2, that $G$ is a connected real semisimple Lie group.

The group $G$ acts by isometries on its associated symmetric space
$K\backslash G$, which is a Riemannian manifold of nonpositive sectional
curvature. (The $p$-adic case uses a Bruhat-Tits building, instead of a
symmetric space.) Applying a random walk in $G$ to a point in $K\backslash G$
yields a sequence of points, and it can be shown that the negative curvature
of $K\backslash G$ implies that this sequence almost surely moves along a
geodesic, up to a sublinear error:

######  Theorem 3.46 (Karlsson-Margulis [24, Thm. 2.1]).

For $\mu_{G}^{\infty}$-a.e. sequence $(g_{i})_{i=1}^{\infty}$ of elements of
$G$, there is a geodesic $\mathsf{a}_{t}$ in $K\backslash G$, with
$\mathsf{a}_{0}=K$, such that

$\frac{d\bigl{(}Kg_{n}g_{n-1}\cdots g_{1},\
\mathsf{a}_{n}\bigr{)}}{n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

######  Definition 3.47.

Fix a nonempty, open subset $\mathcal{O}$ of $G$. Then $\mathcal{O}$ generates
$G$, so we can define the corresponding _word length_ : for $g\in G$, we let

$\ell_{O}(g)=\min\\{\,r\in\mathbb{N}\mid g\in\mathcal{O}^{r}\,\\},$

where $\mathcal{O}^{r}=\\{\,h_{1}h_{2}\cdots
h_{r}\mid\text{$h_{i}\in\mathcal{O}$ or $h_{i}^{-1}\in\mathcal{O}$}\,\\}$.

Geodesics in $K\backslash G$ come from one-parameter subgroups of
($K$-conjugates of) $A$, and every element of $A$ is $K$-conjugate to an
element of $\mathcal{W}_{P}$, so theorem 3.46 can be restated as follows:

######  Corollary 3.48 ([14, Thm. 3.18]).

For $\mu_{G}^{\infty}$-a.e. sequence $(g_{i})_{i=1}^{\infty}$ of elements of
$G$, there exist $k\in K$ and
$\widehat{a_{P}}\in\mathcal{W}_{P}\smallsetminus\\{\mathbf{1}\\}$, such that

$\frac{\ell_{O}(g_{n}g_{n-1}\cdots
g_{1}\,k\,\widehat{a_{P}}^{-n})}{n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

Furthermore, as $(g_{i})_{i=1}^{\infty}$ varies, the element $k$ is uniformly
distributed in $K$ (with respect to the Haar measure $m_{K}$).

We also have the following important technical result:

######  Proposition 3.49 (Benoist-Quint [3, Thm. 10.9(b), p. 158]).

The element $\widehat{a_{P}}$ is a well-defined element of $\mathcal{W}_{P}$
that is independent of $(g_{i})_{i=1}^{\infty}$.

Combining these results has the following consequence:

######  Proposition 3.50 ([14, Prop. 7.6]).

For $\mu_{G}^{\infty}$-a.e. sequence $(g_{i})_{i=1}^{\infty}$ of elements of
$G$, $\mu_{X}^{P}$-a.e. $x\in X$ and every $y\in x+\mathbb{R}$, we have

$\widehat{a_{P}}^{n}\,x\ -\ \widehat{a_{P}}^{n}\,y\ \stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\ 0.$

,

###### Idea of proof.

We will ignore some technical issues, including the difference between
$\mu_{X}$ and $\mu_{X}^{P}$. (4.21 explains how to get from $\mu_{X}$ to
$\mu_{X}^{P}$, by using the fact that $U^{+}_{a}$ is contained in $P$, and
therefore preserves $\mu_{X}^{P}$.)

Since $\mu_{X}$ is $K$-invariant, we see that, for $m_{K}$-a.e. $k\in K$, the
conclusion of proposition 3.45 holds with $kx$ and $ky$ in the place of $x$
and $y$. Then the uniform distribution in the conclusion of corollary 3.48
implies that we can choose $k$ and $(g_{i})_{i=1}^{\infty}$ so that the
conclusions of proposition 3.45 and corollary 3.48 hold simultaneously. That
is,

$|g_{n}g_{n-1}\cdots g_{1}kx\ -\ g_{n}g_{n-1}\cdots
g_{1}ky|\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0$
exponentially fast,

and the difference between $g_{n}g_{n-1}\cdots g_{1}\,k$ and
$\widehat{a_{P}}^{n}$ is the product of a sublinear number of elements of
$\mathcal{O}$. Since elements of $\mathcal{O}$ are uniformly Lipschitz, we
conclude that $|\widehat{a_{P}}^{n}x\ -\ \widehat{a_{P}}^{n}y|$ also tends to
$0$ (exponentially fast). ∎

We have established that the single element $\widehat{a_{P}}$ has global
contraction. We can use this to prove that the elements in a halfspace of $A$
also contract entire $\mathbb{R}$-orbits. In other words, we now prove
proposition 3.33.

###### Proof of proposition 3.33.

Fix $a\in A$. For each $x\in X$, let

$f(x)=\sup\\{\,t\in\mathbb{R}^{\geq 0}\mid a^{n}(x+t)-a^{n}x\
\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\ 0\,\\}.$

For any $n\in\mathbb{Z}^{+}$, the element $\widehat{a_{p}}^{n}$ commutes with
$a$ (and is uniformly Lipschitz on the $\mathbb{R}$-orbits by lemma 3.21), so
we have

$\displaystyle\widehat{a_{p}}^{n}\bigl{(}x+f(x)\bigr{)}=\widehat{a_{p}}^{n}x+f(\widehat{a_{p}}^{n}\,x).$

If $f(x)<\infty$, this implies

$\displaystyle
f(\widehat{a_{p}}^{n}\,x)\quad=\quad\widehat{a_{p}}^{n}\bigl{(}x+f(x)\bigr{)}\
-\ \widehat{a_{p}}^{n}x\quad\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\quad 0$ $\displaystyle\text{(by
\lx@cref{creftype~refnum}{ExistsContractingOverview})}.$

We can now conclude from the Pointwise Ergodic Theorem (2.7) that
$\int_{X}f\,d\mu_{X}^{P}=0$, so $f(x)=0$ $\mu_{X}^{P}$-a.e. However, if
$\chi_{P}(a)<0$, then we see from lemma 3.32 that $f(x)>0$ for
$\mu_{X}^{P}$-a.e. $x$. This is a contradiction, and completes the proof of
proposition 3.33. ∎

##  4\. Some additional explanations

For the interested reader, we provide some arguments that were omitted from
Section 3.

###  4A. Construction of the almost-periodic space $Z$

######  Theorem 4.1 (proof of theorem 3.15, [13, Thm. 1.1], [14, Thm. 5.4]).

There is a (nonempty) compact metrizable space $Z$, such that:

  1. (1)

$\Gamma$ acts on $Z$ (by homeomorphisms) with no global fixed point,

  2. (2)

$\mathbb{R}$ has a continuous action on $Z$ that is free,

  3. (3)

each $\mathbb{R}$-orbit is $\Gamma$-invariant, and

  4. (4)

the action of $\Gamma$ on each $\mathbb{R}$-orbit is orientation preserving.

Furthermore, two additional nice properties (4.4) and (4.5) hold.

###### Sketch of proof.

Fix a finite, symmetric subset $S$ that generates $\Gamma$ and contains
$\mathbf{1}$. A general action of $\Gamma$ on $\mathbb{R}$ may be very badly
behaved, but it has been shown [15, Props. 8.1 and 8.4] that if an action
exists, then there is an action
$\varphi\colon\Gamma\to\operatorname{Homeo}^{+}(\mathbb{R})$, and a constant
$C$, such that for all $s,t\in\mathbb{R}$, and all $\gamma\in S$:

  * •

(Lipschitz) $|\varphi_{\gamma}(s)-\varphi_{\gamma}(t)|\leq C|s-t|$, and

  * •

(bounded displacement) $|\varphi_{\gamma}(t)-t|\leq C$.

Thus, we can fix a constant $C$ such that the following set is nonempty:

$Z_{0}\coloneqq\left\\{\,\begin{matrix}\varphi\colon\Gamma\to\operatorname{Homeo}^{+}(\mathbb{R})\\\
\text{(homomorphism)}\end{matrix}\mathrel{\left|\vphantom{\left\\{\begin{matrix}\varphi\colon\Gamma\to\operatorname{Homeo}^{+}(\mathbb{R})\\\
\text{(homomorphism)}\end{matrix}\mid\begin{matrix}\text{for all $\gamma\in S$
and all $s,t\in\mathbb{R}$,}\\\ \text{we have \
}|\varphi_{\gamma}(s)-\varphi_{\gamma}(t)|\leq C|s-t|\\\ \text{and \
}|\varphi_{\gamma}(t)-t|\leq
C\end{matrix}\right\\}}\right.}\begin{matrix}\text{for all $\gamma\in S$ and
all $s,t\in\mathbb{R}$,}\\\ \text{we have \
}|\varphi_{\gamma}(s)-\varphi_{\gamma}(t)|\leq C|s-t|\\\ \text{and \
}|\varphi_{\gamma}(t)-t|\leq C\end{matrix}\,\right\\}.$

The trivial action is one of the elements of $Z_{0}$. To eliminate this, we
add a nontriviality condition:

(4.2) $\displaystyle Z=\left\\{\,\varphi\in
Z_{0}\mathrel{\left|\vphantom{\left\\{\varphi\in Z_{0}\mid\forall
t\in\mathbb{R},\
\int_{\Gamma}\bigl{(}\varphi_{\gamma}(t)-t\bigr{)}^{2}\,d\mu_{\Gamma}(\gamma)>\frac{1}{C}\right\\}}\right.}\forall
t\in\mathbb{R},\
\int_{\Gamma}\bigl{(}\varphi_{\gamma}(t)-t\bigr{)}^{2}\,d\mu_{\Gamma}(\gamma)>\frac{1}{C}\,\right\\}.$

If $1/C$ is small enough, then $Z$ will be nonempty.

For each $\gamma\in\Gamma$, the bounded displacement condition implies that
$\\{\,\varphi_{\gamma}\mid\varphi\in Z\,\\}$ is uniformly bounded on any
compact interval, and the Lipschitz condition implies that this set of
functions is equicontinuous. Therefore, we see from the Arzelà–Ascoli Theorem
that $Z$ is compact.

This completes the construction of the space $Z$. We now verify that it has
the desired properties (1–4).

(2) The group $\mathbb{R}$ acts on $Z$ by via conjugation by translations: for
$t\in\mathbb{R}$, let

$(t*\varphi)_{\gamma}(s)=\varphi_{\gamma}(s+t)-t.$

Suppose this $\mathbb{R}$-action is not free, so there is some nonzero
$t\in\mathbb{R}$ and some $\varphi\in Z$, such that $t*\varphi=\varphi$. This
means that for all $s\in\mathbb{R}$ and all $\gamma\in\Gamma$, we have

$\varphi_{\gamma}(s+t)-t=\varphi_{\gamma}(s),$

so the action of $\Gamma$ centralizes the translation by $t$. Hence, the
action of $\Gamma$ on $\mathbb{R}$ factors through to a well-defined action on
the quotient $\mathbb{R}/t\mathbb{Z}$. Then theorem 2.4(3) tells us that this
action must have a fixed point $s+t\mathbb{Z}$ (after replacing $\Gamma$ with
a finite-index subgroup). So the subset $s+t\mathbb{Z}$ of $\mathbb{R}$ is
$\Gamma$-invariant. The group $\Gamma$ must act on this set by order-
preserving permutations, which means that it acts by translations. But the
group of translations is abelian, and the abelianization of $\Gamma$ is finite
(see theorem 2.4(2)). So we conclude that $\Gamma$ fixes $s$ (and also fixes
all of the other points in $s+t\mathbb{Z}$. More precisely,
$\varphi_{\gamma}(s)=s$ for every $\gamma\in\Gamma$. This contradicts the
nontriviality condition in (4.2).

(1) We define an action of $\Gamma$ on $Z$ by letting an element $\alpha$ of
$\Gamma$ act on the point $\varphi$ of $Z$ via the action of the element
$\varphi_{\alpha}(0)$ of $\mathbb{R}$, so
${}^{\alpha}\\!\varphi=\varphi_{\alpha}(0)*\gamma$. More concretely:

${}^{\alpha}\\!\varphi_{\gamma}(t)=\bigl{(}\varphi_{\alpha}(0)*\varphi)_{\gamma}(t)=\varphi_{\gamma}\bigl{(}t+\varphi_{\alpha}(0)\bigr{)}-\varphi_{\alpha}(0).$

It may not be obvious that this defines an action, so we provide a
verification. Noting that, for $\alpha\in\Gamma$ and $t\in\mathbb{R}$, we have

(4.3)
$\displaystyle{}^{\alpha}(t*\varphi)=(t*\varphi)_{\alpha}(0)*(t*\varphi)=[\varphi_{\alpha}(0+t)-t]*(t*\varphi)=\varphi_{\alpha}(t)*\varphi,$

we see that

${}^{\alpha}\bigl{(}{}^{\beta}\\!\varphi\bigr{)}={}^{\alpha}(\varphi_{\beta}(0)*\varphi)=\varphi_{\alpha}\bigl{(}\varphi_{\beta}(0)\bigr{)}*\varphi=\varphi_{\alpha\beta}(0)*\varphi={}^{\alpha\beta}\varphi.$

Suppose this $\Gamma$-action has a fixed point: ${}^{\alpha}\varphi=\varphi$
for every $\alpha\in\Gamma$. This means $\varphi_{\alpha}(0)*\varphi=\varphi$,
so $\varphi_{\alpha}(0)=0$ (because the $\mathbb{R}$-action is free). Since
this holds for every $\alpha\in\Gamma$, we conclude that $0$ is a fixed point
of the action given by $\varphi$. This contradicts the nontriviality condition
in (4.2).

(3) Since elements of $\Gamma$ act on each point via an element of
$\mathbb{R}$, it is clear that the each $\mathbb{R}$-orbit is
$\Gamma$-invariant.

(4) If $s<t$, then $\varphi_{\alpha}(s)<\varphi_{\alpha}(t)$ (because
$\varphi_{\alpha}$ is orientation preserving). So we see from (4.3) that the
action of $\Gamma$ on each $\mathbb{R}$-orbit is orientation preserving. ∎

######  Lemma 4.4.

The action of $\Gamma$ on each $\mathbb{R}$-orbit is Lipschitz. More
precisely, for all $\alpha\in S$, all $t\in\mathbb{R}$, and all $\varphi\in
Z$, there is some $s\in\mathbb{R}$, such that

${}^{\alpha}(t*\varphi)=s*{}^{\alpha}\varphi\qquad\text{and\/ $|s|\leq
C|t|$}.$

###### Proof.

By (4.3), we may take $s=\varphi_{\alpha}(t)$, and then we see from the
bounded displacement property in the definition of $Z_{0}$ that

$|s|=|\varphi_{\alpha}(t)|\leq|\varphi_{\alpha}(t)-t|+|t|\leq
C|t|+|t|=(C+1)|t|.\qed$

###  4B. The measures $\mu_{Z}$ and $\mu_{X}$ are stationary

It is an immediate consequence of work of B. Deroin, V. Kleptsyn, A. Navas,
and K. Parwani [15, Prop. 8.1] that left-orderable groups have an action on
$\mathbb{R}$ in which the mean displacement of every point is $0$. Therefore,
the almost-periodic space $Z$ can be constructed to have this property on
every $\mathbb{R}$-orbit:

(4.5) $\displaystyle\forall z\in Z,\quad\int_{\Gamma}(\gamma
z-z)\,d\mu_{\Gamma}(\gamma)=0.$

######  Remark 4.6.

It is shown in [15, Prop. 8.4] that any action that satisfies (4.5) is
Lipschitz and has bounded displacement. Indeed, the existence of an action
with the latter two properties (which was fundamental in the construction of
the almost-periodic space $Z$) is obtained as a consequence of the existence
of an action satisfying (4.5).

From (4.5), we obtain the following fact.

######  Proposition 4.7 (proof of proposition 3.26, [14, Lem. 5.6]).

The measure $\mu_{Z}$ is $\mu_{\Gamma}$-stationary.

###### Proof.

For a.e. $z\in Z$, we have

$\displaystyle\int_{\Gamma}\frac{d\gamma_{*}\mu_{Z}}{d\mu_{Z}}(z)\,d\mu_{\Gamma}(\gamma)$
$\displaystyle=\int_{\Gamma}\frac{d\gamma^{-1}_{*}\mu_{Z}}{d\mu_{Z}}(z)\,d\mu_{\Gamma}(\gamma)$
$\displaystyle=\int_{\Gamma}D^{\text{leaf}}_{\gamma}(z)\,d\mu_{\Gamma}(\gamma)$
$\displaystyle=\int_{\Gamma}\lim_{t\to
0}\frac{\gamma(z+t)-z}{t}\,d\mu_{\Gamma}(\gamma)$
$\displaystyle=1+\int_{\Gamma}\lim_{t\to
0}\frac{\gamma(z+t)-(z+t)}{t}\,d\mu_{\Gamma}(\gamma)$
$\displaystyle=1+\lim_{t\to
0}\frac{\int_{\Gamma}\bigl{(}\gamma(z+t)-(z+t)\bigr{)}\,d\mu_{\Gamma}(\gamma)}{t}$
$\displaystyle\begin{pmatrix}\text{the integral is actually a}\\\ \text{finite
sum (see~{}\lx@cref{creftype~refnum}{nuGammaDefn})}\end{pmatrix}$
$\displaystyle=1+\lim_{t\to 0}\frac{0}{t}$ (4.5) $\displaystyle=1.$

So

$\int_{\Gamma}\gamma_{*}\mu_{Z}\,d\mu_{\Gamma}(\gamma)=\int_{\Gamma}\frac{d\gamma_{*}\mu_{Z}}{d\mu_{Z}}\,\mu_{Z}\,d\mu_{\Gamma}(\gamma)=\left(\int_{\Gamma}\frac{d\gamma_{*}\mu_{Z}}{d\mu_{Z}}\,d\mu_{\Gamma}(\gamma)\right)\cdot\mu_{Z}=1\cdot\mu_{Z}=\mu_{Z}.\qed$

######  Corollary 4.8 (proof of corollary 3.27, [14, Prop. 5.11]).

$\mu_{X}$ is $\mu_{G}$-stationary.

###### Proof.

Let $[(k_{0},z_{0})]\in X$, with $k_{0}\in K$ and $z_{0}\in Z$. Then

$\displaystyle\int_{G}\frac{dg_{*}\mu_{X}}{d\mu_{X}}$
$\displaystyle\bigl{(}[(k_{0},z_{0})]\bigr{)}\,d\mu_{G}(g)$
$\displaystyle=\int_{G}\frac{dg^{-1}_{*}\mu_{X}}{d\mu_{X}}\bigl{(}[(k_{0},z_{0})]\bigr{)}\,d\mu_{G}(g)$
$\displaystyle=\int_{G}D^{\text{leaf}}_{g}\bigl{(}[(k_{0},z_{0})]\bigr{)}\,d\mu_{G}(g)$
$\displaystyle=\int_{G}D^{\text{leaf}}_{gk_{0}}\bigl{(}[(\mathbf{1},z_{0})]\bigr{)}\cdot
D^{\text{leaf}}_{k_{0}^{-1}}\bigl{(}[(k_{0},z_{0})]\bigr{)}\,d\mu_{G}(g)$
(Chain Rule)
$\displaystyle=\int_{G}D^{\text{leaf}}_{gk_{0}}\bigl{(}[(\mathbf{1},z_{0})]\bigr{)}\cdot
1\,d\mu_{G}(g)$
$\displaystyle=\int_{G}D^{\text{leaf}}_{g}\bigl{(}[(\mathbf{1},z_{0})]\bigr{)}\,d\mu_{G}(g)$
($\mu_{G}$ is right $K$-invariant (3.11(2b))
$\displaystyle=\int_{\Gamma}\int_{K}D^{\text{leaf}}_{k\gamma}\bigl{(}[(\mathbf{1},z_{0})]\bigr{)}\,dm_{K}(k)\,d\mu_{\Gamma}(\gamma)$
(definition of $\mu_{\Gamma}$)
$\displaystyle=\int_{\Gamma}\int_{K}D^{\text{leaf}}_{\gamma}\bigl{(}[(\mathbf{1},z_{0})]\bigr{)}\,dm_{K}(k)\,d\mu_{\Gamma}(\gamma)$
(Chain Rule and $D^{\text{leaf}}_{k}=1$)
$\displaystyle=\int_{\Gamma}D^{\text{leaf}}_{\gamma}\bigl{(}[(\mathbf{1},z_{0})]\bigr{)}\,d\mu_{\Gamma}(\gamma)$
(integral of a constant)
$\displaystyle=\int_{\Gamma}D^{\text{leaf}}_{\gamma}(z_{0})\,d\mu_{\Gamma}(\gamma)$
($\gamma\bigl{(}[(\mathbf{1},z)]\bigr{)}=[(\gamma,z)]=[(\mathbf{1},\gamma
z)]$) $\displaystyle=1$
$\displaystyle\text{(\lx@cref{creftype~refnum}{nuGammaStationary})}.\qed$

######  Warning 4.9.

Corollary 4.8 is not a trivial consequence of proposition 4.7: a stationary
measure for an action of a lattice subgroup does not usually lead in an
obvious way to a stationary measure for the induced action of the ambient
group. However, D. Creutz [12, §3] has found a general construction.

######  Lemma 4.10 (proof of proposition 3.29(2)).

If $\mu_{X}$ is $G$-invariant, then $\mu_{Z}$ is $\Gamma$-invariant.

###### Proof.

Given $\gamma\in\Gamma$ and $Z_{0}\subseteq Z$, we wish to show
$\mu_{Z}(Z_{0})=\mu_{Z}(\gamma Z_{0})$. Since $K$ is open, there is an open
neighbourhood $K_{0}$ of $\mathbf{1}$ in $K$, such that $\gamma
K_{0}\gamma^{-1}\subseteq K$. Then

$\gamma[K_{0}\times Z_{0}]=[\gamma K_{0}\times Z_{0}]=[\gamma
K_{0}\gamma^{-1}\times\gamma Z_{0}],$

so

$\displaystyle m_{K}(K_{0})\cdot\mu_{Z}(Z_{0})$
$\displaystyle=\mu_{X}\bigl{(}[K_{0}\times Z_{0}]\bigr{)}$ (definition of
$\mu_{X}$) $\displaystyle=\mu_{X}\bigl{(}\gamma[K_{0}\times Z_{0}]\bigr{)}$
(we are assuming that $\mu_{X}$ is $G$-invariant)
$\displaystyle=\mu_{X}\bigl{(}[\gamma K_{0}\gamma^{-1}\times\gamma
Z_{0}]\bigr{)}$ (definition of equivalence relation on $G\times Z$)
$\displaystyle=m_{K}(\gamma K_{0}\gamma^{-1})\cdot\mu_{Z}(\gamma Z_{0})$
$\displaystyle\text{(definition of~{}$\mu_{X}$)}.$

However,

$m_{K}(\gamma K_{0}\gamma^{-1})=m_{G}(\gamma
K_{0}\gamma^{-1})=m_{G}(K_{0})=m_{K}(K_{0}).$

(The middle equality is because the semisimple group $G$ is unimodular.)
Therefore, by cancelling a factor of $m_{K}(K_{0})$, we conclude that
$\mu_{Z}(Z_{0})=\mu_{Z}(\gamma Z_{0})$. ∎

###  4C. Contraction on the $\mathbb{R}$-orbits

######  Lemma 4.11 (resolution of the technical issue in 3.30, cf. [14, Lem.
5.13]).

$D^{\text{leaf}}_{\\!a}(x)$ exists for $\mu_{X}^{P}$-a.e. $x\in X$.

###### Proof.

Let

$\text{$B=\\{\,x\in X\mid\text{$D^{\text{leaf}}_{\\!a}(x)$ does not
exist}\,\\}$ \quad(the ``bad'' set)}.$

Suppose $[(k,z)]\in B$, and write $ak=k^{\prime}\gamma$ with $k^{\prime}\in K$
and $\gamma\in\Gamma$. Then $a$ acts on the entire $\mathbb{R}$-orbit
$[(k,z+\mathbb{R})]$ via the element $\gamma$ of $\Gamma$ (see Note 3.18(4)).
Since $[(k,z)]\in B$, this implies that $D^{\text{leaf}}_{\gamma}(z)$ does not
exist. However, $K\gamma$ is open, so there is a neighbourhood $\mathcal{O}$
of $\mathbf{1}$ in $K$, such that $a\mathcal{O}k\subseteq K\gamma$. Then $a$
acts via $\gamma$ on every $\mathbb{R}$-orbit in $[\mathcal{O}k\times Z]$.
This implies $\mathcal{O}[(k,z)]\subseteq B$.

For each $x\in B$, we have shown there is a neighbourhood $\mathcal{O}$ of
$\mathbf{1}$ in $K$, such that $\mathcal{O}\,x\subseteq B$. If
$\mu_{X}^{P}(B)>0$, then there are a subset $B^{\prime}$ of positive measure,
and a (small) neighbourhood $\mathcal{O}^{\prime}$ of $\mathbf{1}$ in $K$,
such that $\mathcal{O}^{\prime}B^{\prime}\subset B$. Then, since
$\mu_{X}=m_{K}\times\mu_{Z}$, we have

$\mu_{X}(B)\geq\mu_{X}(\mathcal{O}^{\prime}B^{\prime})\geq
m_{K}(\mathcal{O}^{\prime})\cdot\mu_{Z}\bigl{\\{}\,z\in
Z\mid\bigl{(}K\times\\{z\\}\bigr{)}\cap B^{\prime}\neq\emptyset\,\\}\neq 0.$

This contradicts the fact that $D^{\text{leaf}}_{\\!a}(x)$ exists for
$\mu_{X}$-a.e. $x\in X$. ∎

######  Lemma 4.12 (proof of lemma 3.31, [14, §7.1, p. 43]).

$\chi_{P}$ is a homomorphism.

###### Proof.

We have

$\displaystyle\chi_{P}(a_{1}a_{2})$ $\displaystyle=\int_{X}\log
D^{\text{leaf}}_{\\!a_{1}a_{2}}(x)\,d\mu_{X}^{P}(x)$
$\displaystyle=\int_{X}\log
D^{\text{leaf}}_{a_{1}}(a_{2}x)\,d\mu_{X}^{P}(x)+\int_{X}\log
D^{\text{leaf}}_{\\!a_{2}}(x)\,d\mu_{X}^{P}(x)$ (Chain Rule)
$\displaystyle=\int_{X}\log
D^{\text{leaf}}_{a_{1}}(a_{2}x)\,d\mu_{X}^{P}(x)+\chi_{P}(a_{2}).$

Since $A\subseteq P$, we know that $\mu_{X}^{P}$ is $a_{2}$-invariant, so

$\int_{X}\log D^{\text{leaf}}_{\\!a_{1}}(a_{2}x)\,d\mu_{X}^{P}(x)=\int_{X}\log
D^{\text{leaf}}_{\\!a_{1}}(x)\,d\mu_{X}^{P}(x)=\chi_{P}(a_{1}).\qed$

Since the measure $\mu_{Z}$ is ergodic for the $\mathbb{R}$-action, it can be
shown that the measure $\mu_{X}^{P}$ is ergodic for the action of $P$:

######  Lemma 4.13 ([14, Lem. 6.5]).

The $P$-invariant measure $\mu_{X}^{P}$ is ergodic.

###### Idea of proof.

Suppose $\mu_{X}^{P}$ is not ergodic, so $\mu_{X}^{P}=s\mu_{1}+t\mu_{2}$ is a
convex combination of two different $P$-invariant probability measures (with
$st\neq 0$). Then

$\mu_{X}=\int_{K}k_{*}\mu_{X}^{P}\,dm_{K}=s\int_{K}k_{*}\mu_{1}\,dm_{K}+t\int_{K}k_{*}\mu_{2}\,dm_{K}$

is a convex combination of two $G$-stationary measures. Also, the uniqueness
in theorem 3.28 implies that neither of these stationary measures is equal to
$\mu_{X}$. So $\mu_{X}$ is not ergodic as a stationary measure.

Hence, there is a $G$-invariant function $f$ that is not constant (a.e.). The
Pointwise Ergodic Theorem (2.7) has an analogue for random walks with a
stationary measure [14, Cor. 4.8] (cf. the proof of proposition 4.23).
Combining this with proposition 4.20 (contraction on $\mathbb{R}$-orbits)
implies that $f$ is essentially constant on a.e. $\mathbb{R}$-orbit in $X$.
Since the $\mathbb{R}$-action on $Z$ is ergodic, this implies that
$f\bigl{(}[(h,z)]\bigr{)}$ is essentially independent of $z$. But the
$G$-invariance implies that it is also essentially independent of $h$. So $f$
is constant (a.e.). This is a contradiction. ∎

######  Lemma 4.14 (proof of lemma 3.32, [14, Lem. 7.1]).

If $a\in A$, such that $\chi_{P}(a)<0$, then for $\mu_{X}^{P}$-a.e. $x\in X$,
there exists $\epsilon=\epsilon(x)>0$, such that

$\text{$a^{n}(x-\epsilon)\ -\ a^{n}(x+\epsilon)\ \to\ 0$ \quad as
$n\to\infty$}.$

###### Proof.

For (small) $\eta>0$, let

$D_{\eta}(x)=\sup_{0<\epsilon<\eta}\log\left(\frac{a(x+\epsilon)-a(x)}{\epsilon}\right).$

These functions are uniformly bounded, because we know that $a$ is uniformly
Lipschitz on the $\mathbb{R}$-orbits (see lemma 3.21). Also, the functions
converge pointwise to $D^{\text{leaf}}_{a}$ as $\eta\to 0$. By the Dominated
Convergence Theorem, then

(4.15)
$\displaystyle\textstyle\int_{X}D_{\eta}\,d\mu_{X}^{P}\stackrel{{\scriptstyle\eta\to
0}}{{\longrightarrow}}\chi_{P}(a).$

Now, a technical point. We will apply the Pointwise Ergodic Theorem (2.7) to
the action of $a$. This requires an ergodic measure. We know from lemma 4.13
that the measure $\mu_{X}^{P}$ is ergodic for the $P$-action, but we cannot
expect it to be ergodic for the action of $a$. Therefore, we should consider
the ergodic components of the measure. Fortunately, it can be shown that the
integral in 3.30 that defines $\chi_{P}(a)$ has the same value for each
ergodic component [14, Prop. 7.2], so this is not a serious problem.
Therefore, we will simply ignore it, and act as if the measure $\mu_{X}^{P}$
is ergodic for the action of $a$.

Note that, for $\epsilon,\delta>0$, $\eta$ small, and $n$ large, we have

$\displaystyle\frac{1}{n}\log\left(\frac{a^{n}(x+\epsilon)-a^{n}x}{\epsilon}\right)$
$\displaystyle=\frac{1}{n}\bigl{[}\log\bigl{(}a^{n}(x+\epsilon)-a^{n}x\bigr{]}-\log\epsilon\bigr{]}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\log\left(\frac{a^{i}(x+\epsilon)-a^{i}x}{a^{i-1}(x+\epsilon)-a^{i-1}x}\right)$
(telescoping sum)
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}D_{\eta}(a^{i-1}x)$
$\displaystyle\begin{pmatrix}\text{if $a^{i-1}(x+\epsilon)-a^{i-1}x<\eta$}\\\
\text{for $1\leq i<n$}\end{pmatrix}$
$\displaystyle<\delta+\int_{X}D_{\eta}\,d\mu_{X}^{P}$ (Pointwise Ergodic
Theorem (2.7)) $\displaystyle<2\delta+\chi_{P}(a)$
$\displaystyle\text{{(\ref{chiPLocalPf-toChi})}}.$

So

$\text{$a^{n}(x+\epsilon)-a^{n}x\ <\ \epsilon\,e^{nc}$ \quad where
$c=2\delta+\chi_{P}(a)$}.$

We can choose $\delta$ small enough that $2\delta+\chi_{P}(a)<0$, so
$e^{nc}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0$. Then we may
choose $\epsilon$ small enough that $e^{nc}<\eta$ for all $n$. Then we
conclude, by induction, that $a^{n}(x+\epsilon)-a^{n}x<\eta$ for all $n$. So
the above estimate applies for all $n$. Hence, we have shown that
$a^{n}(x+\epsilon)-a^{n}x\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0$ (exponentially fast).

A similar argument with negative $\epsilon$ completes the proof. ∎

###  4D. $\mu_{X}^{P}$ is $G$-invariant

######  Remark 4.16 (further details for the proof of corollary 3.35).

The proof of corollary 3.35 strongly uses the fact that $\mu_{X}^{P}$ is
$U_{a}^{+}$-invariant, and also uses the fact that $\pi_{*}(c_{*}\mu_{X}^{P})$
is a ${}^{c}\\!P$-invariant probability measure, but the latter is needed only
to conclude that the support of this measure is all of $G/\Gamma$.

Fortunately, both of these properties are true of the ergodic components.
Firstly, it follows from the Mautner Lemma [2, Lem. II.3.6, p. 61] that if a
probability measure is invariant under both $a$ and its expanding
horospherical subgroup $U_{a}^{+}$, then each $a$-ergodic component $\mu$ of
the measure is invariant under $U_{a}^{+}$. Furthermore, it can be shown (by
an argument very similar to the proof of proposition 2.8(2)) that $G/\Gamma$
has no nonempty, proper, closed subsets that are invariant under $\langle
a,U_{a}^{+}\rangle$. This implies that the support of $\pi_{*}\mu$ is all of
$G/\Gamma$.

######  Corollary 4.17 (proof of the general case of corollary 3.42, [14, Thm.
8.1 (and §9)]).

The measure $\mu_{X}^{P}$ is $G$-invariant.

###### Proof.

Let $\mathcal{Q}_{\text{good}}$ be the set of all minimal parabolic subgroups
$Q$ containing $A$, such that $\mu_{X}^{P}$ is $Q$-invariant (or,
equivalently, such that $\mu_{X}^{Q}=\mu_{X}^{P}$). Also let

$\mathcal{H}=\\{\,a\in A\mid\chi_{P}(a)<0\,\\}.$

The arguments given in the “Idea of proof” of corollary 3.42 show that if
$Q_{1}\in\mathcal{Q}_{\text{good}}$, and $\mathcal{W}_{Q_{2}}$ is a Weyl
chamber, such that $\mathcal{H}$ contains some nonidentity element of
$\mathcal{W}_{Q_{1}}\cap\mathcal{W}_{Q_{2}}$, then
$Q_{2}\in\mathcal{Q}_{\text{good}}$. Therefore, if we let

$A_{\text{good}}\
\coloneqq\bigcup_{Q\in\mathcal{Q}_{\text{good}}}\mathcal{W}_{Q},$

it is not difficult to show that $\mathcal{H}\subseteq A_{\text{good}}$.

Furthermore, if $w$ is any element of the Weyl group, such that
$w(\mathcal{H})\cap A_{\text{good}}$ is nonempty, then one can repeat the
argument with $w(\mathcal{H})$ in the place of $\mathcal{H}$, to conclude that
$A_{\text{good}}$ also contains $w(\mathcal{H})$. For every element $w$ of the
Weyl group, this implies that either $w(A_{\text{good}})=A_{\text{good}}$ or
$w(A_{\text{good}})$ is disjoint from the interior of $A_{\text{good}}$.

Case 1. Assume $A_{\text{good}}$ is invariant under the Weyl group.  Since the
Weyl group acts transitively on the set of Weyl chambers, this implies that
every Weyl chamber is in $A_{\text{good}}$, so $A_{\text{good}}=A$. This means
that every minimal parabolic subgroup is in $\mathcal{Q}_{\text{good}}$, so
$\mu_{X}^{P}$ is invariant under every minimal parabolic subgroup. Since the
minimal parabolic subgroups generate $G$, we conclude that $\mu_{X}^{P}$ is
$G$-invariant, as desired.

Case 2. Assume $A_{\text{good}}$ is not invariant under the Weyl group.
Recall that $A_{\text{good}}$ contains the halfspace $\mathcal{H}$, and also
that, for every element $w$ of the Weyl group, either $w(A_{\text{good}})=A$,
or $w(A_{\text{good}})$ is disjoint from the interior of $A_{\text{good}}$.
Then the assumption of this case implies that
$A_{\text{good}}=\overline{\mathcal{H}}$ (the closure of $\mathcal{H}$) and,
for every element of the Weyl group, either
$w(\overline{\mathcal{H}})=\overline{\mathcal{H}}$ or
$w(\overline{\mathcal{H}})=\overline{\mathcal{H}^{\text{op}}}$, the closure of
the opposite halfspace $\\{\,a\in A\mid\chi_{P}(a)>0\,\\}$.

The action of the Weyl group of a simple group is irreducible, so the only way
this can happen is if:

  * •

$G$ has a simple factor $G_{1}$ with
$\operatorname{rank}_{\mathbb{Q}_{p}}G_{1}=1$, and

  * •

$\chi_{P}=c\,\alpha$, where $\alpha$ is the (unique) simple root of $G_{1}$
(for some ordering of the roots), and $c\in\mathbb{R}^{+}$.

Write $G=G_{1}\times H$, and choose some $a\in A\cap G_{1}$, such that
$\alpha(a)<0$. Then $a$ centralizes $H$, so we see from corollary 3.35 that
$\mu_{X}^{P}$ is $H$-invariant. The subgroup $H$ is normal, so this implies
that $k_{*}\mu_{X}^{P}$ is also $H$-invariant, for every $k\in K$:

$h_{*}(k_{*}\mu_{X}^{P})=k_{*}(k^{-1}hk)_{*}\mu_{X}^{P}=k_{*}\mu_{X}^{P}.$

Since $\mu_{X}$ is the integral of these $H$-invariant measures (see theorem
3.28), we conclude that $\mu_{X}$ is $H$-invariant. And we know that $\mu_{X}$
is $K$-invariant (by definition). So $\mu_{X}$ is invariant under $KH$. Thus,
letting $\Gamma_{0}\coloneqq\Gamma\cap KH$, we know that $\mu_{X}$ is
invariant under $\Gamma_{0}$. For every $\gamma\in\Gamma$, this implies that
$D^{\text{leaf}}_{\gamma}=1$ $\mu_{X}$-a.e. (see lemma 3.25), so $\gamma$
commutes with the $\mathbb{R}$-flow. Therefore, $\Gamma_{0}$ acts by
translations on each $\mathbb{R}$-orbit in $Z$. Since the group of
translations is abelian, this implies that the commutator subgroup
$[\Gamma_{0},\Gamma_{0}]$ acts trivially on $Z$. On the other hand, $\Gamma$
acts faithfully on $Z$. So we conclude that $\Gamma_{0}$ is abelian.

Let $\pi\colon G\to G_{1}$ be the projection with kernel $H$. Since $\Gamma$
is irreducible, we know that $\pi(\Gamma)$ is dense in $G_{1}$. Since $K$ is
open, this implies that $\pi(\Gamma_{0})$ is dense in $\pi(K)$, which is an
open subgroup of $G_{1}$. Since nontrivial open subgroups are Zariski dense,
we conclude that $\pi(\Gamma_{0})$ is Zariski dense in $G_{1}$. Since $G_{1}$
is not abelian, this implies that $\pi(\Gamma_{0})$ is not abelian, which
contradicts the conclusion of the preceding paragraph. ∎

######  Remark 4.18.

For real groups, Section 4D of the proof of corollary 4.17 is much longer [14,
§9, pp. 55–64]. This is because $K$ is not open, so the intersection
$\Gamma\cap KH$ may be abelian, or could even be trivial, so a different idea
is needed.

###  4E. More proof of proposition 3.33

######  Proposition 4.19 (proof of proposition 3.44, [14, Prop. 6.3]).

For every action of $\Gamma$ on $\mathbb{R}$ without fixed points, there is a
compact subset $I$ of $\mathbb{R}$, such that for every $\epsilon>0$ and all
$x,y\in\mathbb{R}$, there exists $\gamma\in\Gamma$, such that

$\text{$|\gamma x-\gamma y|<\epsilon$ \ and \ $\\{\gamma x,\gamma y\\}\subset
I$}.$

###### Proof.

It is a general fact about orientation-preserving actions of finitely
generated groups on the real line that one of the following possibilities must
hold [15, Thm. 7.1]:

  1. (1)

the action has a fixed point, or

  2. (2)

there is a nontrivial homomorphism $\Gamma\to\mathbb{R}$, or

  3. (3)

there is a fixed-point-free homeomorphism $\varphi$ of $\mathbb{R}$ that
commutes with every element of $\Gamma$, or

  4. (4)

the action has the desired contraction property.

Since we are assuming that (1) does not hold, and theorem 2.4(2) tells us that
(2) does not hold, we may assume that (3) holds.

We may assume that $\varphi$ is the translation $\varphi(t)=t+1$. Then
$\Gamma$ acts on $\mathbb{R}/\mathbb{Z}=S^{1}$. By theorem 2.4(3), this action
has a finite orbit, so, by passing to a finite-index subgroup of $G$, we may
assume that it has a fixed point. Assuming, without loss of generality, that
the fixed point is $0+\mathbb{Z}$, this implies that the set $\mathbb{Z}$ is
$\Gamma$-invariant. Now, since the action of $\Gamma$ is orientation-
preserving, the action on $\mathbb{Z}$ must be by translations. However, the
group of translations is isomorphic to the abelian group $\mathbb{Z}$, whereas
the abelianization of $\Gamma$ is finite (see theorem 2.4(2) again). Hence,
every point in $\mathbb{Z}$ is fixed by $\Gamma$, which contradicts an
assumption of this proposition. ∎

######  Proposition 4.20 (proof of proposition 3.45, [14, Props. 6.4 and
6.8]).

There is a constant $\delta<1$, such that, for $\mu_{X}$-a.e. $x\in X$, for
all $y\in\mathbb{R}x$, and for $\mu_{G}^{\infty}$-a.e. sequence
$(g_{i})_{i=1}^{\infty}$ of elements of $G$, we have

$\text{$|g_{n}g_{n-1}\cdots g_{1}x\ -\ g_{n}g_{n-1}\cdots g_{1}y|<\delta^{n}$
\quad for all sufficiently large~{}$n$}.$

###### Proof.

For convenience, let $\pi_{n}=g_{n}g_{n-1}\cdots g_{1}$. For any $\ell>0$ and
$x\in X$, it follows from proposition 4.19 that if $n$ is sufficiently large,
then there is a positive probability that $\pi_{n}(x+\ell)-\pi_{n}x$ is small.
Then an argument similar to the proof of lemma 4.14 shows that this difference
almost surely tends to $0$. Hence, we see from the compactness of $X$ that,
for every $\ell,\epsilon>0$, there exist $p>0$ and $N\in\mathbb{N}$, such
that, for every $x\in X$, we have

$\mu_{G}^{\infty}\bigl{(}\\{\,(g_{i})_{i=1}^{\infty}\mid\pi_{N}(x+\ell)-\pi_{N}x\leq\epsilon\bigr{\\}}\bigr{)}\geq
p.$

For $x\in X$, let

$C_{x}=\\{\,g\in G\mid g(x+\ell)-g(x-\ell)<\epsilon\,\\},$

so $\mu_{G}^{*N}(C_{x})\geq p$ (where $\mu_{G}^{*N}$ is the $N$-fold
convolution $\mu_{G}*\mu_{G}*\cdots*\mu_{G}$).

We claim for each $x\in X$ and $\mu_{G}^{\infty}$-a.e. sequence
$(g_{i})_{i=1}^{\infty}$ of elements of $G$, there are infinitely many $n$,
such that $\pi_{n+N}\pi_{n}^{-1}\in C_{\pi_{n}x}$. To see this, let $c(x)$ be
the probability that this happens at least once. Note that:

  * •

if $\pi_{N}\in C_{x}$, then we may let $n=0$, and

  * •

otherwise, we may ignore $n=0,1,2,\ldots,N-1$, and start our search at $n=N$.

Therefore

$c(x)\geq\mu_{G}^{*N}(C_{x})+\int_{G\smallsetminus
C_{x}}c(gx)\,d\mu_{G}^{*N}(g).$

Therefore, the infinum $c_{-}$ of $c(x)$ satisfies

$c_{-}\geq p+(1-p)c_{-}=c_{-}+(1-c_{-})p,$

so $c_{-}=1$. Hence, we have $c(x)=1$ for all $x$. Then, since the random walk
is a Markov process, we conclude that the event happens an infinite number of
times.

Now, the Martingale Convergence Theorem tells us that for every $x\in X$,
every $t\in\mathbb{R}$, and $\mu_{G}^{\infty}$-a.e. $(g_{i})_{i=1}^{\infty}$,
the sequence $\pi_{n}(x+t)-\pi_{n}(x-t)$ converges to a finite limit. For
$\ell\in\mathbb{R}^{+}$, let $\mathcal{E}_{\ell}=\mathcal{E}_{\ell}(x,t)$ be
the set of sequences for which this limit is strictly less than $\ell$. Then,
for $n$ sufficiently large, the length of the interval is less than $\ell$.
Then, by the claim of the preceding paragraph, we know (almost surely) that
there are infinitely many $n$, such that $\pi_{g_{N}g_{N-1}\cdots g_{n+1}}\in
C_{\pi_{n}x}$. This implies

$\pi_{N}(x+t)-\pi_{N}(x-t)<\epsilon.$

Hence, since we know that the sequence $\pi_{n}(x+t)-\pi_{n}(x-t)$ converges,
we conclude that it converges to $0$.

An argument similar to the proof of lemma 4.14 (but using (3.43) in place of
the assumption that $\chi_{P}(a)<0$) shows that the contraction is
exponentially fast [14, Lem. 6.7 and Prop. 6.8]. ∎

As mentioned in Section 3, the next part of the argument is easier for real
groups, so we will switch to that setting for the remainder of the Section:

Assume, contrary to 3.2, that $G$ is a connected real semisimple Lie group.

######  Remark 4.21 (a technical point in the proof of proposition 3.50).

For convenience, let $a=\widehat{a_{P}}$. Also let

$X_{0}=\\{\,x\in X\mid\text{for all $u\in x+\mathbb{R}$, we have
$\widehat{a_{P}}^{n}\,x\ -\ \widehat{a_{P}}^{n}\,y\ \stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\ 0\,\\}$}.$

The “Idea of proof” of proposition 3.50 established that $\mu_{X}(X_{0})=1$,
and we wish to show that $\mu_{X}^{P}(X_{0})=1$.

The measure $\mu_{X}$ is $K$-invariant, so $k_{*}\mu_{X}(X_{0})=1$ for all
$k\in K$. By Fubini’s Theorem, we conclude, for $\mu_{X}$-a.e. $x\in X$, that
we have $kx\in X_{0}$ for $m_{K}$-a.e. $k\in K$. Since
$\mu_{X}=\int_{K}k_{*}\mu_{X}^{P}\,dk$, we can change $\mu_{X}$ to
$\mu_{X}^{P}$:

(4.22) $\displaystyle\text{for $\mu_{X}^{P}$-a.e.\ $x\in X$, for $m_{K}$-a.e.\
$k\in K$, we have $kx\in X_{0}$}.$

Fix $\mu_{X}^{P}$-a.e. $x\in X$. Also fix a neighbourhood $U$ of $\mathbf{1}$
in $U^{+}_{a}$, and let $\mathcal{O}^{-}$ be a small neighbourhood of
$\mathbf{1}$ in the set

$\\{\,g\in G\mid\text{$\\{a^{n}ga^{-n}$ is bounded}\,\\}.$

We see from (4.22) that for $m_{U_{a}^{+}}$-a.e. $u\in U$, we can choose some
$g\in\mathcal{O}^{-}$, such that $gux\in X_{0}$. Letting $g_{n}\coloneqq
a^{n}ga^{-n}$, we have

$|g_{n}a^{n}ux-g_{n}a^{n}uy|=|a^{n}gux-a^{n}guy|\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

Since $\\{g_{n}\\}$ is bounded, this implies

$|a^{n}ux-a^{n}uy|\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0.$

So $ux\in X_{0}$. This (and Fubini’s Theorem) establishes that
$u_{*}\mu_{X}^{P}(X_{0})=1$ for a.e. $u\in U$. However, $\mu_{X}^{P}$ is
$U$-invariant (since $U\subseteq U^{+}_{a}\subseteq P$). So we conclude that
$\mu_{X}^{P}(X_{0})=1$.

We can also show that $\chi_{P}$ is nontrivial. More precisely:

######  Proposition 4.23 (proof of Note 3.41, [14, Lem. 7.4]).

For $\widehat{a_{P}}$ as in corollary 3.48, we have
$\chi_{P}(\widehat{a_{P}})<0$ and $U_{\widehat{a_{P}}}^{+}\subseteq P$.

###### Idea of proof.

Since $\widehat{a_{P}}\in\mathcal{W}_{P}$, we know that
$U_{\widehat{a_{P}}}^{+}\subseteq P$. Therefore, we only need to prove the
other statement.

The Pointwise Ergodic Theorem (2.7) has an analogue for random walks with a
stationary measure [14, Cor. 4.8]. (This was also mentioned in the proof of
lemma 4.13.) Applying this (and the Chain Rule) to the function $\log
D^{\text{leaf}}_{g}(x)$ on $G\times X$ yields: for $\mu_{G}^{\infty}$-a.e.
sequence $(g_{i})_{i=1}^{\infty}$ of elements of $G$, and $\mu_{X}$-a.e. $x\in
X$, if we let $\pi_{n}=g_{n}g_{n-1}\cdots g_{1}$, then

(4.24) $\displaystyle\frac{1}{n}\log
D^{\text{leaf}}_{\pi_{n}}(x)=\frac{1}{n}\sum_{i=0}^{n-1}\log
D^{\text{leaf}}_{g_{i+1}}(\pi_{i}x)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{G}\int_{X}\log
D^{\text{leaf}}_{g}\,d\mu_{X}(x)\,d\mu_{G}(g)<0,$

where the final inequality is from (3.43).

For convenience, let $a=\widehat{a_{P}}$. As in the proof of corollary 3.35,
there is a technical issue that the measure $\mu_{X}^{P}$ might not be ergodic
for the action of $a$, even though lemma 4.13 tells us that it is ergodic for
the action of $P$. An additional technical issue is that the above inequality
is valid for $\mu_{X}$-a.e. $x$, but the definition of
$\chi_{P}(\widehat{a_{P}})$ uses the measure $\mu_{x}^{P}$. We will ignore
both of these issues. (Roughly speaking, they are handled by proving that
statements we prove for $x$ remain valid for $kx$, for a.e. $k\in K$.)

Assuming that $\mu_{X}^{P}$ is ergodic with respect to $a$, we have, for
$\mu_{X}^{P}$-a.e. $x\in X$:

(4.25) $\displaystyle\frac{1}{n}\log D^{\text{leaf}}_{a^{n}}(x)$
$\displaystyle=\frac{1}{n}\sum_{k=0}^{n-1}\log D^{\text{leaf}}_{a}(a^{k}x)$
(Chain Rule) $\displaystyle\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{X}\log D^{\text{leaf}}_{a}\,d\mu_{X}^{P}$
(Pointwise Ergodic Theorem (2.7)) $\displaystyle=\chi_{P}(a)$
$\displaystyle\text{(definition of~{}$\chi_{P}$)}.$

Also, it follows from corollary 3.48 that

$\frac{\log D^{\text{leaf}}_{a^{n}}(x)-\log
D^{\text{leaf}}_{\pi_{n}}(x)}{n}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.$

Therefore, if we ignore the technical issue that (4.25) is for
$\mu_{X}^{P}$-a.e. $x$, but (4.24) is for $\mu_{X}$-a.e. $x$, we can equate
the two (in the limit), and conclude that $\chi_{P}(a)<0$. ∎

##  5\. $K\Gamma$ might not be all of $G$

We have been assuming that $G=K\Gamma$ (see 3.6). This makes it easy to extend
any function $f$ on $\Gamma$ to a function $f_{G}$ on $G$, simply by making it
left $K$-invariant: $f_{G}(k\gamma)=f(\gamma)$. Without this assumption, there
is usually no canonical way to extend a function on $\Gamma$ to a function on
$G$. However, we can do this for “harmonic” functions.

######  Definition 5.1 (Furstenberg [20, Defn. 4.1]).

Assume $\mu_{H}$ is a probability measure on a topological group $H$, and
$f\colon H\to\mathbb{R}$. We say that $f$ is _$\mu_{H}$ -harmonic_ if, for all
$x\in H$, we have

$f(x)=\int_{H}f(hx)\,d\mu_{H}(h).$

That is, the value of $f$ at any point is equal to the average (with respect
to $\mu_{H}$) of the values of $f$ at certain other points.

######  Lemma 5.2 ([20, p. 363]).

Every $\mu_{G}$-harmonic function $f$ on $G$ is left $K$-invariant. Therefore,
$f$ corresponds to a well-defined function $\overline{f}$ on $K\backslash G$.

###### Proof.

For $k\in K$, we have

$\displaystyle f(kx)$ $\displaystyle=\int_{G}f(gkx)\,d\mu_{G}(g)$ ($f$ is
$\mu_{G}$-harmonic) $\displaystyle=\int_{G}f(gx)\,d\mu_{G}(g)$ ($\mu_{G}$ is
right $K$-invariant (see 3.11(2))) $\displaystyle=f(x)$
$\displaystyle\text{($f$ is $\mu_{G}$-harmonic)}.\qed$

######  Remark 5.3.

Choose some $x_{0}\in G$, and construct the random walk
$x_{0},x_{1},x_{2},\ldots$ in $G$, starting at $x_{0}$, with law $\mu_{G}$.
(This means that, for every $n\geq 1$, the distribution of the $n$th step
$x_{n}x_{n-1}^{-1}$ is given by the measure $\mu_{G}$.) We define a stopping
time

$N=N(x_{0},x_{1},\ldots)=\min\\{\,n\geq 0\mid x_{n}\in K\Gamma\,\\}.$

Now, suppose $f\colon G\to\mathbb{R}$ is $\mu_{G}$-harmonic. Then the sequence
$f(x_{0}),f(x_{1}),f(x_{2}),\ldots$ is a _martingale_. More precisely,
$f(x_{n})$ is the expected value of $f(x_{n+1})$, given
$x_{0},x_{1},\ldots,x_{n}$. So it is not difficult to see that $f(x_{0})$ is
the expected value of $f(x_{N})$ (where $N$ is the above-mentioned stopping
time, or any other stopping time that satisfies mild conditions). Since
$x_{N}\in K\Gamma$ and $f$ is left $K$-invariant (see lemma 5.2), we can
conclude that $f(x_{0})$ is determined by the values of $f$ on $\Gamma$. Since
$x_{0}$ is arbitrary, this means that the value of $f$ at every point of the
group $G$ is determined by its values on the subgroup $\Gamma$. Of course,
this argument relies on the assumption that $f$ is $\mu_{G}$-harmonic.

Conversely, given an appropriate function on $\Gamma$, we will explain how to
extend it to a $\mu_{G}$-harmonic function that is defined on all of $G$.

######  Definition 5.4.

  1. (1)

For each $x\in K\backslash G$, we define an atomic probability measure
$\omega_{x}$ on $K\backslash G$ by:

$\text{$\omega_{x}(y)=\mu_{G}(Khg^{-1})$ \qquad if $x=Kg$ and $y=Kh$}.$

This is clearly independent of the choice of the representative $h$ of $y$,
and the following calculation shows that it is also independent of the choice
of the representative $g$ of $x$:

$\mu_{G}\bigl{(}Kh(kg)^{-1}\bigr{)}=\mu_{G}\bigl{(}Khg^{-1}k^{-1}\bigr{)}=\mu_{G}\bigl{(}Khg^{-1}\bigr{)},$

where the final equality follows from our assumption that $\mu_{G}$ is right
$K$-invariant (see 3.11(2b)) .

  2. (2)

We consider the random walk $\omega=\omega(x_{0})$ on $K\backslash G$ that
starts at a point $x_{0}$, and has transition probability $\omega_{x}(y)$ from
$x$ to $y$.

  3. (3)

Let $\widehat{x}=K$ be the basepoint of $K\backslash G$. For $x\in K\backslash
G$ and $\gamma\in\Gamma$, let $\mu_{x}(\gamma)$ be the probability that
$\widehat{x}\gamma$ is the first point of $\widehat{x}\,\Gamma$ that is
encountered by the random walk $\omega(x)$ that starts at $x$.

  4. (4)

Let $\mu_{K\backslash G}$ be the (atomic) measure on $K\backslash G$ that is
induced by $\mu_{G}$. More precisely, if $\pi\colon G\to K\backslash G$ is the
natural quotient map, then $\mu_{K\backslash G}=\pi_{*}\mu_{G}$.

  5. (5)

Define a measure $\mu_{\Gamma}$ on $\Gamma$ by:

$\mu_{\Gamma}(\gamma)=\int_{K\backslash G}\mu_{x}(\gamma)\,d\mu_{K\backslash
G}(x).$

We will need an estimate of the following type:

######  Lemma 5.5.

We have $\mu_{x}(\Gamma)=1$ for all $x\in K\backslash G$.

Indeed, if we let $\ell(\gamma)$ be the word length of $\gamma$, with respect
to some finite generating set of $\Gamma$, then there exists $\delta<1$, such
that

$\sum_{\ell(\gamma)<n}\mu_{x}(\gamma)>1-\delta^{n}\qquad\text{for all
large~{}$n$.}$

###### Proof.

The random walk $\omega$ on $K\backslash G$ factors through to a random walk
$\overline{\omega}$ on $K\backslash G/\Gamma$: for $x,y\in K\backslash G$, the
transition probability from $x\Gamma$ to $y\Gamma$ is
$\sum_{\gamma\in\Gamma}\omega_{x}(y\gamma)$. This is a random walk (or “Markov
chain”) on a finite set (see Note 3.5), so there is some $r\in\mathbb{N}$ and
$\epsilon>0$, such that, for every starting vertex, the walk has a probability
$>\epsilon$ of reaching the point $\widehat{x}\,\Gamma$ within $r$ steps.
Therefore, the probability that the stopping time $N$ is greater than $n$ is
less than $(1-\epsilon)^{n/r}$.

To complete the proof, combine this observation with the fact that, since
$\Gamma$ is cocompact, the word length metric $\ell_{\Gamma}$ on $\Gamma$ is
quasi-isometric to the metric on $\widehat{x}\,\Gamma$ that is obtained by
restricting the metric on $K\backslash G$ [4, Prop. I.8.19, p. 140]. ∎

######  Proposition 5.6 ([14, Thm. 3.12]).

Every positive (or bounded) $\mu_{\Gamma}$-harmonic function on $\Gamma$
extends to a unique $\mu_{G}$-harmonic function on $G$. Specifically, the
$\mu_{G}$-harmonic extension $f_{G}$ of $f$ is defined by:

$\overline{f_{G}}(x)=\sum_{\gamma\in\Gamma}\mu_{x}(\gamma)\,f(\gamma)\qquad\text{for
$x\in K\backslash G$}.$

###### Proof.

To simplify the estimates, we assume that $f$ is bounded. Then it is clear
from lemma 5.5 that the sum defining $\overline{f_{G}}(x)$ converges, so
$f_{G}$ is a function from $G$ to $\mathbb{R}$.

The uniqueness of the extension is a consequence of the argument in 5.3 (if
the extension exists).

For $x=K\gamma\in\widehat{x}\,\Gamma$, the measure $\mu_{x}$ is the point mass
at $\gamma$. So it is immediate from the definition of $\overline{f_{G}}$ that
$\overline{f_{G}}(K\gamma)=f(\gamma)$. Therefore the formula for $f_{G}$ does
provide an extension of $f$.

To complete the proof, we show that $f_{G}$ is $\mu_{G}$-harmonic.

For $x_{0}\notin\widehat{x}\,\Gamma$, we see from the definition that

$\mu_{x_{0}}(\gamma)=\int_{K\backslash
G}\mu_{x_{1}}(\gamma)\,d\mu_{K\backslash G}(x_{1}),$

so

$\overline{f_{G}}(x_{0})=\sum_{\gamma\in\Gamma}\mu_{x_{0}}(\gamma)\,f(\gamma)=\int_{K\backslash
G}\sum_{\gamma\in\Gamma}\mu_{x_{1}}(\gamma)\,f(\gamma)\,d\mu_{G}(x_{1})=\int_{K\backslash
G}\overline{f_{G}}(x_{1})\,d\mu_{G}(x_{1}),$

so $f_{G}$ is $\mu_{G}$-harmonic at the points of $G$ that are in $x_{0}$.

To see that $f_{G}$ is also $\mu_{G}$-harmonic at points of
$\widehat{x}\,\Gamma$, note that

$\displaystyle\overline{f_{G}}(K\lambda)$ $\displaystyle=f(\lambda)$ ($f_{G}$
is an extension of $f$)
$\displaystyle=\sum_{\gamma\in\Gamma}\mu_{\Gamma}(\gamma)\,f(\gamma\lambda)$
($f$ is $\mu_{\Gamma}$-harmonic) $\displaystyle=\int_{K\backslash
G}\sum_{\gamma\in\Gamma}\mu_{x}(\gamma)\,f(\gamma\lambda)\,d\mu_{K\backslash
G}(x)$ (definition of $\mu_{\Gamma}$) $\displaystyle=\int_{K\backslash
G}\sum_{\gamma\in\Gamma}\mu_{x\lambda}(\gamma\lambda)\,f(\gamma\lambda)\,d\mu_{K\backslash
G}(x)$ $\displaystyle\begin{pmatrix}\text{definition of~{}$\mu_{x}$, and
the}\\\ \text{random walk~{}$\omega$ is $G$-invariant}\end{pmatrix}$
$\displaystyle=\int_{K\backslash
G}\sum_{\gamma\in\Gamma}\mu_{x\lambda}(\gamma)\,f(\gamma)\,d\mu_{K\backslash
G}(x)$ (change of variables in the sum) $\displaystyle=\int_{K\backslash
G}\overline{f_{G}}(x\lambda)\,d\mu_{K\backslash G}(x)$
$\displaystyle\text{(definition of $\overline{f_{G}}$)}.\qed$

Properties of harmonic functions (including proposition 5.6) play key roles in
the Deroin-Hurtado proof.

######  Example 5.7.

Harmonic functions are the foundation of the construction of the measure
$\mu_{X}$ when $G\neq K\Gamma$. Let $\operatorname{Prob}(Z)$ be the space of
probability measures on $Z$, and define
$f\colon\Gamma\to\operatorname{Prob}(Z)$ by
$f(\gamma)=\gamma_{*}^{-1}\mu_{Z}$. Then $f$ is $\mu_{\Gamma}$-harmonic
(because $\mu_{Z}$ is $\mu_{\Gamma}$-stationary, and $\mu_{\Gamma}$ is
_symmetric_ , i.e., $\mu_{\Gamma}(\gamma^{-1})=\mu_{\Gamma}(\gamma)$). So it
extends to a unique $\mu_{G}$-harmonic function $f_{G}\colon
G\to\operatorname{Prob}(Z)$. Also note that $f$ is right $\Gamma$-equivariant,
in the sense that $f(\gamma\lambda)=\lambda^{-1}_{*}f(\gamma$), so the
uniqueness implies that $f_{G}$ is also right $\Gamma$-equivariant. Hence,
$f_{G}$ represents a section of the bundle
$\bigl{(}G\times\operatorname{Prob}(Z)\bigr{)}/\Gamma\to G/\Gamma$.
Integrating this section with respect to the $G$-invariant probability measure
on $G/\Gamma$ yields a probability measure $\mu_{X}$ on $(G\times
Z)/\Gamma=X$. Since $f_{G}$ is $\mu_{G}$-harmonic (and the measure on
$G/\Gamma$ is $G$-invariant), the measure $\mu_{X}$ is $\mu_{G}$-stationary.

######  Remark 5.8.

Harmonic functions are left $K$-invariant (see lemma 5.2), so the section
$f_{G}$ of 5.7 is constant on $K$. If $X\simeq K\times Z$ (as stated in Note
3.18(2)), this implies that $\mu_{X}=m_{K}\times\mu_{Z}$, which agrees with
3.22.

######  Remark 5.9 (cf. [14, 5.3.2, 5.3.4, and 5.10(3)]).

Here is a slightly different way to look at the construction of $\mu_{X}$.
Define $D\colon\Gamma\times Z\to\mathbb{R}^{+}$ by
$D(\gamma,z)=D^{\text{leaf}}_{\gamma}(z)$. For $\lambda,\gamma\in\Gamma$, we
see from the Chain Rule that

(5.10) $\displaystyle D(\lambda\gamma,z)=D(\gamma,z)\,D(\lambda,\gamma z).$

Since $\mu_{Z}$ is $\mu_{\Gamma}$-stationary (see proposition 4.7), this
implies

$\sum_{\lambda\in\Gamma}\mu_{\Gamma}(\lambda)\,D(\lambda\gamma,z)=D(\gamma,z)\sum_{\lambda\in\Gamma}\mu_{\Gamma}(\lambda)\,D(\lambda,\gamma
z)=D(\gamma,z)\cdot 1=D(\gamma,z).$

This means that the function $\gamma\mapsto D(\gamma,z)$ is
$\mu_{\Gamma}$-harmonic, and therefore extends to a unique $\mu_{G}$-harmonic
function on $G$. Thus, we obtain a function $D_{G}\colon G\times
Z\to\mathbb{R}^{\geq 0}$. (This function is $\mu_{G}$-harmonic in the
$G$-coordinate, and defined for $\mu_{Z}$-a.e. $z$.) Let

$\widetilde{\mu_{X}}=D_{G}\cdot(m_{G}\times\mu_{Z}),$

so $\widetilde{\mu_{X}}$ is a measure on $G\times Z$ that is in the measure
class of $m_{G}\times\mu_{Z}$.

For any $g\in G$, the uniqueness of $D_{G}$ implies that (5.10) still holds if
we replace $D$ with $D_{G}$, and replace $\lambda$ with $g$. Then for
$\gamma\in\Gamma$ (acting on the right via
$(g,z)\mapsto(g\gamma,\gamma^{-1}z)$), we can use the fact that
$d\gamma^{-1}_{*}\mu_{Z}/d\mu_{Z}=D_{G}(\gamma,z)$ (see lemma 3.25) to
conclude that

$\frac{d\gamma_{*}\widetilde{\mu_{X}}}{d\widetilde{\mu_{X}}}(g,z)=\frac{D_{G}(g\gamma^{-1},\gamma
z)}{D_{G}(g,z)}\cdot\frac{d\gamma_{*}m_{G}}{dm_{G}}(g)\cdot\frac{d\gamma^{-1}_{*}\mu_{Z}}{d\mu_{Z}}(z)=\frac{D_{G}(g\gamma^{-1},\gamma
z)}{D_{G}(g,z)}\cdot 1\cdot D_{G}(\gamma,z)=1.$

This means that $\widetilde{\mu_{X}}$ is invariant for the action of $\Gamma$,
so we can construct a well-defined measure $\mu_{X}$ on $X$ from
$\widetilde{\mu_{X}}$, by restricting the quotient map $G\times Z\to X$ to any
fundamental domain of the action. The harmonicity of $D$ in the $G$-coordinate
implies that $\mu_{X}$ is $\mu_{G}$-stationary.

The advantage of this approach is that it shows $\mu_{X}$ is in the same
measure class as $m_{G}\times\mu_{Z}$ (restricted to a fundamental domain).

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