# Effective discreteness radius of stabilisers for stationary actions

T. Gelander and A. Levit and G.A. Margulis

###### Abstract.

We prove an effective variant of the Kazhdan–Margulis theorem generalized to
stationary actions of semisimple groups over local fields: the probability
that the stabilizer of a random point admits a non-trivial intersection with a
small $r$-neighborhood of the identity is at most $\beta r^{\delta}$ for some
explicit constants $\beta,\delta>0$ depending only the group. This is a
consequence of a key convolution inequality. We deduce that vanishing at
infinity of injectivity radius implies finiteness of volume. Further
applications are the compactness of the space of discrete stationary random
subgroups and a novel proof of the fact that all lattices in semisimple groups
are weakly cocompact.

> Dedicated to Gopal Prasad on the occasion of his 75th birthday.

##  1\. Introduction

Let $G$ be a semisimple real Lie group without compact factors. Recall

######  Theorem 1.1 (Kazhdan–Margulis [KM68]).

There exists an identity neighborhood $V\subset G$ such that if $\Gamma$ is a
discrete subgroup of $G$ then $\Gamma^{g}\cap V=\\{e\\}$ for some $g\in G$.

This theorem establishes the existence of a _single_ point $g\Gamma$ in the
probability space $G/\Gamma$ whose stabilizer
$\mathrm{Stab}_{G}(g\Gamma)=\Gamma^{g}$ intersects trivially a given identity
neighborhood. In fact, by [Gel18], the same conclusion holds true for _most_
points in any probability space admitting a measure preserving action of $G$.
Our main results make this statement quantitative, and apply more generally to
stationary measures rather than just invariant one.

Let $K$ be a maximal compact subgroup of the Lie group $G$. Let $\mu$ be a
bi-$K$-invariant probability measure on the group $G$ whose support
$\mathrm{supp}(\mu)$ is generating. Recall that a probability $G$-space
$(Z,\nu)$ is called _$\mu$ -stationary_ if $\mu*\nu=\nu$. In particular any
$G$-invariant probability measure is such.

Fix a left-invariant Riemannian metric on the Lie group $G$ and denote by
$\mathrm{B}_{r}$ the $r$-ball at the identity element of $G$ with respect to
this metric.

######  Theorem 1.2.

There are constants $\beta,\rho,\delta>0$ such that every $\mu$-stationary
probability $G$-space $(Z,\nu)$ with $\nu$-almost everywhere discrete
stabilizers $G_{z}$ satisfies

(1.1) $\nu(\\{z\in
Z\>:\>G_{z}\cap\mathrm{B}_{r}\neq\\{\mathrm{id}_{G}\\}\\})\leq\beta
r^{\delta}\quad\forall r<\rho.$

The constants $\beta$ and $\rho$ depend on the choice of the metric on the Lie
group $G$ but are independent of the fixed probability measure $\mu$ and of
the particular $\mu$-stationary $G$-space $(Z,\nu)$. The constant $\delta$
depends only on the Lie group $G$ and admits the explicit lower bound

(1.2)
$\delta\geq(3\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}G)^{-(\mathrm{rank}(K)+1)}$

where $\mathrm{rank}(K)$ is the rank of the maximal compact subgroup $K$ and
$\mathrm{ht}(\mathfrak{g})$ is the largest height111The family
$\mathrm{A}_{n}$ has largest height $1$. The other classical families
$\mathrm{B}_{n}$, $\mathrm{C}_{n}$, $\mathrm{BC}_{n}$ and $\mathrm{D}_{n}$ all
have largest height $2$. The exceptional semisimple Lie algebras have largest
height $6$ at most. of any positive root in the relative root system
associated to $G$.

The non-effective analogue of Theorem 1.2 is called _weak uniform
discreteness_. It was previously established in [Gel18] for invariant
probability measures by an abstract compactness argument in the space of
invariant random subgroups. Clearly one recovers Theorem 1.1 for any lattice
$\Gamma$ in the Lie group $G$ as a special case of [Gel18, Theorem 1.3] (or of
Theorem 1.2) by applying it to the probability space $G/\Gamma$ with respect
to any sufficiently small radius $r>0$.

We remark that assuming the measure $\nu$ is invariant and non-atomic, if the
Lie group $G$ is simple, or more generally if the $G$-action is irreducible
(i.e. every non-central normal subgroup acts ergodically), then $\nu$-almost
every stabilizer subgroup will be discrete [ABB+17].

While the conclusion of Theorem 1.2 is indeed independent of the choice of the
probability measure $\mu$, our proof relies on the properties of a specific
and carefully chosen such probability measure, see Theorem 1.5 below.

######  Remark 1.3.

The constant $\rho$ appearing in the statement of Theorem 1.2 is strictly
speaking redundant, in the sense that Equation (1.1) holds for _all_ values of
$r>0$ provided that the constant $\beta$ is replaced by
$\max\\{\beta,\rho^{-\delta}\\}$. We prefer to keep our sharper statement as
is, since the constant $\rho$ plays an essential role in the proof.

### The positive characteristic case

Let $k$ be a non-Archimedean local field and $\mathbb{G}$ a connected simply-
connected semisimple $k$-algebraic linear group without $k$-anisotropic
factors. Take $G=\mathbb{G}(k)$ so that $G$ is a $k$-analytic group.

If $\textrm{char}(k)=0$ then $G$ has no small discrete subgroups [Ser09, Part
II, Chapter V.9, Theorem 5], namely there is a compact open subgroup $U\leq G$
such that every discrete subgroup $\Gamma$ satisfies $\Gamma\cap
U=\\{\mathrm{id}_{G}\\}$. This fact is already much stronger than any
potential analogue of Theorems 1.1 or 1.2.

Assume therefore that $\textrm{char}(k)$ is positive and is a good
prime222Good primes are discussed e.g. in [SS70, §4]. The only primes which
fail to be good for some semisimple group $\mathbb{G}$ are $2,3$ and $5$. for
$\mathbb{G}$. Our main result naturally generalizes to the $k$-analytic group
$G$, answering in the positive [Gel18, Question 4.3] and extending Theorem 1.1
to this setting.

We now state our main result in the positive characteristic case. Let
$\mathcal{O}$ be the ring of integers of the non-Archimedean local field $k$
and $\mathfrak{m}$ be the maximal ideal of $\mathcal{O}$. Let
$\mathbb{G}(\mathfrak{m}^{i})$ denote the congruence subgroup333The exact
definition of each congruence subgroup $G(\mathfrak{m}^{i})$ depends (up to
finite index) on the particular matrix realization of the $k$-algebraic group
$\mathbb{G}$. of the compact group $\mathbb{G}(\mathcal{O})$ modulo the ideal
$\mathfrak{m}^{i}$ for every $i\in\mathbb{N}$. The subgroups
$\mathbb{G}(\mathfrak{m}^{i})$ form a basis of identity neighborhoods for the
topology of $G$.

Let $\mu$ be a bi-$\mathbb{G}(\mathcal{O})$-invariant probability measure on
the $k$-analytic group $G$ such that the support of $\mu$ generates the group
$G$.

######  Theorem 1.4.

There are constants $\beta,\rho,\delta>0$ such that every $\mu$-stationary
probability $G$-space $(Z,\nu)$ with $\nu$-almost everywhere discrete
stabilizers $G_{z}$ satisfies

(1.3) $\nu(\\{z\in
Z\>:\>G_{z}\cap\mathbb{G}(\mathfrak{m}^{i})\neq\\{\mathrm{id}_{G}\\}\\})\leq\beta\left|\nicefrac{{\mathcal{O}}}{{\mathfrak{m}}}\right|^{-\delta
i}$

for all $i\in\mathbb{N}$ with
$\left|\nicefrac{{\mathcal{O}}}{{\mathfrak{m}}}\right|^{-i}<\rho$.

The constants $\beta,\rho$ and $\delta$ are independent of the particular
probability measure $\mu$ as well as of the $\mu$-stationary probability
$G$-space $(Z,\nu)$. The constant $\delta$ admits the explicit lower bound

(1.4) $\delta\geq\frac{1}{\mathrm{ht}(\mathfrak{g})^{2}\dim^{5}_{k}G}.$

Alternatively, one may consider the norm $\|\cdot\|_{\mathfrak{m}}$ on the
group $\mathbb{G}(\mathfrak{m})$ given by

(1.5)
$\|g\|_{\mathfrak{m}}=\inf\>\\{\left|\nicefrac{{\mathcal{O}}}{{\mathfrak{m}}}\right|^{-i}\>:\>\text{$i\in\mathbb{N}$
and $g\in\mathbb{G}(\mathfrak{m}^{i})$}\\}$

for all elements $g\in\mathbb{G}(\mathfrak{m})$ and reformulate Equation (1.3)
in terms of $r$-balls with respect to $\|\cdot\|_{\mathfrak{m}}$ in a manner
analogous to Equation (1.1).

Just as in the Archimedean case, if $(Z,\nu)$ is a non-atomic irreducible
probability measure preserving $G$-space then $\nu$-almost every stabilizer
subgroup is discrete [GL18, Theorem 1.9].

We remark that while an analog of the Kazhdan–Margulis theorem over positive
characteristic local fields was proved for products of rank-one groups in
[Rag89, Lub90, Lub91] and for simply connected Chevalley groups of any rank in
[Gol09], the general case has not been available in the literature until now.

### The key inequality

A fundamental part of this work is the inequality stated in Theorem 1.5 below.
The above results Theorems 1.2 and 1.4 are both derived in a rather
straightforward manner from this inequality, which is of independent interest
and admits various other applications.

To fix notations, let $k$ be a local field and $\mathbb{G}$ a connected
simply-connected semisimple $k$-algebraic linear group. Denote
$G=\mathbb{G}(k)$ so that $G$ is $k$-analytic. Let $K$ be a maximal compact
subgroup of $G$ (if $k$ is non-Archimedean assume moreover the subgroup $K$ is
good).

Let $\mu_{s}$ be the following bi-$K$-invariant probability measure on the
$k$-analytic group $G$

(1.6) $\mu_{s}=\eta_{K}*\delta_{s}*\eta_{K}$

where $\eta_{K}$ is the normalized Haar probability measure of the compact
group $K$ and $\delta_{s}$ is an atomic probability measure supported on some
sufficiently expanding regular semisimple element $s\in G$, see §8 for
details. Note that $\mu_{s}$ is compactly supported.

To state the key inequality we consider the _discreteness radius_ function
$\mathcal{I}_{G}(\Gamma)$ defined for every discrete subgroup $\Gamma$ of the
$k$-analytic group $G$ by

(1.7) $\mathcal{I}_{G}(\Gamma)=\sup\\{0\leq
r\leq\rho\>:\>\Gamma\cap\mathrm{B}_{r}=\\{\mathrm{id}_{G}\\}\\}$

for some constant $\rho>0$ (determined in §7). Here
$\mathrm{B}_{r}=\exp(\\{X\in\mathfrak{g}:\|X\|\leq r\\})$ with respect to some
norm $\|\cdot\|$ on the Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$ in the
Archimedean case and
$\mathrm{B}_{r}=\\{g\in\mathbb{G}(\mathfrak{m})\>:\|g\|_{\mathfrak{m}}\leq
r\\}$ in the non-Archimedean case.

######  Theorem 1.5 (The key inequality).

There are constants $0<c<1$ and $b,\delta>0$ such that every discrete subgroup
$\Gamma$ of the group $G$ satisfies

(1.8) $\int_{G}\mathcal{I}_{G}^{-\delta}(\Gamma^{g})\>\mathrm{d}\mu_{s}(g)\leq
c\mathcal{I}_{G}^{-\delta}(\Gamma)+b.$

The constant $\delta$ depends only on the analytic group $G$ in question.
Moreover $\delta$ and $\rho$ coincide with the eponymous constants appearing
in Theorems 1.2 and 1.4.

### The space of discrete stationary random subgroups is compact

We consider the space $\mathrm{Sub}\left(G\right)$ of all closed subgroups of
the group $G$ equipped with the Chabauty topology where $G$ is a $k$-analytic
group as above. This is a compact space on which $G$ acts continuously by
conjugation. Let $\mathrm{Sub}_{\text{d}}\left(G\right)$ be the subspace444If
the local field $k$ is Archimedean then the subspace
$\mathrm{Sub}_{\text{d}}\left(G\right)$ is Chabauty open. of
$\mathrm{Sub}\left(G\right)$ consisting of discrete subgroups. It is
$G$-invariant but no longer compact. A _discrete random subgroup_ of $G$ is a
Borel probability measure $\nu$ on the Chabauty space
$\mathrm{Sub}\left(G\right)$ satisfying
$\nu(\mathrm{Sub}_{\text{d}}\left(G\right))=1$.

Any probability $G$-space $(Z,\nu)$ with $\nu$-almost surely discrete
stabilizers determines a discrete random subgroup of $G$ by pushing forward
the measure $\nu$ via the stabilizer map $Z\to\text{Sub}(G),~{}z\mapsto
G_{z}$. Vice versa, every discrete random subgroup of $G$ can be realized in
this way with respect to some probability $G$-space555See [ABB+17, Theorem
2.6] for a proof of the analogue fact for invariant random subgroups. The same
proof applies to any locally compact group $G$ and any measure $\nu$ on
$\mathrm{Sub}\left(G\right)$ as long as $\nu$-almost every subgroup is
unimodular..

This alternative view point allows Theorems 1.2, 1.4 and 1.5 to be
reformulated as results about discrete stationary random subgroups. In
particular, the weak uniform discreteness of the discrete stationary random
subgroups can be stated as a compactness result.

######  Corollary 1.6.

Let $\mu$ be a bi-$K$-invariant probability measure on the group $G$. Then the
set of $\mu$-stationary probability measures on the space
$\mathrm{Sub}_{\text{d}}\left(G\right)$ is compact in the weak-$*$ topology.

We remark that in the Archimedean case it is possible to go in the opposite
direction and deduce weak uniform discreteness from the weak-$*$ compactness
of the set of discrete invariant random subgroups, see [Gel18].

### Evanescence and finiteness of volume

Let $M$ be a Riemannian manifold of non-positive sectional curvature. The
following notion is introduced in [BGS13]: the manifold $M$ has
$\text{InjRad}\to 0$ (hereafter _evanescent_) if its injectivity radius
$\textrm{Inj-Rad}_{M}$ vanishes at infinity. In other words $M$ is evanescent
if and only if its $\varepsilon$-thick part $M_{\geq\varepsilon}=\\{x\in
M\,:\,\textrm{Ind-Rad}_{M}(x)\geq\varepsilon\\}$ is compact for every
$\varepsilon>0$. This is equivalent to the finiteness of the
$\varepsilon$-essential volume666We use $\varepsilon$-essential volume in the
sense of [BGS13, §12.B]. for every $\varepsilon>0$.

Gromov proved that an evanescent analytic manifold of bounded non-positive
sectional curvature is diffeomorphic to the interior of a compact manifold
with boundary [BGS13, Theorem 2].

It is clear that finite volume implies evanescence. The converse of this fact
for locally symmetric spaces is a consequence of our key inequality (Theorem
1.5).

######  Theorem 1.7.

A locally symmetric manifold is evanescent if and only if it has finite
volume.

### Weak cocompactness of lattices

A lattice $\Gamma$ in a locally compact group $H$ is _weakly cocompact_ if the
quasiregular unitary representation of $H$ in the Hilbert space
$L^{2}_{0}(H/\Gamma)$ does not admit almost invariant vectors, or in other
words the $H$-space $H/\Gamma$ has spectral gap.

Cocompact lattices are weakly cocompact [Mar91, III.1.8]. It was asked in
[Mar91, III.1.12] whether all lattices are weakly cocompact. This is true for
Lie groups [Bek98, BC10], simple algebraic groups over non-Archimedean local
fields [BL11, Theorem 1] and groups with Kazhdan’s property (T).

Some locally compact groups however do admit non-weakly cocompact lattices. A
first explicit example in the group of tree automorphisms was constructed in
[BL11, Theorem 2]. A lattice in an amenable second countable locally compact
group is cocompact if and only if it is weakly cocompact [BCGM19, Proposition
1.8]. Thus, the non-uniform lattices constructed in [BCGM19, Theorem 1.11]
provide further non-weakly cocompact examples.

We obtain the following result relying on the key inequality (Theorem 1.5) and
on the methods developed towards its proof.

######  Theorem 1.8.

Let $k$ be a local field. Let $\mathbb{G}$ be a connected simply-connected
semisimple $k$-algebraic group without $k$-anisotropic factors. Then every
lattice in the locally compact group $\mathbb{G}(k)$ is weakly cocompact.

The conclusion of Theorem 1.8 has been previously known in most (but not all)
cases. Its novelty lies in obtaining a unified and a direct proof.

Interestingly Theorem 1.8 is deduced in a rather straightforward manner from a
stronger and a more general statement, namely Theorem 9.3 below. Roughly
speaking, this last theorem says that the norm of a convolution operator
restricted to the ”thin part” of any probability measure preserving action of
the group $\mathbb{G}(k)$ is bounded away from one uniformly over all such
actions. This result is novel even in the presence of Kazhdan’s property (T).

##  2\. Averaging operators

Let $G$ be a locally compact second countable group and $\mu$ a probability
measure on $G$. Let $Z$ be a locally compact topological space admitting a
continuous action of the group $G$.

We study the averaging operator $A_{\mu}$ acting on continuous functions on
the space $Z$. This operator is used to bound the $\nu$-measure of super-level
sets of continuous functions satisfying a certain key inequality, where $\nu$
is any $\mu$-stationary measure on the space $Z$.

The current discussion uses ideas from [EMM98, Mar04]. See also the works
[EM01, EM02, Ath06, EK09, MM10, BGHM10, BQ12, KLKM14, EMM15, HLMK16, MO20] for
related ideas and further applications.

### Bounds on super-level sets

Consider the averaging operator $A_{\mu}$ corresponding to the probability
measure $\mu$. That is, given any continuous function
$F:Z\to\left[0,\infty\right]$ we define the function
$A_{\mu}F:Z\to\left[0,\infty\right]$ by

(2.1) $A_{\mu}F(z)=\int_{G}F(gz)\,\mathrm{d}\mu(g)\quad\forall z\in Z.$

Let $\nu$ be a $\mu$-stationary Borel probability measure on $Z$.

######  Lemma 2.1.

Assume that the space $Z$ is $\sigma$-compact. Let $F:Z\to(0,\infty)$ be a
continuous function. If there are constants $0<c<1$ and $b>0$ such that

(2.2) $A_{\mu}F(z)\leq cF(z)+b$

then

(2.3) $\nu(\\{z\in Z\>:\>F(z)\geq M\\})\leq\frac{b}{(1-c)M}\quad\forall M>0.$

###### Proof.

Consider the sequence of probability measures
$\mu_{m}=\frac{1}{m}\sum_{i=1}^{m}\mu^{*i}$ on the group $G$ for all
$m\in\mathbb{N}$. Note that the probability measure $\nu$ is
$\mu_{m}$-stationary for all $m\in\mathbb{N}$, namely $\mu_{m}*\nu=\nu$.

Since the space $Z$ is assumed to be $\sigma$-compact it is possible to write
$Z=\bigcup_{n\in\mathbb{N}}Z_{n}$ where each $Z_{n}$ is a compact subspace and
$Z_{n}\subset Z_{n+1}$ for all $n\in\mathbb{N}$. Fix an index $n\in\mathbb{N}$
and denote $\nu_{n}=\nu_{|Z_{n}}$. Decompose the probability measure $\nu$ as

(2.4) $\nu=\mu_{m}*\nu=\mu_{m}*\nu_{n}+\mu_{m}*(\nu-\nu_{n}).$

Up to passing twice to a subsequence, we may assume that
$\mu_{m}*\nu_{n}\xrightarrow{m\to\infty}\nu_{n}^{\infty}$ and
$\mu_{m}*(\nu-\nu_{n})\xrightarrow{m\to\infty}\eta_{n}^{\infty}$ in the
weak-$*$ topology for some positive $\mu$-stationary measures
$\nu_{n}^{\infty}$ and $\eta_{n}^{\infty}$ on the space $Z$. Generally
speaking, the Portmanteau theorem says that
$\nu_{n}^{\infty}(Z)\leq\nu_{n}(Z)$ and
$\eta_{n}^{\infty}(Z)\leq(\nu-\nu_{n})(Z)=1-\nu_{n}(Z)$. Note however that
passing to the limit in Equation (2.4) gives
$\nu=\nu_{n}^{\infty}+\eta_{n}^{\infty}$. Therefore it must be the case that
$\nu_{n}^{\infty}(Z)=\nu_{n}(Z)$ and $\eta_{n}^{\infty}(Z)=1-\nu_{n}(Z)$. In
particular

(2.5)
$\inf_{n}\eta_{n}^{\infty}(Z)=1-\sup_{n}\nu_{n}^{\infty}(Z)=1-\sup_{n}\nu_{n}(Z)=0.$

Clearly $\sup_{Z_{n}}F<\infty$ as the subspace $Z_{n}$ is compact and the
function $F$ is continuous. Iterating Equation (2.2) gives

(2.6)
$A_{\mu}^{i}F(z)\leq\frac{b}{1-c}+c^{i}F(z)\leq\frac{b}{1-c}+\sup_{Z_{n}}F\cdot
c^{i}$

for all points $z\in Z_{n}$. Therefore

(2.7)
$\displaystyle\begin{split}\int_{Z}F\mathrm{d}\nu^{\infty}_{n}&=\lim_{m\to\infty}\int_{Z}F\;\mathrm{d}(\mu_{m}*\nu_{n})=\lim_{m\to\infty}\frac{1}{m}\sum_{i=1}^{m}\int_{Z}A_{\mu}^{i}F\;\mathrm{d}\nu_{n}\leq\\\
&\leq\frac{b}{1-c}+\sup_{Z_{n}}F\cdot\lim_{m\to\infty}\frac{1}{m}\sum_{i=1}^{m}c^{i}=\frac{b}{1-c}.\end{split}$

Markov’s inequality applied with respect to the measure $\nu_{n}^{\infty}$ for
each $n\in\mathbb{N}$ and combined with Equation (2.5) implies that

(2.8) $\begin{split}\nu(\\{z\in Z\>:\>F(z)\geq
M\\})=\inf_{n}\;(\nu_{n}^{\infty}+\eta_{n}^{\infty})(\\{z\in Z\>:\>F(z)\geq
M\\})\leq\\\
\leq\frac{b}{(1-c)M}+\inf_{n}\eta_{n}^{\infty}(Z)=\frac{b}{(1-c)M}\end{split}$

for all $M>0$ as required. ∎

######  Remark.

It is possible to prove an analog of Lemma 2.1 in the measurable setting and
without assuming the continuity of the function $F$. This alternative relies
on the von Neumann ergodic theorem and requires the averaging operator
$A_{\mu}$ to admit no non-trivial invariant vectors.

### From expansion to contraction

Let $f:Z\to\left(0,\infty\right)$ be a non-negative continuous function. Our
goal is to show that, under suitable assumptions, if $f$ is pointwise expanded
by the averaging operator $A_{\mu}$ on some subset of large measure, then the
function $F=f^{-\delta}$ is pointwise contracted by $A_{\mu}$ for some
sufficiently small exponent $\delta>0$. We begin with an elementary
calculation.

######  Calculation 2.2.

Let $a_{1}>1>a_{2}>0$ and $0<p<1$. The real function $\varphi$ given by

(2.9) $\varphi(\delta)=pa_{1}^{-\delta}+(1-p)a_{2}^{-\delta}$

has a global minimum point at

(2.10)
$\delta_{0}=\delta_{0}(a_{1},a_{2};p)=-\frac{\ln\left(-\frac{1-p}{p}\frac{\ln
a_{2}}{\ln a_{1}}\right)}{\ln\left(\frac{a_{1}}{a_{2}}\right)}.$

Moreover if

(2.11) $(1-p)\ln\frac{1}{a_{2}}<p\ln a_{1}$

then $\delta_{0}>0$ and $\varphi(\delta_{0})<1$.

###### Proof.

Note that $\varphi(0)=1$ and
$\lim_{\delta\to\pm\infty}\varphi(\delta)=+\infty$. The derivative
$\varphi^{\prime}$ of the function $\varphi$ is given by

(2.12) $\varphi^{\prime}(\delta)=-p\ln a_{1}\cdot a_{1}^{-\delta}-(1-p)\ln
a_{2}\cdot a_{2}^{-\delta}.$

A real number $\delta_{0}$ is a critical point of the function $\varphi$ if
and only if $\varphi^{\prime}(\delta_{0})=0$. This condition is equivalent to

(2.13) $\left(\frac{a_{1}}{a_{2}}\right)^{-\delta_{0}}=-\frac{1-p}{p}\frac{\ln
a_{2}}{\ln a_{1}}>0.$

Therefore $\varphi$ has a unique critical point
$\delta_{0}=\delta(a_{1},a_{2};p)$ whose exact value is given by Equation
(2.10). As $a_{1}/a_{2}>1$ this critical point satisfies $\delta_{0}>0$ if and
only if

$-\frac{1-p}{p}\frac{\ln a_{2}}{\ln a_{1}}<1\quad\Leftrightarrow\quad-(1-p)\ln
a_{2}<p\ln a_{1}.$

Moreover, in this case $\varphi(\delta_{0})<\varphi(0)=1$. ∎

######  Proposition 2.3.

Let $a_{1}>1>a_{2}>0$ and $0<p<1$ be such that Equation (2.11) holds. Let
$\rho_{0}>0$ be such that every point $z\in Z$ satisfies

  1. (1)

if $f(z)<\rho_{0}$ then the subset

$L_{z}=\\{g\in G\>:\>f(gz)\geq a_{1}f(z)\\}$

satisfies $\mu(L_{z})\geq p$ and

  2. (2)

$f(gz)\geq a_{2}f(z)$ holds for $\mu$-almost every element $g\in G$.

Then there are constants $0<c<1$ and $b>0$ such that

(2.14) $A_{\mu}f^{-\delta}\leq cf^{-\delta}+b$

where $\delta=\delta_{0}(a_{1},a_{2};p)$ is as given by Equation (2.10).

###### Proof.

Let $\delta=\delta_{0}(a_{1},a_{2};p)$ be as in Equation (2.10). Take the
constant $c$ to be

$c=pa_{1}^{-\delta}+(1-p)a_{2}^{-\delta}.$

According to Calculation 2.2 this constant satisfies $0<c<1$.

Consider the partition of the topological space $Z$ into a pair of sublevel
and superlevel sets

$Z=Z_{<\rho_{0}}\amalg Z_{\geq\rho_{0}}$

where

$Z_{<\rho_{0}}=\\{z\in Z\>:\>f(z)<\rho_{0}\\}\quad\text{and}\quad
Z_{\geq\rho_{0}}=\\{z\in Z\>:\>f(z)\geq\rho_{0}\\}.$

The subsets $Z_{<\rho_{0}}$ and $Z_{\geq\rho_{0}}$ are respectively open and
closed in the space $Z$.

On the one hand, take the constant $b=(a_{2}\rho_{0})^{-\delta}$ and observe
that every point $z\in Z_{\geq\rho_{0}}$ and $\mu$-almost every element $g\in
G$ satisfy

$f(gz)\geq a_{2}f(z)\quad\Rightarrow\quad
f^{-\delta}(gz)\leq(a_{2}f(z))^{-\delta}\leq(a_{2}\rho_{0})^{-\delta}=b.$

On the other hand, observe that every point $z\in Z_{<\rho_{0}}$ satisfies

$\displaystyle A_{\mu}f^{-\delta}(z)$
$\displaystyle=\int_{G}f^{-\delta}(gz)\;\mathrm{d}\mu(g)=$
$\displaystyle=\int_{L_{z}}f^{-\delta}(gz)\;\mathrm{d}\mu(g)+\int_{G\setminus
L_{z}}f^{-\delta}(gz)\;\mathrm{d}\mu(g)\leq$ $\displaystyle\leq
pa_{1}^{-\delta}f^{-\delta}(z)+(1-p)a_{2}^{-\delta}f^{-\delta}(z)=cf^{-\delta}(z).$

We conclude that the two inequalities $A_{\mu}f^{-\delta}\leq b$ and
$A_{\mu}f^{-\delta}\leq cf^{-\delta}$ hold true on the two subsets
$Z_{\geq\rho_{0}}$ and $Z_{<\rho_{0}}$ respectively. Since $f^{-\delta}\geq 0$
it follows that the inequality given in Equation (2.14) holds at every point
$z\in Z$. ∎

### The limiting behavior of $\delta$

We estimate the asymptotic behavior of the function
$\delta_{0}(a_{1},a_{2};p)$ within a certain regime of the parameters
$a_{1},a_{2}$ and $p$ that will be used below, see e.g. Proposition 6.12 and
Probability 7.3. This is needed in order to compute the explicit value of the
constant $\delta$ appearing in our main results.

######  Calculation 2.4.

Let $h,\alpha,\zeta,a_{0}>0$ as well as $a_{1}>1$ be some fixed constants. For
all $\lambda>1$ denote

(2.15) $a_{2,\lambda}=a_{0}\lambda^{-h}\quad\text{and}\quad
p_{\lambda}=1-\zeta\lambda^{-\alpha}.$

Then for all sufficiently large $\lambda>1$ we have that Equation (2.11) holds
and the critical point

(2.16) $\delta_{\lambda}=\delta_{0}(a_{1},a_{2,\lambda};p_{\lambda})$

determined by Equation (2.10) satisfies $\delta_{\lambda}>0$. Moreover

(2.17) $\lim_{\lambda\to+\infty}\delta_{\lambda}=\frac{\alpha}{h}.$

###### Proof.

Recall that Equation (2.11) requires

(2.18) $(1-p_{\lambda})\ln\frac{1}{a_{2,\lambda}}<p_{\lambda}\ln a_{1}.$

Consider the behavior of Equation (2.18) in the limit as $\lambda\to+\infty$.
Note that

$\lim_{\lambda\to+\infty}p_{\lambda}\ln a_{1}=\ln a_{1}.$

On the other hand

$\lim_{\lambda\to\infty}(1-p_{\lambda})\ln(1/a_{2,\lambda})=\lim_{\lambda\to\infty}\zeta\lambda^{-\alpha}\ln(a_{0}\lambda^{h})=0.$

Therefore the inequality given in Equation (2.18) is satisfied and
$a_{2,\lambda}<1$ for all $\lambda$ sufficiently large. As shown in
Calculation 2.2, it follows that

(2.19)
$\delta_{\lambda}=-\ln_{\left(\frac{a_{1}}{a_{2,\lambda}}\right)}\left(-\frac{1-p_{\lambda}}{p_{\lambda}}\frac{\ln
a_{2,\lambda}}{\ln a_{1}}\right)>0$

for all $\lambda$ sufficiently large. Lastly, the following computation

(2.20)
$\lim_{\lambda\to\infty}\delta_{\lambda}=\lim_{\lambda\to\infty}-\frac{\ln\left(-\frac{1-p_{\lambda}}{p_{\lambda}}\frac{\ln
a_{2,\lambda}}{\ln
a_{1}}\right)}{\ln(\frac{a_{1}}{a_{2,\lambda}})}=\lim_{\lambda\to\infty}-\frac{\ln\left(-\frac{\zeta\lambda^{-\alpha}\ln(a_{0}\lambda^{-h})}{\ln
a_{1}}\right)}{\ln\left(a_{1}a_{0}^{-1}\lambda^{h}\right)}=\frac{\alpha}{h}$

establishes Equation (2.17). ∎

##  3\. The order of an analytic function on a manifold

Let $k$ be a local field. Consider a $k$-analytic777 We refer the reader to
[Bou07] and [Ser09, Part II] for basic information on $k$-analytic functions
and manifolds. manifold $M$ of dimension $d\in\mathbb{N}$. Let
$\mathcal{A}(M,k^{m})$ denote the $k$-vector space of all $k$-analytic maps
$f:M\to k^{m}$ for each dimension $m\in\mathbb{N}$.

The _order_ $\mathrm{ord}_{p}f$ of any given $k$-analytic function
$f\in\mathcal{A}(M,k)$ at the point $p\in M$ is defined as follows. Fix an
arbitrary local chart $\varphi:U\to V$ where $p\in U\subset M$ and $0\in
V\subset k^{d}$ are open subsets such that $\varphi(p)=0$. The $k$-analytic
function $f$ can be written in local coordinates as

$(f\circ\varphi^{-1})(x)=\sum_{\beta}c_{\beta}x^{\beta}\quad\forall x\in V$

where the $\beta$’s are multi-indices and the coefficients $c_{\beta}$ belong
to the local field $k$. Then

$\mathrm{ord}_{p}f=\inf_{\beta}\\{|\beta|\>:\>c_{\beta}\neq 0\\}.$

If the function $f$ is identically zero on a neighborhood of the point $p$
then all the coefficients $c_{\beta}$ are zero and we set
$\mathrm{ord}_{p}f=\infty$. The fact that transition maps are $k$-analytic
isomorphisms implies that the definition of $\mathrm{ord}_{p}f$ is independent
of the local chart.

The _order_ of any $k$-analytic function
$f=(f_{1},\ldots,f_{m})\in\mathcal{A}(M,k^{m})$ is

(3.1)
$\mathrm{ord}_{p}(f)=\inf_{i\in\\{1,\ldots,m\\}}\mathrm{ord}_{p}(f_{i}).$

Let $\mathcal{F}\subset\mathcal{A}(M,k^{m})$ be a family of $k$-analytic
functions. The _order of the family_ $\mathcal{F}$ at the point $p\in M$ is

(3.2) $\mathrm{ord}_{p}\mathcal{F}=\sup_{f\in\mathcal{F}}\mathrm{ord}_{p}f.$

Moreover denote

(3.3) $\mathrm{ord}_{M}\mathcal{F}=\sup_{p\in M}\mathrm{ord}_{p}\mathcal{F}.$

######  Remark.

The order of the product $fg$ of any pair of $k$-analytic functions
$f,g\in\mathcal{A}(M,k)$ satisfies

(3.4)
$\mathrm{ord}_{p}(fg)=\mathrm{ord}_{p}(f)+\mathrm{ord}_{p}(g)\quad\forall p\in
M.$

######  Lemma 3.1.

Let $N$ be an embedded $k$-analytic submanifold of $M$. Then every
$k$-analytic function $f\in\mathcal{A}(M,k^{m})$ satisfies
$\mathrm{ord}_{p}f\leq\mathrm{ord}_{p}{f_{|N}}$ for all points $p\in N$.

###### Proof.

Denote $e=\dim_{k}N$. Consider an arbitrary point $p\in N$. By assumption
there is an relative neighborhood $p\in U_{1}\subset N$ and a local chart
$\varphi_{1}:U_{1}\xrightarrow{\cong}V_{1}\subset k^{e}$ with
$\varphi_{1}(p)=0$ such that the power series expressing the function
$f\circ\varphi_{1}^{-1}$ at the point $0$ admits a non-zero monomial of degree
$\mathrm{ord}_{p}f_{|N}$. The $k$-vector space $k^{d}$ admits a direct sum
decomposition $k^{d}=k^{e}\oplus k^{d-e}$ such that, up to replacing $U_{1}$
and $V_{1}$ by suitable smaller neighborhoods, there is an open neighborhood
$p\in U\subset M$ and a local chart $\varphi:U\to k^{d}$ with $\varphi(p)=0$,
$\varphi(U)=V_{1}\times V_{2}$ for some subset $V_{2}\subset k^{d-e}$ so that
$U_{1}\subset U$ and $\varphi_{|U_{1}}=\varphi_{1}$ [Ser09, Part II, Chapter
III, §10.1]. It is clear that in this situation
$\mathrm{ord}_{p}f\leq\mathrm{ord}_{p}f_{|N}$. As the point $p\in N$ was
arbitrary the conclusion follows. ∎

### Strong order

In working with matrix coefficients of unipotent groups we will find it
convenient to use the following sharper notion.

######  Definition 3.2.

The $k$-analytic function $f\in\mathcal{A}(M,k^{m})$ has _strong order_ at
most $d\in\mathbb{N}$ at the point $p\in M$ is there exists an embedded
$k$-analytic submanifold $N$ admitting a local chart $\varphi:U\to V$ where
$p\in U\subset N$ and $V\subset k^{\dim_{k}N}$ are open subsets and such that
$f\circ\varphi^{-1}:V\to k^{m}$ is a non-zero _polynomial_ mapping of degree
at most $d$.

Lemma 3.1 implies that strong order gives an upper bound on order. Namely the
following is true.

######  Lemma 3.3.

If $f\in\mathcal{A}(M,k^{m})$ has strong order at most $d\in\mathbb{N}$ at the
point $p\in M$ then $\mathrm{ord}_{p}f\leq d$.

### Order of matrix coefficients of compact Lie groups

Let $V$ be a real finite dimensional inner product space. Let $K$ be a compact
connected subgroup of the orthogonal group $\mathrm{O}(V)$. In particular $K$
is a real Lie group. Let $T$ be a maximal compact torus of the group $K$ with
real Lie algebra $\mathfrak{t}=\mathrm{Lie}(T)$.

######  Lemma 3.4.

Let $\Omega\subset i\mathfrak{t}^{*}$ be a finite set of analytically integral
forms. Let $c_{\omega}\in\mathbb{C}$ be an arbitrary coefficient for each
$\omega\in\Omega$. Consider the real analytic function
$\zeta\in\mathcal{A}(T,\mathbb{C})$ given by

(3.5)
$\zeta(\exp(X))=\sum_{\omega\in\Omega}c_{\omega}e^{\omega(X)}\quad\forall
X\in\mathfrak{t}.$

If the function $\zeta$ is not identically zero on the torus $T$ then
$\mathrm{ord}_{T}\zeta<|\Omega|$.

The function $\zeta$ is defined on the real Lie group $T$ and is regarded as a
real analytic mapping into the two-dimensional real vector space
$\mathbb{C}\cong\mathbb{R}^{2}$.

###### Proof of Lemma 3.4.

Fix an arbitrary element $t_{0}\in T$ with $t_{0}=\exp(X_{0})$ for some
$X_{0}\in\mathfrak{t}$. We will compute the order of the function $\zeta$ at
the point $t_{0}$ and show that $\mathrm{ord}_{t_{0}}\zeta<|\Omega|$.

Choose neighborhoods $t_{0}\in U\subset T$ and $0\in V\subset\mathfrak{t}$
such that the mapping $\varphi:U\xrightarrow{\cong}V$ given by
$\varphi:\exp(X_{0}+X)\mapsto X$ for all $X\in V$ is a local chart. For a
given element $t\in U$ with $\varphi(t)=X\in V$ write

$\zeta(t)=\zeta(\exp(X_{0}+X))=\sum_{\omega\in\Omega}c_{\omega}e^{\omega(X_{0})}\sum_{l=0}^{\infty}\frac{\omega(X)^{l}}{l!}=\sum_{l=0}^{\infty}\frac{1}{l!}\sum_{\omega\in\Omega}\bar{c}_{\omega}\omega(X)^{l}$

where $\bar{c}_{\omega}=c_{\omega}e^{\omega(X_{0})}$. The part $\zeta_{0}$ of
the real analytic function $\zeta$ consisting of all monomials with multi-
index $\beta$ satisfying $|\beta|<|\Omega|$ equals

$\zeta_{0}(t)=\sum_{l=0}^{|\Omega|-1}\frac{1}{l!}\sum_{\omega\in\Omega}\bar{c}_{\omega}\omega(X)^{l}$

for each given element $t=\exp(X_{0}+X)\in U$ as above.

The subspace $\ker(\omega_{1}-\omega_{2})$ of the real Lie algebra
$\mathfrak{t}$ is proper for each pair of distinct forms
$\omega_{1},\omega_{2}\in\Omega$. Therefore there is an element $Y\in V$ such
that $\omega_{1}(Y)\neq\omega_{2}(Y)$ for all such pairs of distinct indices.
The Vandermonde determinant formula gives

$\det\left(\frac{\omega(Y)^{l}}{l!}\right)_{\begin{subarray}{c}\omega\in\Omega\\\
l\in\\{0,\ldots,|\Omega|-1\\}\end{subarray}}=\prod_{l=0}^{|\Omega|-1}\frac{1}{l!}\cdot\prod_{\begin{subarray}{c}\omega_{1},\omega_{2}\in\Omega\\\
\omega_{1}\neq\omega_{2}\end{subarray}}\left(\omega_{1}(Y)-\omega_{2}(Y)\right)\neq
0.$

The non-vanishing of the above determinant implies $\zeta_{0}(Y)\neq 0$ so
that $\mathrm{ord}_{t_{0}}\zeta<|\Omega|$ as required. As the point $t_{0}\in
T$ in the above argument was arbitrary we conclude that
$\mathrm{ord}_{T}f<|\Omega|$. ∎

Consider the Hermitian space $V_{\mathbb{C}}=V\otimes\mathbb{C}$ which is the
complexification of the inner product space $V$. The maximal compact torus $T$
preserves the Hermitian form on $V_{\mathbb{C}}$. Let $\Omega=\Omega(T,V)$ be
the weights of the torus $T$ in its representation on the Hermitian space
$V_{\mathbb{C}}$. Every weight $\omega\in\Omega$ is an analytic integral form
belonging to $i\mathfrak{t}^{*}$. Let

$V_{\mathbb{C}}=\bigoplus_{\omega\in\Omega}W_{\omega}$

be the associated weight space decomposition satisfying

(3.6) $\exp(X)w=e^{\omega(X)}w\quad\forall\omega\in\Omega,w\in
W_{\omega},X\in\mathfrak{t}.$

######  Lemma 3.5.

Consider the matrix coefficient $\zeta_{u,v}\in\mathcal{A}(K,\mathbb{R})$
given by

$\zeta_{u,v}:K\to V,\quad\zeta_{u,v}(g)=\left<gu,v\right>\quad\forall g\in K.$

for some pair of vectors $u,v\in V$. If $\zeta_{u,v}$ is not identically zero
on the group $K$ then
$\mathrm{ord}_{K}\zeta_{u,v}<|\Omega(T,V)|\leq\mathrm{dim}_{\mathbb{R}}V$.

###### Proof.

Every element of the compact connected Lie group $K$ belongs to some maximal
torus and all such tori are conjugate [Kna13, Corollary 4.35, Theorem 4.36].
Up to replacing the vectors $u$ and $v$ by $gu$ and $gv$ for some element
$g\in K$, we may assume without loss of generality that the restriction of the
matrix coefficient $\zeta_{u,v}$ to the maximal torus $T$ is not identically
zero.

Denote $\Omega=\Omega(T,V)$ and consider the weight space decompositions
$u=\bigoplus_{\omega\in\Omega}u_{\omega}$ and
$v=\bigoplus_{\omega\in\Omega}v_{\omega}$ where $u_{\omega},v_{\omega}\in
W_{\omega}$. The matrix coefficient $\zeta_{u,v}$ can be expressed at the
point $g=\exp X\in T$ with $X\in\mathfrak{t}$ as

(3.7)
$\zeta_{u,v}(g)=\left<\exp(X)u,v\right>=\left<\exp(X)\sum_{\omega\in\Omega}u_{\omega},\sum_{\omega\in\Omega}v_{\omega}\right>=\sum_{\omega\in\Omega}\left<u_{\omega},w_{\omega}\right>e^{\omega(X)}.$

It follows from Lemmas 3.1 and 3.4 applied at the identity element that

$\mathrm{ord}_{e_{K}}\zeta_{u,v}\leq\mathrm{ord}_{e_{T}}\zeta_{u,v|T}<|\Omega|.$

Observe that $\zeta_{u,v}(gh)=\zeta_{hu,v}(g)$ for all pairs of elements
$g,h\in K$. By repeating the above argument with respect to each matrix
coefficient of the form $\zeta_{hu,v}$ we show that
$\mathrm{ord}_{K}\zeta_{u,v}<|\Omega|$ as required. ∎

Fix some $l\in\mathbb{N}$. Consider the $l$-exterior power inner product space
$\bigwedge^{l}V$. The compact connected group $K$ can naturally be regarded as
a subgroup of $\mathrm{O}(\bigwedge^{l}V)$. The weights
$\Omega(T,\bigwedge^{l}V_{\mathbb{C}})$ of the torus $T$ in its representation
on the Hermitian space $\bigwedge^{l}V_{\mathbb{C}}$ satisfy

$\Omega(T,\bigwedge^{l}V_{\mathbb{C}})\subset\Omega^{l}=\\{\omega_{1}+\cdots+\omega_{l}\>:\>\omega_{1},\ldots,\omega_{l}\in\Omega\\}.$

The following follows immediately from the previous Lemma 3.5.

######  Lemma 3.6.

Consider the matrix coefficient $\zeta^{l}_{u,v}\in\mathcal{A}(K,\mathbb{R})$
given by

$\zeta_{u,v}^{l}(g)=\left<\bigwedge^{l}gu,v\right>\quad\forall g\in K.$

for some pair of vectors $u,v\in\bigwedge^{l}V$. If $\zeta^{l}_{u,v}$ is not
identically zero on the group $K$ then
$\mathrm{ord}_{K}\zeta_{u,v}^{l}<|\Omega(T,\bigwedge^{l}V_{\mathbb{C}})|\leq|\Omega^{l}|$.

######  Remark.

The above proofs of Lemmas 3.5 and 3.6 apply more generally without assuming
the compact group $K$ is connected at each connected component where the
matrix coeffcient in question is non-zero.

### Strong order of unipotent groups in positive characteristic

Assume that $k$ is a non-Archimedean local field with $p=\mathrm{char}(k)>0$.
Let $V$ be a finite dimensional $k$-vector space of dimension $d=\dim_{k}V$.

Consider an arbitrary element $s\in\mathrm{GL}(V)$. Write
$n=s-\mathrm{Id}\in\mathrm{End}(V)$. Note that $s^{p}=\mathrm{Id}$ if and only
if $n^{p}=0$. If this is the case then the Cayley–Hamilton theorem implies
that $n^{d}=0$. In particular

(3.8) $\log(s)=\log(\mathrm{Id}+n)=\sum_{i=1}^{d-1}(-1)^{i+1}\frac{n^{i}}{i}.$

The matrix $x=\log(s)$ is a linear combination of positive powers of the
matrix $n$ and as such $x^{d}=0$. Therefore

(3.9) $s=\exp(x)=\exp(\log s)=\sum_{i=0}^{d-1}\frac{x^{i}}{i!}.$

######  Lemma 3.7.

Let $T$ be a diagonal $k$-split torus subgroup of $\mathrm{GL}(V)$. Let
$\theta:k^{r}\to\mathrm{GL}(V)$ be a $k$-rational representation for some
$r\in\mathbb{N}$ so that $U=\theta(k^{r})\leq\mathrm{GL}(V)$ is a connected
$k$-unipotent subgroup. Assume that $T$ normalizes $U$ and that there is a
$k$-character $\alpha\in X(T)$ so that

(3.10) $t\theta(z)t^{-1}=\theta(\alpha(t)z)\quad\forall t\in T,z\in k^{r}$

Then the inclusion mapping $U\hookrightarrow\mathrm{End}(V)$ has strong order
at most $2drI$ at all points, where $I\in\mathbb{N}$ is the maximal number so
that $I\alpha=\beta-\beta^{\prime}$ for any pair of distinct $k$-characters
$\beta,\beta^{\prime}\in X(T)$ corresponding to diagonal entries of $T$.

###### Proof.

Assume to begin with that $r=1$. In other words $\theta$ is a $k$-rational
representation of the additive group. The explicit form of such
representations is known [DG80, II.§2.2.6]. Namely there is some
$N\in\mathbb{N}\cup\\{0\\}$ and a family of pairwise commuting matrices
$s_{0},\ldots,s_{N}\in\mathrm{GL}(V)$ satisfying $s_{i}^{p}=\mathrm{Id}$ for
all $i\in\\{0,\ldots,N\\}$ such that

(3.11) $\theta(z)=\prod_{i=0}^{N}\exp(z^{p^{i}}\log s_{i})\quad\forall z\in
k.$

Denote $x_{i}=\log s_{i}$. The expression $t\theta(z)t^{-1}$ for each element
$t\in T$ and $z\in k$ can be computed in two different ways. On the one hand
Equation (3.10) gives

(3.12)
$t\theta(z)t^{-1}=\theta(\alpha(t)z)=\prod_{i=0}^{N}\exp((\alpha(t)z)^{p^{i}}x_{i})=\prod_{i=0}^{N}\exp(z^{p^{i}}\alpha(t)^{p^{i}}x_{i})\quad\forall
z\in k.$

On the other hand

(3.13)
$t\theta(z)t^{-1}=\prod_{i=0}^{N}t\exp(z^{p^{i}}x_{i})t^{-1}=\prod_{i=0}^{N}\exp(z^{p^{i}}tx_{i}t^{-1})\quad\forall
z\in k.$

We apply logarithm to both Equations (3.12) and (3.13). Since the matrices
$x_{i}$ pairwise commute we obtain the equality

(3.14)
$\sum_{i=0}^{N}z^{p^{i}}(\alpha(t)^{p^{i}}x_{i})=\sum_{i=0}^{N}z^{p^{i}}(tx_{i}t^{-1})\quad\forall
z\in k,t\in T.$

Note that Equation (3.14) holds true if and only if
$\alpha(t)^{p^{i}}x_{i}=tx_{i}t^{-1}$ for all $i\in\\{0,\ldots,N\\}$. This
only be the case for each given index $i$ provided there are two distinct
$k$-characters $\beta,\beta^{\prime}\in X(T)$ corresponding to diagonal
entries of the torus $T$ such that $p^{i}\alpha=\beta-\beta^{\prime}$.
Therefore the number $N$ must be such that $p^{N}\leq I$ where $I$ is as in
the statement.

The discussion preceding the Lemma shows that each function
$\exp(z^{p^{i}}\log s_{i})$ is a polynomial map in the variable $z$ into the
$k$-vector space $\mathrm{End}(V)$ of degree at most $p^{i}\dim_{k}V$ for
every $i\in\\{0,\ldots,N\\}$. Therefore $\theta$ is a polynomial map in the
variable $z$ into the $k$-vector space $\mathrm{End}(V)$ of degree at most

$\sum_{i=0}^{N}p^{i}\dim_{k}V\leq\frac{p^{N+1}-1}{p-1}\dim_{k}V\leq
2I\dim_{k}V.$

Lastly if $r>1$ then $\theta$ is obtained as a product in the endomorphism
ring $\mathrm{End}(V)$ of $r$-many $k$-rational representations as above. The
result follows. ∎

##  4\. Sublevel set estimates of analytic functions on manifolds

Let $k$ be either the field $\mathbb{R}$ or a non-Archimedean local
field888Our discussion and results easily extend to the complex case via a
restriction of scalars. This has the effect of doubling the dimension of the
domain in Theorem 4.3 and its applications below. and $|\cdot|$ be an absolute
value on $k$. Fix a dimension $m\in\mathbb{N}$. Endow the $k$-vector space
$k^{m}$ with the supremum norm $\|\cdot\|_{\infty}$ given by

(4.1) $\|(x_{1},\ldots,x_{m})\|_{\infty}=\sup_{i}|x_{i}|\quad\forall
x_{1},\ldots,x_{m}\in k.$

Let $M$ be a fixed compact $k$-analytic manifold. Let $\eta$ be a probability
measure in the canonical measure class of $M$. For example, if $M$ is a
compact $k$-analytic group then $\eta$ can be taken to be normalized Haar
measure.

Recall from §3 that $\mathcal{A}(M,k^{m})$ denotes the $k$-vector space of all
$k$-analytic maps $f:M\to k^{m}$. Every non-empty subset $X\subset M$ defines
a seminorm $\|\cdot\|_{X}$ on $\mathcal{A}(M,k^{m})$ by

(4.2) $\|f\|_{X}=\sup_{x\in X}\|f(x)\|_{\infty}\quad\forall
f\in\mathcal{A}(M,k^{m}).$

Endow $\mathcal{A}(M,k^{m})$ with the topology coming from the norm
$\|\cdot\|_{M}$.

The goal of the current §4 is to prove Theorem 4.1. This establishes an upper
bound on the $\eta$-measure of sublevel sets of $k$-analytic maps on the
compact $k$-analytic manifold $M$.

######  Theorem 4.1.

Let $\mathcal{F}\subset\mathcal{A}(M,k^{m})$ be a compact family of
$k$-analytic maps. If $\dim_{k}\mathrm{span}_{k}\mathcal{F}<\infty$ then there
are constants $\kappa_{\mathcal{F}},\varepsilon_{\mathcal{F}}>0$ such that
every map $f\in\mathcal{F}$ satisfies

(4.3) $\eta(\\{x\in
M\>:\>\|f(x)\|_{\infty}<\varepsilon\\})\leq\kappa_{\mathcal{F}}\varepsilon^{\frac{1}{\dim_{k}M\mathrm{ord}_{M}\mathcal{F}}}\quad\forall
0<\varepsilon<\varepsilon_{\mathcal{F}}.$

Note that the dimension $m$ of the target space does not appear in Equation
(4.3).

We deduce Theorem 4.1 from known results on $(C,\alpha)$-good functions that
are discussed below. In fact Theorem 4.1 can be seen as an analogue of Theorem
4.3 in the setting of $k$-analytic manifolds.

### Sublevel set estimates

Let $d\in\mathbb{N}$ be fixed. A _ball_ in the $d$-dimensional $k$-vector
space $k^{d}$ with center point $y\in k^{d}$ and radius $r>0$ is the subset

$B=B(y,r)=\\{x\in k^{d}\>:\>\|x-y\|_{\infty}\leq r\\}.$

Let $V\subset k^{d}$ be a fixed open subset. Given a continuous function
$f:V\to k$ and a ball $B\subset V$ denote

(4.4) $\|f\|_{B}=\sup_{x\in B}|f(x)|.$

In addition, for every $\varepsilon>0$ denote

(4.5) $B^{f,\varepsilon}=\\{x\in B\>:\>|f(x)|<\varepsilon\\}.$

Let $\lambda$ be a Haar measure on the $k$-vector space $k^{d}$ regarded as an
additive locally compact group. The following terminology has been introduced
in [KM98].

######  Definition 4.2.

A continuous function $f:V\to k$ is _$(C,\alpha)$ -good_ if every open ball
$B\subset V$ satisfies

(4.6) $\lambda(B^{f,\varepsilon})\leq
C\left(\frac{\varepsilon}{\|f\|_{B}}\right)^{\alpha}\lambda(B)$

for every $\varepsilon>0$.

Polynomials are $(C,\alpha)$-good with parameters $C$ and $\alpha$ depending
only on degree. In the real case this goes back to the Remez inequality
[Rem36, BG73]. See also [DM93, Lemma 4.1], [KM98, Proposition 3.2] and
[ABRDS15, Proposition 2.8].

Analytic functions are also known to be $(C,\alpha)$-good under suitable
assumptions:

######  Theorem 4.3.

Let $f:V\to k$ be a $k$-analytic function. Assume that there exist constants
$A_{1},A_{2}>0$ and some $l\in\mathbb{N}$ such that

$\|\partial_{\beta}f\|_{V}\leq A_{1}\quad\quad\text{$\forall$multi-index
$\beta$ with $|\beta|\leq l$}$

and

$|\partial_{i}^{l}f(x)|\geq A_{2}\quad\quad\forall x\in V,\;\forall
i=1,\ldots,d.$

Then there is a constant $C>0$ such that $f$ is $(C,\frac{1}{dl})$-good. The
constant $C$ depends only on the values of $A_{1},A_{2},l$ and on the
dimension $d$.

###### Proof.

The real case is precisely [KM98, Lemma 3.3]. Note that our “balls” are
“cubes” in the terminology of [KM98]. The non-Archimedean case is contained in
[KT05, Theorem 3.2]. ∎

The following argument is well-known. In particular, it is used implicitly in
the proof of [KM98, Proposition 3.4, p. 349] and explicitly in [KT05,
Proposition 4.2]. We provide a detailed proof for the reader’s convenience.

######  Proposition 4.4.

Let $f:V\to k$ be a $k$-analytic function. Assume that there is a constant
$A^{\prime}_{1}>0$ and some $l\in\mathbb{N}$ such that

$\|\partial_{\beta}f\|_{V}\leq A^{\prime}_{1}\quad\text{$\forall$multi-index
$\beta$ with $|\beta|\leq l$}.$

Assume moreover that $0\in V$ and that there is a multi-index $\beta_{0}$ with
$|\beta_{0}|=l$ as well as a constant $A^{\prime}_{2}>0$ satisfying

$|\partial_{\beta_{0}}f(0)|\geq A^{\prime}_{2}.$

Then there constants $A_{1},A_{2}>0$ and $K\geq 1$ as well as a linear map
$g\in\mathrm{GL}(k^{d})$ with $1\leq|\det g|\leq K$ such that

$\|\partial_{\beta}(f\circ g)\|_{V}\leq
A_{1}\quad\text{and}\quad|\partial_{i}^{l}(f\circ g)(0)|\geq A_{2}\quad\forall
i\in\\{1,\ldots,d\\}.$

The constants $A_{1},A_{2}$ and $K$ depend only on the two constants
$A^{\prime}_{1}$ and $A^{\prime}_{2}$ and on the local field $k$.

###### Proof.

Let $\beta$ be an arbitrary multi-index with $|\beta|=l$. Consider the value
of the partial derivative $\partial_{\beta}(f\circ g)(0)$ where the element
$g$ varies over the ring of endomorphisms $\mathrm{End}(k^{d})$. Identify the
space $\mathrm{End}(k^{d})$ with the $k$-affine space of matrices
$\mathrm{M}_{d}(k)$. A repeated application of the chain rule shows that the
value of the partial derivative $\partial_{\beta}(f\circ g)(0)$ is given by a
homogeneous polynomial $P_{\beta,f}$ in the matrix entries of $g$. The
coefficients of the polynomial $P_{\beta,f}$ are the partial derivatives
$\partial_{\beta^{\prime}}f(0)$ where $\beta^{\prime}$ varies over _all_
possible multi-indices with $|\beta^{\prime}|=l$. In particular, the
polynomial $P_{\beta,f}$ is non-trivial as one of its coefficients is given by
$\partial_{\beta_{0}}f(0)$.

The above discussion applied with respect to the multi-indices corresponding
to the partial differentiation operators $\partial_{i}^{l}$ implies that there
exists a non-empty Zariski open subset of $\mathrm{End}(k^{d})$ consisting of
endomorphisms $g$ satisfying

(4.7) $\partial_{i}^{l}(f\circ g)(0)\neq 0\quad\forall i\in\\{1,\ldots,d\\}.$

Therefore we may find a linear map $g$ belonging to the Zariski open subset
$\mathrm{GL}(k^{d})$ and satisfying Equation (4.7). As the polynomials
$\mathrm{P}_{\beta,f}$ are all homogeneous we may assume, up to replacing the
matrix $g$ by a scalar multiple, that $1\leq|\det g|=K$ for some constant999In
the Archimedean case $K=1$. Namely we may assume that $|\det g|=1$. $K\geq 1$
depending only on the local field $k$.

The coefficients of the polynomials in the family $P_{\beta,f}$ satisfy
$|\partial_{\beta^{\prime}}f(0)|\leq A^{\prime}_{1}$ and
$|\partial_{\beta_{0}}f(0)|\geq A^{\prime}_{2}$ for some multi-index
$\beta_{0}$. The family of all polynomials satisfying such coefficient bounds
is compact. This implies the existence of the constants $A_{1}$ and $A_{2}$
depending only on the pair of constants $A^{\prime}_{1}$ and $A^{\prime}_{2}$
and the local field $k$. ∎

We are now ready to complete the proof of the main result for §4.

###### Proof of Theorem 4.1.

Fix an arbitrary point $x\in M$. Consider a compatible chart given by an open
subset $U_{x}\subset M$ containing the point $x$ and a map
$\varphi_{x}:U_{x}\to V_{x}$ with $\varphi_{x}(x)=0$ where $V_{x}\subset
k^{d}$ is a ball at the point $0$. Let $\psi_{x}:V_{x}\to U_{x}$ denote the
inverse mapping of $\varphi_{x}$.

For every multi-index $\beta$ consider the two seminorms $\|\cdot\|_{x,\beta}$
and $\|\cdot\|_{x,\beta}^{0}$ on the $k$-vector space $\mathcal{A}(M,k^{m})$
given by

$\|f\|_{x,\beta}=\|\partial_{\beta}(f\circ\psi_{x})\|_{V_{x}}\quad\text{and}\quad\|f\|_{x,\beta}^{0}=\|\partial_{\beta}(f\circ\psi_{x})(0)\|_{\infty}\quad\forall
f\in\mathcal{A}(M,k^{m}).$

Our assumptions that the family $\mathcal{F}$ is compact and that
$\dim\mathrm{span}_{k}(\mathcal{F})<\infty$ imply that there are constants
$A^{\prime}_{1,x},A^{\prime}_{2,x}>0$ such that

$\|f\|_{x,\beta}\leq A^{\prime}_{1,x}\quad\forall
f\in\mathcal{F}\;\;\forall\text{multi-index $\beta$ with
$0\leq|\beta|\leq\mathrm{ord}_{x}\mathcal{F}+1$}$

and

$\|f\|_{x,\beta_{0}}^{0}\geq A^{\prime}_{2,x}\quad\forall
f\in\mathcal{F}\;\;\text{and for \emph{some} multi-index
$\beta_{0}=\beta_{0}(f)$ with
$0\leq|\beta_{0}|\leq\mathrm{ord}_{x}\mathcal{F}$}.$

According to Proposition 4.4 there are constants
$A_{1,x},A^{\prime\prime}_{2,x}>0$ such that for every function
$f\in\mathcal{F}$ there exists a linear isomorphism
$g=g(x,f)\in\mathrm{GL}(k^{d})$ with $\|f\circ g\|_{x,\beta}\leq A_{1}$ and

$\|f\circ g\|_{x,\partial_{i}^{l}}^{0}\geq A^{\prime\prime}_{2,x}\;\;\text{for
some $l=l(f)\in\\{0,\ldots,\mathrm{ord}_{x}f\\}$ and $\forall
i\in\\{1,\ldots,d\\}$}.$

We may assume that the collection of elements $g(x,f)$ for $f\in\mathcal{F}$
is relatively compact.

Finally, there is a constant $A_{2,x}>0$ such that, up to replacing $U_{x}$ by
a smaller neighborhood of the point $x\in M$ and $V_{x}$ by a corresponding
ball of smaller radius, the function $f\circ\psi_{x}\circ g(x,f)$ is well-
defined on the ball $V_{x}$ and satisfies

$\inf_{y\in V_{x}}\|\partial_{i}^{l}(f\circ\psi_{x}\circ
g(x,f))(y)\|_{\infty}\geq A_{2,x}\quad\forall i\in\\{1,\ldots,d\\}$

for all functions $f\in\mathcal{F}$ and for some
$l=l(f)\in\\{0,\ldots,\mathrm{ord}_{x}\mathcal{F}\\}$. In the real case this
follows from the error estimate in the multi-dimensional variant of Taylor’s
series. In the non-Archimedean case this follows from the non-Archimedean
triangle inequality.

The map $\psi_{x}$ pushes forward the measure class of the Haar measure
$\lambda$ on the ball $V_{x}$ to the measure class of $\eta$ on $V$. Let
$0<D_{x}<\infty$ be the supremum of the $L_{\infty}$-norm of the Radon–Nikodym
derivative $\frac{\mathrm{d}(\psi_{x}\circ g)_{*}\lambda}{\mathrm{d}m}$ as $g$
ranges over all possible elements $g=g(f,x)$.

We know from Theorem 4.3 that each map $f\circ\varphi_{x}\circ g:V_{x}\to
U_{x}$ has at least one coordinate which is $(C_{x},\frac{1}{dl})$-good on
$V_{x}$ where $C_{x}>0$ is some constant depending only on the constants
$A_{1,x}$ and $A_{2,x}$. This means that

(4.8) $\eta(V_{x}^{f,\varepsilon})\leq D_{x}\lambda(V_{x}^{f\circ\psi_{x}\circ
g,\varepsilon})\leq C_{x}D_{x}\left(\frac{\varepsilon}{\|f\circ\psi_{x}\circ
g\|_{V_{x}}}\right)^{\frac{1}{dl}}\lambda(V_{x})\quad\forall
f\in\mathcal{F},\forall\varepsilon>0.$

Recall that the $k$-analytic variety $M$ is assumed to be compact. Therefore
there is some $N\in\mathbb{N}$ and some points $x_{1},\ldots,x_{N}\in M$ such
that $M\subset\bigcup_{i=1}^{N}U_{x_{i}}$. Denote $D=\max_{i=1}^{N}D_{x_{i}}$
and $C=\max_{i=1}^{N}C_{x_{i}}$.

We know that $\sup_{f\in\mathcal{F}}\|f\|<\infty$. Moreover, for all
$i\in\\{1,\ldots,N\\}$, we have $\inf_{f\in\mathcal{F}}\|f\|_{U_{x_{i}}}>0$,
as $\mathrm{ord}_{x}\mathcal{F}\leq l$. Therefore there is some constant $B>0$
such that

(4.9)
$\inf_{i\in\\{1,\ldots,N\\}}\inf_{f\in\mathcal{F}}\frac{\|f\circ\psi_{x_{i}}\|_{B_{x_{i}}}}{\|f\|_{M}}=\inf_{i\in\\{1,\ldots,N\\}}\inf_{f\in\mathcal{F}}\frac{\|f\|_{U_{x_{i}}}}{\|f\|_{M}}\geq
B.$

It remains to deduce the global Equation (4.3) from the local information of
Equation (4.8). For every function $p\in\mathcal{F}$ and every $\varepsilon>0$
we have that

(4.10)
$\eta(M^{f,\varepsilon})\leq\sum_{i=1}^{N}\eta(U_{x_{i}}^{f,\varepsilon})\leq
CD\sum_{i=1}^{N}\left(\frac{\varepsilon}{\|f\|_{U_{x_{i}}}}\right)^{\frac{1}{dl}}\lambda(B_{x_{i}}).$

The proof follows for an appropriate choice of the constant
$\kappa_{\mathcal{F}}>0$ by putting together Equations (4.8), (4.9) and
(4.10). ∎

##  5\. Grassmannians of normed vector spaces

Let $k$ be the field $\mathbb{R}$ or a non-Archimedean local field with
absolute value $|\cdot|$. Let $V$ be a $k$-vector space of dimension
$n=\dim_{k}V$ equipped with the norm $\|\cdot\|$. Fix a non-trivial direct sum
decomposition

$V=U\oplus U^{\prime}$

and a pair of projections

$\mathrm{P}:V\to U\quad\text{and}\quad\mathrm{P}^{\prime}:V\to U^{\prime}.$

Fix some $l\in\mathbb{N}$. Let $K$ be a fixed compact $k$-analytic subgroup of
$\mathrm{GL}(V)$ with the Haar probability measure $\eta_{K}$. For each vector
$x\in\bigwedge^{l}V$ consider the $k$-analytic function $Q_{x}$ on the compact
group $K$ defined by

(5.1) $Q_{x}\in\mathcal{A}(K,\bigwedge^{l}U),\quad
Q_{x}(g)=\left(\bigwedge^{l}\mathrm{P}g\right)x\quad\forall g\in K.$

######  Theorem 5.1.

Let $\mathfrak{N}$ be a closed subset of the Grassmannian
$\mathfrak{Gr}(l,V)$. Consider the family
$\mathcal{F}(\mathfrak{N})\subset\mathcal{A}(K,\bigwedge^{l}U)$ of
$k$-analytic functions given by

(5.2) $\mathcal{F}(\mathfrak{N})=\\{Q_{w_{1}\wedge\cdots\wedge
w_{l}}\>:\>\text{$\|w_{i}\|\leq 1$ and
$\mathrm{span}_{k}\\{w_{1},\ldots,w_{l}\\}\in\mathfrak{N}$}\\}.$

Then there are constants $\kappa_{\mathfrak{N}},\varepsilon_{\mathfrak{N}}>0$
such that

(5.3) $\eta_{K}(\\{g\in K\>:\>\inf_{0\neq w\in
gW}\frac{\|\mathrm{P}w\|}{\|w\|}\leq\varepsilon\\})<\kappa_{\mathfrak{N}}\varepsilon^{\frac{1}{\mathrm{dim}_{k}K\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})}}$

for every subspace $W\in\mathfrak{N}$ and every
$0<\varepsilon<\varepsilon_{\mathfrak{N}}$.

In proving Theorem 5.1 we may and will assume without loss of generality that
$\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})<\infty$. In other words, for every
subspace $W\in\mathfrak{N}$ and every open subset $U\subset K$ there is an
element $g\in U$ with $gW\cap U^{\prime}=\\{0\\}$. In particular we may assume
that that $l\leq\dim_{k}U$.

### Bijections and contraction

In §6 below we will be relying on Theorem 5.1 in combination with the
following elementary observation.

######  Lemma 5.2.

Let $L:V\to V$ be a linear bijection that preserves the two subspaces $U$ and
$U^{\prime}$. Then

$\inf_{0\neq w\in gW}\frac{\|Lw\|}{\|w\|}\geq\inf_{0\neq w\in
U}\frac{\|Lw\|}{\|w\|}\inf_{0\neq w\in gW}\frac{\|\mathrm{P}w\|}{\|w\|}$

for all elements $g\in K$.

###### Proof.

Consider a group element $g\in K$. Let $w\in gW$ be any non-zero vector. Write
$w=\mathrm{P}w+\mathrm{P}^{\prime}w$. Therefore
$Lw=L\mathrm{P}w+L\mathrm{P}^{\prime}w$ with $L\mathrm{P}w\in U$ and
$L\mathrm{P}^{\prime}w\in U^{\prime}$. Using the fact that $\mathrm{P}w\in U$
we obtain

$\|Lw\|\geq\|L\mathrm{P}w\|\geq\|\mathrm{P}w\|\inf_{0\neq w\in
U}\frac{\|Lw\|}{\|w\|}\geq\|w\|\inf_{0\neq w\in
U}\frac{\|Lw\|}{\|w\|}\inf_{0\neq w\in gW}\frac{\|\mathrm{P}w\|}{\|w\|}.$

The conclusion follows by taking the infimum over all such non-zero vectors
$w$. ∎

The remainder of §5 is dedicated to setting up the necessary mechanism towards
deducing Theorem 5.1 as a consequence of Theorem 4.1.

### Norms of projections

The statement of Theorem 5.1 is independent of the choice of the norm
$\|\cdot\|$ on the finite dimensional $k$-vector space $V$ up to modifying the
value of the constant $\kappa_{\mathfrak{N}}$. It will be convenient to assume
for the remainder of the current §5 that the norm $\|\cdot\|$ is given as
follows:

Fix a basis $\beta=\\{e_{1},\ldots,e_{n}\\}$ for the $k$-vector space $V$ such
that the subspace $U$ is spanned by $\\{e_{1},\ldots,e_{\dim_{k}U}\\}$. We
will assume that $\|\cdot\|$ is the Euclidean norm making the basis $\beta$
orthonormal in the real case or the maximum norm with respect to the basis
$\beta$ in the non-Archimedean case.

Recall that $l\in\mathbb{N}$ is a fixed integer satisfying $l\leq\dim_{k}U$.
The exterior power $\bigwedge^{l}V$ is a $k$-vector space satisfying
$\dim_{k}\bigwedge^{l}V={n\choose l}$. There is a $k$-linear subspace
$\bigwedge^{l}U\leq\bigwedge^{l}V$ and a projection
$\bigwedge^{l}\mathrm{P}:\bigwedge^{l}V\to\bigwedge^{l}U$. Endow the vector
space $\bigwedge^{l}V$ with the supremum norm $\|\cdot\|_{\beta}$ with respect
to the fixed basis

$\bar{\beta}=\\{e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge
e_{i_{l}}\>:\>1\leq i_{1}<i_{2}<\cdots<i_{l}\leq n\\}.$

Consider the continuous function $q$ on the Grassmannian $\mathfrak{Gr}(l,V)$
given by

(5.4) $q(W)=\sup\\{\|\bigwedge^{l}\mathrm{P}(w_{1}\wedge\cdots\wedge
w_{l})\|_{\beta}\>:\>w_{1},\ldots,w_{l}\in W\quad\text{and}\quad\|w_{i}\|\leq
1\\}.$

for any $l$-dimensional $k$-linear subspace $W\leq V$. Note that $q(W)=0$ if
and only if $\mathrm{rank}_{k}\mathrm{P}_{|W}<l$. This happens if and only if
$\dim_{k}(W\cap U^{\prime})>0$.

######  Proposition 5.3.

Every subspace $W\leq V$ the projection $\mathrm{P}:W\to U$ satisfies

(5.5) $\inf_{0\neq w\in W}\frac{\|\mathrm{P}w\|}{\|w\|}\geq q(W).$

Before proceeding, we will need an upper bound on the absolute value of the
determinant of a matrix in terms of, say, the norms of its column vectors.

######  Lemma 5.4.

Let $A$ be an $l$-by-$l$ matrix over the local field $k$ for some
$l\in\mathbb{N}$. Let $v_{i}\in k^{l}$ denote the $i$-th column of the matrix
$A$ for every $i\in\\{1,\ldots,l\\}$. Then

(5.6) $|\det A|\leq\prod_{i=1}^{l}\|v_{i}\|$

where $\|\cdot\|$ is the Euclidean norm if $k=\mathbb{R}$ and for the supremum
norm if $k$ is non-Archimedean.

###### Proof of Lemma 5.4.

The real case is the Hadamard determinant inequality. In the non-Archimedean
case Equation (5.6) follows immediately from the Leibniz determinant formula
and the non-Archimedean triangle inequality. ∎

###### Proof of Proposition 5.3.

Fix a $k$-linear subspace $W\in\mathfrak{Gr}(l,V)$. Consider any unit vector
$w\in W$ and any ordered basis $\omega=\\{w_{1},\ldots,w_{l}\\}$ for the
subspace $W$ consisting of unit vectors. To establish Equation (5.5) it
suffices to verify that

(5.7) $\|\bigwedge^{l}\mathrm{P}(w_{1}\wedge\cdots\wedge
w_{l})\|_{\beta}\leq\|\mathrm{P}w\|.$

We claim that there is a linear transformation $B\in\mathrm{SL}(W)$ satisfying
$Bw_{1}=xw$ for some $x\in k$ with $|x|\leq 1$ and $\|Bw_{i}\|=\|w_{i}\|=1$
for all $i\in\\{2,\ldots,l\\}$. The claim follows in the real case with $x=1$
as the $\mathrm{SO}(W)$-action is transitive on Euclidean unit vectors in $W$.
To prove the claim in the non-Archimedean case let $\mathcal{O}_{k}$ be the
ring of units and $\pi\in\mathcal{O}_{k}$ a uniformizing element. Let
$i\in\mathbb{N}\cup\\{0\\}$ be a minimal integer such that $\pi^{i}w$ belongs
to the $\mathcal{O}_{k}$-module spanned by $w_{1},\ldots,w_{l}$. Write
$\pi^{i}w=\sum_{j=1}^{l}x_{j}w_{j}$ for some coefficients
$x_{j}\in\mathcal{O}_{k}$. The minimality of $i$ implies that $|x_{j}|=1$ for
some $j$. Assume without loss of generality $|x_{1}|=1$ and write

$(x_{1}^{-1})\pi^{i}w-w_{1}=\sum_{j=2}^{l}x_{1}^{-1}x_{j}w_{j}.$

By considering the suitable product of elementary matrices, we obtain the
matrix $B\in\mathrm{SL}(W)$ taking the vector $w_{1}$ to $xw$ with
$x=x_{1}^{-1}\pi^{i}$ and preserving all other vectors $w_{j}$. This completes
the claim.

Now let $B\in\mathrm{SL}(W)$ be any transformation as provided by the claim.
Up to replacing the ordered basis $\omega$ with the ordered basis $B\omega$
and observing that $w_{1}\wedge\cdots\wedge w_{l}=Bw_{1}\wedge\cdots\wedge
Bw_{l}$ we may assume without loss of generality that $w_{1}=xw$ with some
$x\in k,|x|\leq 1$ and maintain the assumption that $\|w_{i}\|=1$ for all
$i\in\\{2,\ldots,l\\}$.

The coordinates of the $l$-vector
$\bigwedge^{l}\mathrm{P}(w_{1}\wedge\cdots\wedge w_{l})\in\bigwedge^{l}V$ in
the basis $\bar{\beta}$ are the determinants of the corresponding $l$-minors
of the matrix representing the linear transformation $\mathrm{P}_{|W}:W\to U$
with respect to the basis $\omega$ for the subspace $W$ and the basis
$e_{1},\ldots,e_{\dim_{k}U}$ for the subspace $U$. Let therefore
$e_{i_{1}}\wedge\cdots\wedge e_{i_{l}}\in\bar{\beta}$ with $1\leq
i_{1}<\cdots<i_{l}\leq n$ be a particular basis element satisfying

$\|\bigwedge^{l}\mathrm{P}(w_{1}\wedge\cdots\wedge w_{l})\|_{\beta}=|\det
A_{i_{1},\ldots,i_{l}}|$

where $A_{i_{1},\ldots,i_{l}}$ is the corresponding $l$-minor of the above-
mentioned matrix. The determinant bound established in Lemma 5.4 gives

$|\det
A_{i_{1},\ldots,i_{l}}|\leq\prod_{i=1}^{l}\|\mathrm{P}w_{i}\|\leq\|\mathrm{P}w_{1}\|=x\|\mathrm{P}w\|\leq\|\mathrm{P}w\|.$

We have used the fact that $\mathrm{P}$ is contracting so that
$\|\mathrm{P}w_{i}\|\leq 1$ for all $i\in\\{1,\ldots,l\\}$ as well as that
$|x|\leq 1$. ∎

### Proof of the main result

Let $\mathfrak{N}$ be a closed subset of the Grassmannian
$\mathfrak{Gr}(l,V)$. Assume that every subspace $W\in\mathfrak{N}$ satisfies
$gW\cap U^{\prime}=\\{0\\}$ for some element $g\in K$. Consider the family
$\mathcal{F}(\mathfrak{N})$ of $k$-analytic functions as introduced in
Equation (5.2). Denote $\|Q\|_{K}=\sup_{g\in K}\|Q(g)\|_{\beta}$ for every
function $Q\in\mathcal{F}(\mathfrak{N})$.

######  Proposition 5.5.

There is a constant $E>0$ such that every subspace $W\in\mathfrak{N}$
satisfies

$E\leq\sup_{\begin{subarray}{c}w_{i}\in W\\\ \|w_{i}\|\leq
1\end{subarray}}\|Q_{w_{1}\wedge\cdots\wedge w_{l}}\|_{K}\leq 1.$

###### Proof.

The existence of the upper bound follows from Lemma 5.4. To establish the
existence of the lower bound, assume towards contradiction that there is a
sequence of subspaces $W_{i}\in\mathfrak{N}$ such that

$\lim_{i}\sup_{\begin{subarray}{c}w_{j}\in W_{i}\\\ \|w_{j}\|\leq
1\end{subarray}}\|Q_{w_{1}\wedge\cdots\wedge w_{l}}\|_{K}=0.$

Up to passing to a subsequence, we may assume that $W_{i}\to
W_{0}\in\mathfrak{N}\subset\mathfrak{Gr}(l,V)$. Consider an arbitrary ordered
basis $\omega_{0}=\\{w_{1},\ldots,w_{l}\\}$ consisting of unit vectors for the
limiting vector subspace $W_{0}$. There is a sequence $\omega_{i}$ of ordered
bases for $W_{i}$ consisting of unit vectors such that
$\omega_{i}\to\omega_{0}$. It follows that $\|Q_{w_{1}\wedge\cdots\wedge
w_{l}}\|_{K}=0$. Since the basis $\omega_{0}$ was arbitrary it follows that
$q(gW_{0})=0$ for all elements $g\in K$ in the sense of Equation (5.4). This
is a contradiction to the above assumption. ∎

######  Proposition 5.6.

For every constant $E>0$ the subfamily

$\mathcal{F}_{E}(\mathfrak{N})=\\{Q\in\mathcal{F}(\mathfrak{N})\>:\>\|Q\|_{K}\geq
E\\}$

is compact in the topology of uniform convergence on the compact group $K$.

###### Proof.

Consider a sequence of functions $Q_{i}=Q_{w^{i}_{1}\wedge\cdots\wedge
w^{i}_{l}}\in\mathcal{F}_{E}(\mathfrak{N})$ where $w^{i}_{1},\ldots,w^{i}_{l}$
is an ordered basis for some subspace
$W_{i}\in\mathfrak{N}\subset\mathfrak{Gr}(l,V)$. Assume that the sequence
$Q_{i}$ uniformly converges to some continuous function
$\Psi:K\to\bigwedge^{l}U$. Up to passing to a subsequence, we may assume that
$W_{i}\to W_{0}\in\mathfrak{N}$, that $w^{i}_{j}\to w^{0}_{j}$ for each
$j\in\\{1,\ldots,l\\}$ and that $w^{0}_{1},\ldots,w^{0}_{l}$ is an ordered
$l$-tuple of unit vectors contained in $W_{0}$. Denote
$Q_{0}=Q_{w^{0}_{1}\wedge\cdots\wedge w^{0}_{l}}$. There are elements
$g_{i}\in K$ with $\|Q_{i}(g_{i})\|_{\beta}\geq E$. Up to passing to a further
subsequence, we may assume that $g_{i}\to g_{0}\in K$. Therefore
$\|Q_{0}(g_{0})\|_{\beta}\geq E$ and the ordered $l$-tuple
$w^{0}_{1},\ldots,w^{0}_{l}$ is linearly independent, hence a basis of
$W_{0}$. In particular $Q_{0}\in\mathcal{F}_{E}(\mathfrak{N})$. Finally it is
clear that the uniform limit $\Psi$ of the sequence $Q_{i}$ coincides with
$Q_{0}$. ∎

We are ready to complete the proof of the main result of §5.

###### Proof of Theorem 5.1.

Let $\mathcal{F}(\mathfrak{N})$ be the family of polynomial mappings
associated to the subset $\mathfrak{N}$ as introduced in Equation (5.2). Fix a
sufficiently small constant $E>0$ as provided by Proposition 5.5. Namely every
subspace $W\in\mathfrak{N}$ admits some ordered basis $w_{1},\ldots,w_{l}$ of
unit vectors such that $Q_{w_{1}\wedge\cdots\wedge
w_{l}}\in\mathcal{F}_{E}(\mathfrak{N})$. The family
$\mathcal{F}_{E}(\mathfrak{N})$ of $k$-analytic maps from the compact group
$K$ to the $k$-vector space $\bigwedge^{l}U$ is compact in the topology of
uniform convergence according to Proposition 5.6.

The Haar measure $\eta_{K}$ belongs to the canonical measure class of $K$
regarded as a $k$-analytic manifold. Taking into account Proposition 5.3, the
desired Equation (5.3) follows from Theorem 4.1 providing estimates on the
measures of sublevel sets of $k$-analytic functions with respect to the Haar
measure $\eta_{K}$ on the compact $k$-analytic group $K$. ∎

##  6\. Nilpotent subalgebras of semisimple Lie algebras

Let $k$ be either the field $\mathbb{R}$ or a non-Archimedean local field with
absolute value $|\cdot|$. Let $\mathbb{G}$ be a connected simply-connected
semisimple $k$-algebraic linear group without $k$-anisotropic factors. Denote
$G=\mathbb{G}(k)$ so that $G$ is a $k$-analytic group.

In the positive characteristic case we make the additional assumption that
$\mathrm{char}(k)$ is a good prime for the semisimple group $\mathbb{G}$ in
the sense of [SS70, Definition 4.1]. This means that $\mathrm{char}(k)$ does
not divide the coefficient of any simple root in the highest root. We point
out that the only primes which fail to be good for some semisimple group are
$2,3$ and $5$.

### Maximal split torus

Let $\mathbb{T}$ be a maximal $k$-split torus of $\mathbb{G}$. Let
$\Phi=\Phi(\mathbb{T},\mathbb{G})$ be the relative $k$-root system of the
group $\mathbb{G}$ with respect to the torus $\mathbb{T}$.

Choose an ordering on the $k$-root system $\Phi$ and let $\Phi^{+}$ and
$\Phi^{-}=-\Phi^{+}$ denote the subsets of positive and negative roots
respectively. Let $\Phi_{0}^{+}\subset\Phi^{+}$ denote the subset of the
simple positive roots.

Let $\mathbb{B}$ be the minimal parabolic $k$-subgroup corresponding to
$\Phi^{+}$. We have $\mathbb{B}=Z(\mathbb{T})\mathbb{U}$ where $\mathbb{U}$ is
the unipotent radical of $\mathbb{B}$. Every minimal parabolic $k$-subgroup is
conjugate to $\mathbb{B}$ via an element of $G$ [Bor12, §I.21.12]. The
subgroup $\mathbb{U}$ is defined over $k$ [Bor12, §V.21.11].

Fix notations for the corresponding groups of rational points, namely

(6.1) $B=\mathbb{B}(k),\quad T=\mathbb{T}(k)\quad\text{and}\quad
U=\mathbb{U}(k).$

### The relative Weyl group

Let $W=W(\mathbb{T},\mathbb{G})$ be the relative Weyl group with respect to
the maximal $k$-split torus $\mathbb{T}$ in the group $\mathbb{G}$. It is a
finite Coxeter group containing an involution $s_{\alpha}$ for each root
$\alpha\in\Phi$ [Bor12, §21]. In fact the set involutions
$\Sigma=\\{s_{\alpha}\\}_{\alpha\in\Phi^{+}_{0}}$ corresponding to the simple
roots $\Phi^{+}_{0}$ generates the Weyl group $W$.

Let $L_{\Sigma}:W\to\mathbb{N}\cup\\{0\\}$ be the length function on the
relative Weyl group $W$ with respect to the generating set $\Sigma$. Let
$w_{0}\in W$ be the longest element of $W$, i.e. the unique element satisfying
$L_{\Sigma}(w_{0})\geq L_{\Sigma}(w)$ for every other element $w\in W$ [Hum90,
§1.8]. It is the unique element satisfying $w_{0}\Phi^{+}=\Phi^{-}$ and such
that $\mathbb{B}^{w_{0}}$ is the opposite parabolic $k$-subgroup to
$\mathbb{B}$. The longest element $w_{0}$ is independent of the choice of
ordering.

### The compact subgroup $K$

Let $K$ be a maximal compact subgroup of the $k$-analytic group $G$. In the
real case assume that $K$ is compatible with the Cartan involution. In the
positive characteristic case assume that $K$ is a _good_ maximal compact
subgroup, namely it admits a representative of every element of the spherical
Weyl group. In either case, let $\eta_{K}$ denote the Haar measure on the
compact group $K$ normalized so that $\eta_{K}(K)=1$.

The group $G$ admits an Iwasawa decomposition $G=KB$ [BT65]. In particular the
compact subgroup $K$ is transitive in its action on the homogeneous space
$G/B$.

### Unipotent subgroups — positive characteristic case

We state an important theorem concerning unipotent subgroups of simply-
connected semisimple groups in positive characteristic.

######  Theorem 6.1 (Gille).

Assume that the characteristic of $k$ is a good prime for $\mathbb{G}$. Then
every unipotent subgroup $V$ of $G$ is contained in the group of $k$-points of
some minimal parabolic $k$-subgroup.

###### Proof.

This follows from the assumption that $\mathbb{G}$ is simply connected and
relying on the work [Gil02]. See also [Gol09, Theorem 2.7]. ∎

### Intersections with minimal parabolic $k$-subgroups

We show that any torus $S$ in $G$ as well as the unipotent subgroup $U$ admit
a zero-dimensional intersection with a generic minimal parabolic $k$-subgroup.

The proofs are based to a large extent on the ideas of [KM68, Lemma 3]. In
particular, we rely on the Zariski density of $k$-points — namely if
$\mathbb{H}$ is any connected reductive linear algebraic $k$-group then
$\mathbb{H}(k)$ is Zariski dense, see [Bor12, 18.3] or [PR92, §2.1 Theorem
2.2] in the real case. Therefore the $k$-dimension of the group of $k$-points
$\mathbb{H}(k)$ is equal to the dimension of $\mathbb{H}$.

######  Proposition 6.2.

The subset

$\\{g\in G\>:\>\dim(U^{g}\cap B)=0\\}$

is non-empty and Zariski open in $G$.

###### Proof.

By the Zariski density of $k$-points it will suffice to show that

$\Psi=\\{g\in\mathbb{G}\>:\>\dim(\mathbb{U}^{g}\cap\mathbb{B})=0\\}$

is a non-empty Zariski open subset of $\mathbb{G}$. The longest element
$w_{0}$ of the relative Weyl group satisfies
$\mathbb{B}^{w_{0}}\cap\mathbb{B}=\mathbb{T}$. Therefore
$\dim(\mathbb{U}^{w_{0}}\cap\mathbb{B})=0$ and in particular
$\dim(\mathbb{U}^{uw_{0}b}\cap\mathbb{B})=0$ for every pair of elements
$u\in\mathbb{U}$ and $b\in\mathbb{B}$. In other words, the big Bruhat cell
$\mathbb{U}w_{0}\mathbb{B}$ is contained in $\Psi$. This Bruhat cell is
Zariski open and non-empty. ∎

######  Proposition 6.3.

Let $S=\mathbb{S}(k)$ where $\mathbb{S}\leq\mathbb{G}$ is any $k$-torus. Then
the subset

$\\{g\in G\>:\>\dim(S^{g}\cap B)=0\\}$

is non-empty and Zariski open in $G$.

We do not assume that the $k$-torus $\mathbb{S}$ is split.

###### Proof of Proposition 6.3.

By the preceding discussion about the Zariski density of $k$-points it will
suffice to show that

$\Omega=\\{g\in\mathbb{G}\>:\>\dim(\mathbb{S}^{g}\cap\mathbb{B})>0\\}$

is a proper Zariski closed subset of $\mathbb{G}$. Up to conjugation by an
element of $\mathbb{G}$ we may assume that $\mathbb{S}\leq\mathbb{T}$ [Bor12,
§IV.11.3].

Consider any element $g\in\Omega$. There is a Zariski closed subgroup
$\mathbb{S}_{1}\leq\mathbb{S}$ depending on $g$ such that
$\dim\mathbb{S}_{1}>0$ and $\mathbb{S}_{1}^{g}\leq\mathbb{B}$. Since all
maximal tori in the solvable group $\mathbb{B}$ are conjugate there is an
element $h\in\mathbb{B}$ with $\mathbb{S}_{1}^{gh}\leq\mathbb{T}$ [Bor12,
§III.10.16]. As $\mathbb{S}_{1}\leq\mathbb{T}$ this condition implies that
$gh\in\mathrm{Z}_{\mathbb{G}}(\mathbb{S}_{1})$ [Bor12, §III.8.10 Corollary 1].
We conclude that

$\Omega\subset\bigcup_{\begin{subarray}{c}\mathbb{S}_{1}\leq\mathbb{S}\\\
\dim\mathbb{S}_{1}>0\end{subarray}}\mathrm{Z}_{\mathbb{G}}(\mathbb{S}_{1})\mathbb{B}.$

Our assumption that $\mathbb{G}$ has no $k$-anisotropic factors implies that
each $\mathrm{Z}_{\mathbb{G}}(\mathbb{S}_{1})\mathbb{B}$ is a proper standard
parabolic $k$-subgroup [Bor12, §V.21.11]. There are only finitely many such
subgroups. In particular $\Omega$ is proper and Zarsiki closed. ∎

###  Lie algebras

Let $\mathfrak{g}=\mathrm{Lie}(G)$ denote the Lie algebra101010In the
Archimedean case, the Lie algebra $\mathfrak{g}$ taken in the sense of Lie
theory coincides with the real points of the Lie algebra of $\mathbb{G}$ taken
in the sense of algebraic groups [PR92, Lemma §3.3.1]. of the Lie group $G$.
Let $\mathfrak{g}_{\alpha}$ be the relative $k$-root space corresponding to
each relative $k$-root $\alpha\in\Phi$, namely

(6.2) $\mathrm{Ad}(s)X=\alpha(s)X\quad\forall s\in T,\quad\forall
X\in\mathfrak{g}_{\alpha}.$

Consider the two Lie subalgebras

$\mathfrak{u}^{-}=\bigoplus_{\alpha\in\Phi^{-}}\mathfrak{g}_{\alpha}\quad\text{and}\quad\mathfrak{u}^{+}=\bigoplus_{\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha}.$

The Lie algebra $\mathfrak{g}$ admits a direct sum decomposition

$\mathfrak{g}=\mathfrak{b}\oplus\mathfrak{u}^{-}\quad\text{and}\quad\mathfrak{b}=\mathfrak{u}^{+}\oplus\mathfrak{g}_{0}$

where $\mathfrak{b}=\mathrm{Lie}(B)$ and
$\mathfrak{g}_{0}=\mathrm{Lie}(Z_{G}(T))$ [Bor12, §V.21.7]. Fix some
$\mathrm{Ad}(K)$-invariant norm $\|\cdot\|$ on the Lie algebra $\mathfrak{g}$.
In the real case assume that this norm is coming from an
$\mathrm{Ad}(K)$-invariant inner product on $\mathfrak{g}$.

### Nilpotent Lie subalgebras

Let $\mathfrak{N}(\mathfrak{g})$ denote the subset of the Grassmannian
$\mathfrak{Gr}(\mathfrak{g})$ consisting of all the nilpotent Lie subslagebras
of $\mathfrak{g}$.

The subspace of $\mathfrak{Gr}(\mathfrak{g})$ consisting of Lie subalgebras is
closed. This follows from the fact that the Lie bracket operation
$\left[\cdot,\cdot\right]:\mathfrak{Gr}(\mathfrak{g})\times\mathfrak{Gr}(\mathfrak{g})\to\mathfrak{Gr}(\mathfrak{g})$
is lower semi-continuous. By induction on the nilpotency degree we deduce
moreover that $\mathfrak{N}(\mathfrak{g})$ is a closed subset of
$\mathfrak{Gr}(\mathfrak{g})$.

There is an alternative argument in the real case. Namely, recall that
$X\in\mathfrak{g}$ is called _ad-nilpotent_ if
$\mathrm{ad}(X)^{\dim\mathfrak{g}}=0$. Engel’s theorem says that a Lie
subalgebra $\mathfrak{h}\leq\mathfrak{g}$ is nilpotent if and only if every
$X\in\mathfrak{g}$ is ad-nilpotent. The adjoint map
$\mathrm{ad}:\mathfrak{g}\to\mathrm{End}(\mathfrak{g})$ is continuous.
Therefore being ad-nilpotent is a closed condition in $\mathfrak{g}$, and it
follows that $\mathfrak{N}(\mathfrak{g})$ is closed in
$\mathfrak{Gr}(\mathfrak{g})$.

We remark that $\mathfrak{N}(\mathfrak{g})$ is Zarsiki closed in
$\mathfrak{Gr}(\mathfrak{g})$ but we will not need this.

### Algebraic Lie subalgebras

A Lie subalgebra $\mathfrak{h}\leq\mathfrak{g}$ is _algebraic_ if
$\mathfrak{h}=\mathrm{Lie}(\mathbb{H}(k))$ for some algebraic subgroup
$\mathbb{H}$ of $\mathbb{G}$. For more on this notion see [Bor12, §I.7].

We rely on Propositions 6.2 and 6.3 to deduce the following statement
concerning almost every conjugate of an algebraic nilpotent Lie subalgebra.

######  Proposition 6.4.

If $\mathfrak{n}\in\mathfrak{N}(\mathfrak{g})$ is an algebraic nilpotent Lie
subalgebra then the subset

$\\{g\in G\>:\>\mathrm{Ad}(g)\mathfrak{n}\cap\mathfrak{b}=\\{0\\}\\}$

is non-empty and Zariski open in $G$.

###### Proof.

Let $\mathfrak{n}$ be an algebraic nilpotent Lie subalgebra of $\mathfrak{g}$.
Let $N$ be a Zariski closed nilpotent subgroup of $G$ with
$\mathrm{Lie}(N)=\mathfrak{n}$. The nilpotent subgroup $N$ decomposes as a
direct product $N=N_{s}\times N_{u}$ where $N_{s}$ and $N_{u}$ are the semi-
simple and unipotent parts of $N$, respectively [Bor12, 10.6]. Both $N_{s}$
and $N_{u}$ are defined over $k$. The subgroup $N_{s}$ is a $k$-torus.

The intersection $N\cap B^{g}$ is nilpotent and Zariski closed for every
element $g\in G$. By the same reasoning as above, these intersections admit
direct product decompositions

$N\cap B^{g}=N_{s}(g)\times N_{u}(g)$

for some Zariski closed subgroups $N_{s}(g)\leq N_{s}$ and $N_{u}(g)\leq
N_{u}$ depending on the element $g\in G$. By Proposition 6.2 and Proposition
6.3, $\dim N_{s}(g)=\dim N_{u}(g)=0$ for every element $g$ belonging to a non-
empty Zariski open subset of $G$. The result follows. ∎

### Nilpotent Lie subalgebras — real case

Assume that the local field $k$ is $\mathbb{R}$ in the following two
Propositions 6.5 and 6.6.

######  Proposition 6.5.

Every nilpotent Lie subalgebra $\mathfrak{n}\in\mathfrak{N}(\mathfrak{g})$ is
contained in some algebraic nilpotent subalgebra.

###### Proof.

Let $\mathfrak{n}$ be a nilpotent Lie subalgebra of $\mathfrak{g}$. Lie’s
third theorem provides us with a simply connected Lie group $N_{0}$ with Lie
algebra $\mathrm{Lie}(N_{0})\cong\mathfrak{n}$. There is a Lie group
homomorphism $f:N_{0}\to G$ such that the differential
$df:\mathrm{Lie}(N_{0})\xrightarrow{\simeq}\mathfrak{n}$ is an isomorphism.
Let $N$ denote the Zariski closure of the nilpotent subgroup $f(N_{0})$ inside
$G$. The subgroup $N$ is nilpotent and Zariski closed. Its Lie algebra is
algebraic by definition and contains $\mathfrak{n}$. ∎

Let $\mathrm{rank}(K)$ denote the _compact rank_ of the maximal compact group
$K$, in other words $\mathrm{rank}(K)$ is the dimension of any maximal compact
torus of $K$. Let $\mathrm{P}$ denote be a fixed projection from the Lie
algebra $\mathfrak{g}$ to its Lie subalgebra $\mathfrak{u}^{-}$.

######  Proposition 6.6.

Let $\mathfrak{n}\in\mathfrak{N}$ be any nilpotent Lie subalgebra with basis
$\beta_{\mathfrak{n}}$. Then the real analytic function
$q_{\mathfrak{n}}\in\mathcal{A}(G,\bigwedge^{\dim_{\mathbb{R}}\mathfrak{n}}\mathfrak{u}^{-})$
given by

(6.3) $\quad
q_{\mathfrak{n}}(g)=\bigwedge_{X\in\beta_{\mathfrak{n}}}(\mathrm{P}\circ\mathrm{Ad})(g)X\quad\forall
g\in G$

satisfies
$\mathrm{ord}_{K}q_{\mathfrak{n}}<(6\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}\mathfrak{n}+1)^{\mathrm{rank}(K)}$.

###### Proof.

The real analytic function $q_{\mathfrak{n}}$ is non-zero on a Zariski open
subset of $G$ as can be seen by combining Proposition 6.4 and Proposition 6.5.
Note that for any pair of elements $g\in G$ and $b\in B$ the function
$q_{\mathfrak{n}}$ satisfies $q_{\mathfrak{n}}(g)=0$ if and only if
$q_{\mathfrak{n}}(bg)=0$. The Iwasawa decomposition $G=KB$ shows that the
function $q_{\mathfrak{n}}$ is not identically zero on each connected
component of the compact subgroup $K$.

We wish to bound the order $\mathrm{ord}_{K}q_{\mathfrak{n}}$ relying on Lemma
3.6. With this goal in mind, let $T_{K}$ be a maximal compact torus of $K$ and
$\Lambda_{\text{analytic}}$ be the lattice of analytically integral forms on
$\mathrm{Lie}(T_{K})$. Note that
$\Lambda_{\text{analytic}}\cong\mathbb{Z}^{\mathrm{rank}(K)}$.

Let $S=\mathbb{S}(\mathbb{R})$ be a maximal $\mathbb{R}$-torus containing the
compact torus $T_{K}$ and $\Delta$ be the absolute root system associated to
$S$ in the semisimple group $G$. The restrictions to $T_{K}$ of the simple
positive roots (with respect to some notion of positivity on $\Delta$) are
analytic and are a $\mathbb{Z}$-basis for some lattice
$\Lambda\leq\Lambda_{\text{analytic}}$, see Proposition A.1 of the appendix.
It follows that the weights of the $T_{K}$-action on the complexification of
the vector space $\bigwedge^{\dim_{\mathbb{R}}\mathfrak{n}}\mathfrak{g}$ are
contained in a ball of radius
$\mathrm{ht}(\Delta)\dim_{\mathbb{R}}\mathfrak{n}$ in the lattice $\Lambda$.

A comparison of the possibilities for absolute and restricted root systems in
the real case [OV12, Reference Chapter, Table 9] shows111111 If an absolute
root system $\Psi$ is classical then each real restricted root system $\Phi$
obtained from it is classical as well. More generally $\mathrm{ht}(\Psi)\leq
2\mathrm{ht}(\Phi)$ unless $\Psi$ and $\Phi$ have types E6 and F4
respectively. that $\mathrm{ht}(\Delta)\leq
3\mathrm{ht}(\Phi)=3\mathrm{ht}(\mathfrak{g})$. The size of the ball of radius
$3\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}\mathfrak{n}$ in the lattice
$\Lambda\cong\mathbb{Z}^{\mathrm{rank}(K)}$ is at most
$(6\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}\mathfrak{n}+1)^{\mathrm{rank}(K)}$.
This concludes the proof. ∎

### Nilpotent Lie subalgebras — positive characteristic case

Assume that the local field $k$ has positive characteristic. Let $\mathrm{P}$
be a projection from the Lie algebra $\mathfrak{g}$ to its Lie subalgebra
$\mathfrak{u}^{-}$. We establish the following the following order estimate.

######  Proposition 6.7.

Let $\beta^{+}$ be a $k$-basis for $\mathfrak{u}^{+}$. Then the $k$-analytic
function

$q_{\mathfrak{u}}\in\mathcal{A}(K,\bigwedge^{\dim_{k}\mathfrak{u}^{-}}\mathfrak{u}^{-})$

given by

$q_{\mathfrak{u}}(g)=\bigwedge_{X\in\beta^{+}}(\mathrm{P}\circ\mathrm{Ad})(g)X\quad\forall
g\in K$

satisfies

$\mathrm{ord}_{K}q_{\mathfrak{u}}\leq
4\mathrm{ht}(\mathfrak{g})\dim^{3}_{k}U\dim_{k}\mathfrak{g}.$

Some preparation is needed before giving the proof of Proposition 6.7. To
begin with, recall that the relative $k$-root system $\Phi$ may not be reduced
in general. For each relative $k$-root $\alpha\in\Phi$ denote

(6.4) $\bar{\alpha}=\begin{cases}2\alpha&2\alpha\in\Phi,\\\
\alpha&2\alpha\notin\Phi\end{cases}$

and consider the subset $(\alpha)\subset\Phi$ given by

(6.5)
$(\alpha)=\mathbb{N}\alpha\cap\Phi=\\{\alpha,\bar{\alpha}\\}=\begin{cases}\\{\alpha,2\alpha\\}&2\alpha\in\Phi,\\\
\\{\alpha\\}&2\alpha\notin\Phi.\end{cases}$

Generally speaking, the group $G$ admits a metabelian unipotent $k$-subgroup
$U_{(\alpha)}$ whose center is the connected unipotent $k$-subgroup
$U_{(\bar{\alpha})}$. The groups $U_{(\alpha)}$ and $U_{(\bar{\alpha})}$ are
normalized by the maximal $k$-split torus $T$ [Bor12, Proposition 21.9].

The following result uses the notion of strong order introduced in Definition
3.2.

######  Proposition 6.8.

Let $\alpha\in\Phi$ be a relative $k$-root. Then the adjoint representation of
the group $U_{(\bar{\alpha})}$ regarded as a $k$-analytic map

$\mathrm{Ad}\in\mathcal{A}(U_{(\bar{\alpha})},\mathrm{End}(\mathfrak{g}))$

has strong order at most
$4\mathrm{ht}(\mathfrak{g})\dim_{k}U_{(\bar{\alpha})}\dim_{k}\mathfrak{g}$ at
each point of $U_{(\bar{\alpha})}$.

###### Proof.

The connected unipotent $k$-subgroup $U_{(\bar{\alpha})}$ is abelian and
admits a $k$-isomorphism $\theta_{\alpha}:k^{n}\to U_{(\bar{\alpha})}$ where
$n=\dim_{k}U_{(\bar{\alpha})}$ [Bor12, Lemma 21.17]. Since the adjoint
representation $\mathrm{Ad}:G\to\mathrm{GL}(\mathfrak{g})$ is a central
isogeny onto its image, the $k$-rational representation
$\mathrm{Ad}\circ\theta_{\alpha}:k^{n}\to\mathrm{GL}(\mathfrak{g})$ must be a
$k$-isomorphism onto its image [Bor12, Proposition 22.4]. It follows that the
map
$\mathrm{Ad}\circ\theta_{\alpha}\in\mathcal{A}(k^{n},\mathrm{End}(\mathfrak{g}))$
is an immersion of $k$-analytic manifolds [Mar91, §I.2.5.3].

We wish to conclude relying on Lemma 3.7. Indeed, the subgroup
$\mathrm{Ad}(T)$ is a $k$-split torus of $\mathrm{GL}(\mathfrak{g})$ [Bor12,
Corollary 8.4]. It satisfies

$\mathrm{Ad}(t)\mathrm{Ad}(\theta_{\alpha}(z))\mathrm{Ad}(t^{-1})=\mathrm{Ad}(t\theta_{\alpha}(z)t^{-1})=\mathrm{Ad}(\theta_{\alpha}(\bar{\alpha}(t)z))\quad\forall
t\in T,z\in k^{n}.$

The diagonal entries of the subgroup $\mathrm{Ad}(T)$ are given by characters
corresponding to the restricted roots $\Phi\cup\\{0\\}$. In particular, the
condition $I\bar{\alpha}=\beta-\beta^{\prime}$ for some pair of roots
$\beta,\beta^{\prime}\in\Phi\cup\\{0\\}$ and some $I\in\mathbb{N}$ implies
that $I\leq 2\mathrm{ht}(\mathfrak{g})$. ∎

The next Lemma concerns the combinatorics of the Bruhat cells. We use the set
of generators $\Sigma$ as well as the length function
$L_{\Sigma}:W\to\mathbb{N}\cup\\{0\\}$ on the relative Weyl group $W$
introduced above. Recall that $w_{0}$ is the longest element of $W$.

######  Lemma 6.9.

Let $w$ be any element of the relative Weyl group $W$. Write

(6.6) $w^{-1}w_{0}=s_{1}\cdots s_{L_{\Sigma}(w^{-1}w_{0})}$

as a reduced word, where the $s_{i}=s_{\alpha_{i}}$ are generators belonging
to $\Sigma$. Then

(6.7) $BwBu_{1}\cdots u_{n}\subset Bw_{0}B$

for any choice of _non-trivial_ unipotent elements $u_{i}\in
U_{(-\alpha_{i})}$.

###### Proof.

As $w_{0}$ is the longest element it follows that
$L_{\Sigma}(w^{-1}w_{0})=L_{\Sigma}(w_{0})-L_{\Sigma}(w)$ [Hum90, p. 16,
Equation (2)]. Therefore

$L_{\Sigma}(ws_{1}\cdots s_{i+1})>L_{\Sigma}(ws_{1}\cdots s_{i})\quad\forall
i\in\\{1,\ldots,L_{\Sigma}(w^{-1}w_{0})-1\\}.$

For any pair of elements $w^{\prime}\in W$ and $s\in\Sigma$ the condition
$L_{\Sigma}(w^{\prime}s)>L_{\Sigma}(w^{\prime})$ implies that
$Bw^{\prime}BsB\subset Bw^{\prime}sB$ [Bor12, Proposition 21.22]. Observe that

$U_{(-\alpha)}\setminus\\{e\\}\subset Bs_{\alpha}B$

for all simple positive roots $\alpha\in\Phi^{+}_{0}$, see [Bor12, Theorem
21.15] and its proof. Applying these facts inductively shows that

$\displaystyle Bwu_{1}\cdots u_{i-1}Bu_{i}\subset Bwu_{1}\cdots u_{i}B$

for all $i\in\\{1,\ldots,L_{\Sigma}(w^{-1}w_{0})\\}$ provided that the
unipotent elements $u_{i}\in U_{(-\alpha_{i})}$ are all non-trivial. The final
induction step gives

$BwBu_{1}\cdots u_{L_{\Sigma}(w^{-1}w_{0})}\subset Bwu_{1}\cdots
u_{L_{\Sigma}(w^{-1}w_{0})}B=Bw(w^{-1}w_{0})B=Bw_{0}B$

as required. ∎

We are ready to complete the following argument.

###### Proof of Proposition 6.7.

Fix an arbitrary element $g\in K$. There is a unique relative Weyl group
element $w(g)\in W$ satisfying $g\in Bw(g)B$ [Bor12, Theorem 21.15]. Denote

$w^{\prime}=w^{\prime}(g)=w(g)^{-1}w_{0}$

where $w_{0}\in W$ is the longest element of $W$. Further denote
$L=L_{\Sigma}(w^{\prime}(g))$. Express the Weyl group element $w^{\prime}$ as
a reduced word $w^{\prime}=s_{1}\cdots s_{L}$ where each $s_{i}$ belongs to
the generating set $\Sigma$ with $s_{i}=s_{\alpha_{i}}$ for some root
$\alpha_{i}\in\Phi^{+}_{0}$.

For each relative $k$-root $\alpha\in\Phi$ recall that $U_{(\bar{\alpha})}$ is
a connected $k$-unipotent group $k$-isomorphic to a $k$-vector space. Let
$\mathrm{Id}_{\alpha},\mathrm{Fr}_{\alpha}:U_{(\bar{\alpha})}\to
U_{(\bar{\alpha})}$ be the identity and the Frobenius maps, respectively. For
each index $i\in\\{1,\ldots,L\\}$ define a map121212Strictly speaking, the
objects $f_{g,i},\Phi_{g},H_{g}$ and $F_{g}$ all depend on the reduced word
$w^{\prime}(g)$ rather than on the element $g$.
$f_{g,i}:U_{(-\bar{\alpha}_{i})}\to U_{(-\bar{\alpha}_{i})}$ by

$f_{g,i}=\begin{cases}\mathrm{Id}_{\alpha_{i}}&\text{$s_{j}\neq s_{i}$ for all
$j<i$},\\\ \mathrm{Fr}_{\alpha_{i}}&\text{otherwise}.\\\ \end{cases}$

Consider the subset of simple positive roots

$\Phi_{g}=\\{\alpha\in\Phi^{+}_{0}:\text{$s_{i}=s_{\alpha_{i}}$ for some
$1\leq i\leq L$}\\}$

and the $k$-analytic product group
$H_{g}=\prod_{\alpha\in\Phi_{g}}U_{(-\bar{\alpha})}$. We are interested in the
$k$-analytic map $F_{g}:H_{g}\to\mathrm{GL}(\mathfrak{g})$ given by

$F_{g}:(u_{\alpha})_{\alpha\in\Phi_{g}}\mapsto gf_{g,1}(u_{\alpha_{1}})\cdots
f_{g,L}(u_{\alpha_{L}}).$

The differential of the map $\mathrm{Ad}\circ F_{g}$ at the identity element
$e\in H_{g}$ can be computed relying on the chain-rule formula [Bor12, §3.2].
Namely

$\mathrm{d}\left(\mathrm{Ad}\circ
F_{g}\right)_{|e}(X_{\alpha})_{\alpha\in\Phi_{g}}=\mathrm{Ad}(g)\sum_{\alpha\in\Phi_{g}}\mathrm{ad}(X_{\alpha}).$

Note that
$\mathrm{ad}(X_{\alpha})\subset\mathrm{ad}(\mathfrak{g}_{-\bar{\alpha}})$ for
all roots $\alpha\in\Phi_{g}$. Therefore the differentials
$\mathrm{ad}(X_{\alpha})$ are linearly independent provided that the
$X_{\alpha}$’s are non-zero. It follows that the map $\mathrm{Ad}\circ F_{g}$
is a $k$-analytic immersion at the identity element of $H_{g}$.

Let $H_{0}\leq H$ be a sufficiently small open subgroup such that
$(\mathrm{Ad}\circ F)_{g|H_{0}}$ is a $k$-analytic embedding into the compact
subgroup $\mathrm{Ad}(K)$. We know from Lemma 6.9 that every element of
$F_{g}(H_{0})$ obtained as a product of non-trivial elements in each
coordinate of $H_{0}$ belongs to the big Bruhat cell $Bw_{0}B$. In particular
$q_{\mathfrak{u}|F_{g}(H_{0})}$ is not identically zero according to
Proposition 6.2 and its proof.

The $k$-analytic maps $\mathrm{Ad}\circ
f_{g,i}:U_{(-\bar{\alpha}_{i})}\to\mathrm{End}(\mathfrak{g})$ have strong
order at most
$4\mathrm{ht}(\mathfrak{g})\dim_{k}U_{(\bar{\alpha}_{i})}\dim_{k}\mathfrak{g}$
at each point of their domain and for all $i\in\\{1,\ldots,L\\}$, see
Proposition 6.8. As the map $(\mathrm{Ad}\circ F)_{g|H_{0}}$ is a
$k$-embedding it has strong order at most

$\sum_{i=1}^{L}4\mathrm{ht}(\mathfrak{g})\dim_{k}U_{(\bar{\alpha}_{i})}\dim_{k}\mathfrak{g}\leq
4\mathrm{ht}(\mathfrak{g})\dim^{2}_{k}U\dim_{k}\mathfrak{g}$

at each point of $H_{0}$. The above computation relies on the fact that

$L\leq L_{\Sigma}(w_{0})=|\Phi^{+}|\leq\dim_{k}U.$

To conclude we estimate the order of the map $q_{\mathfrak{u}}$ at the element
$g\in K$. Denote $\dim_{k}\mathfrak{u}^{+}=|\beta^{+}|=l$ and enumerate
$\beta^{+}=\\{X_{1},\ldots,X_{l}\\}$. Note that

$(q_{\mathfrak{u}}\circ F_{g})(h)=\det\begin{pmatrix}\vdots&&\vdots\\\
(\mathrm{Ad}\circ F_{g})(h)X_{1}&\cdots&(\mathrm{Ad}\circ F_{g})(h)X_{l}\\\
\vdots&&\vdots\end{pmatrix}\quad\forall h\in H_{0}.$

In light of Lemma 3.3 we therefore have that

$\mathrm{ord}_{g}q_{\mathfrak{u}}\leq\mathrm{ord}_{g}q_{\mathfrak{u}|F(H_{0})}\leq
4\mathrm{ht}(\mathfrak{g})\dim^{3}_{k}U\dim_{k}\mathfrak{g}.$

Since the point $g\in K$ was arbitrary this concludes the proof. ∎

### Conjugation by semisimple elements

Let $s\in T$ be any element. The linear operator $\mathrm{Ad}(s)$ as well as
its inverse $\mathrm{Ad}(s^{-1})$ preserve the direct sum decomposition
$\mathfrak{g}=\mathfrak{u}^{-}\oplus\mathfrak{b}$. It is easy to compute the
norms of the restrictions of these operators to, say, $\mathfrak{u}^{-}$.

######  Proposition 6.10.

Let $s\in T$ be such that $|\alpha(s)|<1$ for all $\alpha\in\Phi^{+}$. Then

(6.8)
$\|\mathrm{Ad}(s)\|=\|\mathrm{Ad}(s^{-1})\|=\max_{\alpha\in\Phi^{-}}|\alpha(X)|$

and

(6.9)
$\|\mathrm{Ad}(s^{-1})_{|\mathfrak{u}^{-}}\|=\frac{1}{\min_{\alpha\in\Phi^{-}}|\alpha(X)|}.$

###### Proof.

The operator $\mathrm{Ad}(s)$ is diagonizable, and the corresponding weights
for its action on the Lie subalgebra $\mathfrak{g}_{\alpha}$ are given by the
relative $k$-root $\alpha\in\Phi$, see Equation (6.2). Therefore

$\|\mathrm{Ad}(s)\|=\max_{\alpha\in\Phi}|\alpha(s)|=\max_{\alpha\in\Phi^{-}}|\alpha(s)|.$

This implies that Equation (6.8) holds. Since $\Phi=-\Phi$, the same reasoning
shows that $\|\mathrm{Ad}(s)^{-1}\|=\|\mathrm{Ad}(s)\|$. The linear
isomorphism $\mathrm{Ad}(s)$ preserves $\mathfrak{u}^{-}$. Therefore

$\|\mathrm{Ad}(s^{-1})_{|\mathfrak{u}^{-}}\|=\max_{\alpha\in\Phi^{-}}\frac{1}{|\alpha(s)|}=\frac{1}{\min_{\alpha\in\Phi^{-}}|\alpha(s)|}.$

The validity of Equation (6.9) follows. ∎

Recall that $\Phi_{0}^{+}$ is the subset of the simple positive roots. Every
positive root $\alpha\in\Phi^{+}$ can be written as an integral linear
combination $\alpha=\sum_{\alpha_{0}\in\Phi_{0}^{+}}n_{\alpha_{0}}\alpha_{0}$
of simple roots in a unique way. The height $\mathrm{ht}(\alpha)$ is the sum
of the coefficients $n_{\alpha_{0}}$ in the above expression. Denote

(6.10)
$\mathrm{ht}(\mathfrak{g})=\sup_{\alpha\in\Phi^{+}}\mathrm{ht}(\alpha).$

######  Proposition 6.11.

There is a semisimple element $s_{0}\in T$ and some $\lambda_{0}>1$ such that

(6.11)
$\|\mathrm{Ad}(s_{0}^{n})\|=\lambda_{0}^{n\mathrm{ht}(\mathfrak{g})}\quad\text{and}\quad\|\mathrm{Ad}(s_{0}^{-n})_{|\mathfrak{u}^{-}}\|=\lambda_{0}^{-n}$

for all $n\in\mathbb{N}$.

###### Proof.

For each simple relative root $\alpha\in\Phi_{0}^{+}$ there is a one-parameter
subgroup $\chi_{\alpha}$ of the torus $T$ so that

$\left<\alpha,\chi_{\beta}\right>=N\delta_{\alpha\beta}$

for some fixed $N\in\mathbb{N}$ and all simple relative $k$-roots
$\beta\in\Phi_{0}^{+}$ [Bor12, Proposition 8.6]. Fix any element $x_{0}\in k$
with $|x_{0}|<1$. Consider the semisimple element $s_{0}\in T$ given by

$s_{0}=\prod_{\alpha\in\Phi_{0}^{+}}\chi_{\alpha}(x_{0}).$

Note that $\alpha(s_{0}^{n})=|x_{0}|^{nN\mathrm{ht}(\alpha)}<1$ for every
positive root $\alpha\in\Phi^{+}$ and all $n\in\mathbb{N}$. The result follows
relying on Proposition 6.10 and taking $\lambda_{0}=|x_{0}|^{-N}$. ∎

### Norms of restrictions to nilpotent subalgebras

Consider the Lie algebra $\mathfrak{g}$ with its direct sum decomposition
$\mathfrak{g}=\mathfrak{u}^{-}\oplus\mathfrak{b}$. The current setup fits into
the framework of our discussion in §5, where

  * •

the Lie algebra $\mathfrak{g}$ plays the role of the vector space $V$,

  * •

the subgroup $\mathrm{Ad}(K)$ plays the role of the compact subgroup of
$\mathrm{GL}(V)$,

  * •

the two Lie subalgebras $\mathfrak{u}^{-}$ and $\mathfrak{b}$ respectively
play the roles of the two direct summands $U$ and $U^{\prime}$,

  * •

any fixed nilpotent subalgebra $\mathfrak{n}$ plays the role of the fixed
subspace $W$, and

  * •

the linear operator $\mathrm{Ad}(s)$ for any fixed semisimple element $s\in T$
plays the role of the linear isomorphism $L$ preserving both direct summands
$U$ and $U^{\prime}$ as in Lemma 5.2.

In light of this correspondence, Theorem 5.1 can be reformulated to obtain the
following result. Consider the subset
$\mathfrak{N}\subset\mathfrak{Gr}(\mathfrak{g})$ consisting of all

  * •

the non-zero nilpotent Lie subalgebras
$\mathfrak{n}\in\mathfrak{N}(\mathfrak{g})$ if $k=\mathbb{R}$, or

  * •

the Lie subalgebras $\mathrm{Ad}(g)\mathfrak{u}^{+}$ for some $g\in K$ if $k$
is non-Archimedean.

The collection $\mathfrak{N}$ of Lie subalgebras is closed in the Grassmannian
$\mathfrak{Gr}(\mathfrak{g})$. This follows from the preceding discussion in
the real case and from the compactness of the group $K$ in the non-Archimedean
case. Let $\mathcal{F}(\mathfrak{N})$ be the family on $k$-analytic functions
from the group $K$ associated to the collection $\mathfrak{N}$ as considered
in Equations (5.1) and (5.2). Namely

(6.12)
$\mathcal{F}(\mathfrak{N})=\\{g\mapsto\bigwedge_{i=1}^{l}\mathrm{P}gw_{i}\>:\>\text{$\|w_{i}\|\leq
1$ and $\mathrm{span}_{k}\\{w_{1},\ldots,w_{l}\\}\in\mathfrak{N}$}\\}$

where $\mathrm{P}:\mathfrak{g}\to\mathfrak{u}^{-}$ is some fixed projection.

######  Proposition 6.12.

There is a constant $\zeta>0$ with the following property — for all
sufficiently large $\lambda>1$ there exists a semisimple element
$s_{\lambda}\in T$ such that

(6.13) $\eta_{K}\left(\\{g\in K\>:\>\inf_{0\neq
X\in\mathrm{Ad}(g)\mathfrak{n}}\frac{\|\mathrm{Ad}(s_{\lambda})X\|}{\|X\|}\geq
2\\}\right)\geq
1-\zeta\lambda^{-\frac{1}{\dim_{k}K\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})}}$

and

(6.14) $\|\mathrm{Ad}(s_{\lambda})\|\leq\lambda^{\mathrm{ht}(\mathfrak{g})}$

for all Lie subalgebras $\mathfrak{n}\in\mathfrak{N}$.

###### Proof.

Consider the semisimple element $s_{0}\in T$ and the constant $\lambda_{0}>1$
given in Proposition 6.11. We have

(6.15)
$\|\mathrm{Ad}(s_{0}^{n})\|=\lambda_{0}^{n\mathrm{ht}(\mathfrak{g})}\quad\text{and}\quad\|\mathrm{Ad}(s_{0}^{-n})_{|\mathfrak{u}^{-}}\|=\lambda_{0}^{-n}$

for all $n\in\mathbb{N}$.

Given any sufficiently large fixed $\lambda>1$ there is an integer
$n_{0}\in\mathbb{N}$ such that
$\lambda_{0}^{n_{0}}\leq\lambda\leq\lambda_{0}^{n_{0}+1}$. Take
$s_{\lambda}=s_{0}^{n_{0}}$. Observe that Equation (6.14) holds with respect
to the element $s_{\lambda}$ as
$\|\mathrm{Ad}(s_{\lambda})\|=\lambda_{0}^{n_{0}\mathrm{ht}(\mathfrak{g})}\leq\lambda^{\mathrm{ht}(\mathfrak{g})}$.

We are now in a position to apply our results from the previous §5. According
to Theorem 5.1 combined with Lemma 5.2 there exists a constant
$\kappa_{\mathfrak{N}}>0$ such that every Lie subalgebra
$\mathfrak{n}\in\mathfrak{N}$ and every constant $\varepsilon>0$ satisfy

(6.16) $\eta_{K}(\\{g\in K\>:\>\inf_{0\neq
X\in\mathrm{Ad}(g)\mathfrak{n}}\frac{\|\mathrm{Ad}(s_{\lambda})X\|}{\|X\|}\geq\varepsilon\inf_{0\neq
X\in\mathfrak{u}^{-}}\frac{\|\mathrm{Ad}(s_{\lambda})X\|}{\|X\|}\\})\geq\\\
\geq
1-\kappa_{\mathfrak{N}}\varepsilon^{\frac{1}{\mathrm{dim}_{k}K\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})}}.$

Note that Equation (6.15) implies

(6.17) $\varepsilon\inf_{0\neq
X\in\mathfrak{u}^{-}}\frac{\|\mathrm{Ad}(s_{\lambda})X\|}{\|X\|}\geq\varepsilon\lambda_{0}^{n}\geq\varepsilon\frac{\lambda}{\lambda_{0}}.$

Take $\varepsilon=\frac{2\lambda_{0}}{\lambda}$. We conclude from the two
Equations (6.16) and (6.17) that the desired Equation (6.13) holds true with
the constant
$\zeta=\kappa_{\mathfrak{N}}(2\lambda_{0})^{\frac{1}{\mathrm{dim}_{k}K\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})}}$.
∎

### The parameters $\lambda(G),\delta(G)$ and $s(G)$.

We now fix some parameters for the remainder of this work. Let the constant
$\zeta$ be as provided by Proposition 6.12. According to Calculation 2.4 there
is a sufficiently large value

(6.18) $\lambda=\lambda(G)>0$

such that the parameters

(6.19) $a_{1}=2,\quad
a_{2}(G)=\lambda(G)^{-\mathrm{ht}(\mathfrak{g})}\quad\text{and}\quad
p(G)=p_{\lambda}=1-\zeta\lambda^{-\frac{1}{\dim_{k}K\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})}}$

satisfy Equation (2.11) and such that the corresponding constant

(6.20) $\delta=\delta(G)=\delta_{0}(a_{1},a_{2,\lambda};p_{\lambda})$

is bounded from below as follows131313The constant $2$ appearing in the first
line of Equation (6.21) can be replaced by any real number in the large
$(1,\infty)$. The integral value of $2$ is only a matter of convenience.

(6.21) $\displaystyle\delta^{-1}$ $\displaystyle\leq
2\mathrm{ht}(\mathfrak{g})\dim_{k}K\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})\leq$
$\displaystyle\leq\begin{cases}2\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}K(6\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}U+1)^{\mathrm{rank}(K)}&\text{$k$
is $\mathbb{R}$,}\\\
8\mathrm{ht}(\mathfrak{g})^{2}\dim_{k}K\dim_{k}G\dim_{k}^{3}U&\text{$k$ is
non-Arch.}\end{cases}\leq$
$\displaystyle\leq\begin{cases}(3\mathrm{ht}(\mathfrak{g})\dim_{\mathbb{R}}G)^{(\mathrm{rank}(K)+1)}&\text{$k$
is $\mathbb{R}$,}\\\ \mathrm{ht}(\mathfrak{g})^{2}\dim^{5}_{k}G&\text{$k$ is
non-Arch.}\end{cases}$

The upper bounds on $\mathrm{ord}_{K}\mathcal{F}(\mathfrak{N})$ follow from
Propositions 6.6 and 6.7 respectively. Lastly, for the remainder of this work
fix the semisimple element

(6.22) $s=s(G)=s_{\lambda}\in T$

such that the two Equations (6.13) and (6.14) of Proposition 6.12 are
satisfied.

##  7\. Subgroups generated by small elements

We continue using the notations introduced in §6 so that in particular $G$ is
a semisimple $k$-analytic group over the local field $k$. The goal of this
section is to study discrete subgroups of $G$ generated by small elements. We
introduce a function $\mathcal{I}_{G}$ defined on the Chabauty space of
discrete subgroups $\mathrm{Sub}_{\text{d}}\left(G\right)$ which, roughly
speaking, assigns to every discrete subgroup $\Gamma$ of $G$ the norm of its
smallest non-trivial element.

Let $\mathfrak{g}=\mathrm{Lie}(G)$ be the Lie algebra of the $k$-analytic
group $G$. We consider $\mathfrak{g}$ with some $\mathrm{Ad}(K)$-invariant
norm $\|\cdot\|$ as in §6. It will be useful to introduce the following
notation

$\mathrm{B}_{\mathfrak{g}}(r)=\\{X\in\mathfrak{g}\>:\>\|X\|\leq r\\}$

for every radius $r>0$.

### The real case. Zassenhaus lemma.

Assume that the local field $k$ is $\mathbb{R}$ so that the group $G$ is a
semisimple real Lie group without compact factors.

The exponential map $\exp$ sets up a local diffeomorphism from
$0\in\mathfrak{g}$ to $\mathrm{id}_{G}\in G$. We will use $\log$ to denote the
inverse of the exponential map $\exp$ where defined.

We rely on the classical lemma due to Zassenhaus [Zas37]. It was proved also
in [KM68, Lemma 2]. Roughly speaking, it says that a discrete subgroup
generated by small elements of $G$ is nilpotent. The precise formulation is as
follows.

######  Lemma 7.1 (Zassenhaus).

There is a radius $R=R(G)>0$ and an identity neighborhood
$V=\exp\mathrm{B}_{\mathfrak{g}}(R)$ such that for every discrete subgroup
$\Gamma$ of $G$ the group generated by $\Gamma\cap V$ is contained in a
connected nilpotent Lie subgroup.

For every discrete subgroup $\Gamma$ of $G$ let $N(\Gamma)$ denote the minimal
connected nilpotent Lie subgroup of $G$ containing $\left<\Gamma\cap
V\right>$. Denote $\mathfrak{n}(\Gamma)=\mathrm{Lie}(N(\Gamma))$.

### The non-Archimedean case. Springer isomorphism.

Assume that $k$ is a non-Archimedean local field of positive characteristic
and that $\mathrm{char}(k)$ is a good prime for the semisimple group
$\mathbb{G}$.

Let $\mathcal{O}$ be the ring of integers of the non-Archimedean local field
$k$. Let $\mathfrak{m}$ be the maximal ideal of $\mathcal{O}$. Let
$\mathbb{G}(\mathfrak{m}^{i})$ denote the congruence subgroup of the compact
group $\mathbb{G}(\mathcal{O})$ with resect to the ideal $\mathfrak{m}^{i}$
for every $i\in\mathbb{N}$.

We define a norm $\|\cdot\|_{\mathfrak{m}}$ on the group
$\mathbb{G}(\mathfrak{m})$ by letting

(7.1)
$\|g\|_{\mathfrak{m}}=\inf\;\\{\left|\nicefrac{{\mathcal{O}}}{{\mathfrak{m}}}\right|^{-i}\>:\>\text{$i\in\mathbb{N}$
and $g\in G(\mathfrak{m}^{i})$}\\}$

for all elements $g\in\mathbb{G}(\mathfrak{m})$.

It is well-known that every discrete subgroup of $\mathbb{G}(\mathfrak{m})$ is
finite and unipotent. Indeed the subgroup $\mathbb{G}(\mathfrak{m})$ is a
pro-$p$-group [PR92, Lemma §3.3.8]. Therefore every discrete subgroup $\Gamma$
of $\mathbb{G}(\mathfrak{m})$ is a finite $p$-group and is in particular
nilpotent. We obtain

$(g-\mathrm{id}_{G})^{p^{l}}=g^{p^{l}}-\mathrm{id}_{G}^{p^{l}}=g^{p^{l}}-\mathrm{id}_{G}=0$

for some $l\in\mathbb{N}$ and all $g\in\Gamma$. Therefore $g-\mathrm{id}_{G}$
is nilpotent and $g$ is unipotent.

Let $\mathfrak{U}$ be the subset of $\mathbb{G}$ consisting of all unipotent
elements. Then $\mathfrak{U}$ is an irreducible closed subvariety of
$\mathbb{G}$ defined over $k$ and
$\dim\mathfrak{U}=\dim\mathbb{G}-\mathrm{rank}\mathbb{G}$ [Spr69, Proposition
1.1]. Likewise, the subset $\mathfrak{B}\subset\mathrm{Lie}(\mathbb{G})$ of
all nilpotent elements is an irreducible closed subvariety of
$\mathrm{Lie}(\mathbb{G})$ defined over $k$ [Spr69, Proposition 2.1].

######  Theorem 7.2 (Springer isomorphism).

Assume that the characteristic of $k$ is a good prime for $\mathbb{G}$. Then
there exists a $\mathbb{G}$-equivariant $k$-isomorphism
$\Sigma:\mathfrak{U}\to\mathfrak{B}$. Moreover $\Sigma(U)=\mathfrak{u}^{+}$.

Here $U$ is the group of $k$-rational points of the unipotent radical of a
minimal parabolic $k$-subgroup. The Lie subalgebra $\mathfrak{u}^{+}$ is
discussed in §6.

###### Proof of Theorem 7.2.

Springer constructed in [Spr69, Theorem 3.1] a $\mathbb{G}$-equivariant
$k$-morphism $\Sigma:\mathfrak{U}\to\mathfrak{B}$ such that
$\Sigma:\mathfrak{U}(\overline{k})\to\mathfrak{B}(\overline{k})$ is a
homeomorphism. This statement was strengthened in [BR85, Corollary 9.3.4]
where $\Sigma$ is shown to be a $k$-isomorphism. The fact that
$\Sigma(U)=\mathfrak{u}^{+}$ follows from the proof. ∎

Note that $\Sigma(\mathfrak{U}(k))=\mathfrak{B}(k)$. The restriction of
$\Sigma$ to the compact Hausdorff space $\mathfrak{U}(\mathfrak{m})$ is a
$k$-morphism which is a homeomorphism onto its image. Therefore there is a
constant $\sigma>1$ such that

(7.2)
$\sigma^{-1}\|u\|_{\mathfrak{m}}\leq\|\Sigma(u)\|\leq\sigma\|u\|_{\mathfrak{m}}$

for every unipotent element $u\in\mathfrak{U}(\mathfrak{m})$.

_Abusing notation, we will from now on use $\log$ and $\exp$ to denote the
Springer isomorphism $\Sigma$ and its inverse where defined, even through
$\Sigma$ is not a logarithmic map in the strict sense._

### The function $\mathcal{I}_{G}$

In the Archimedean case let $R>0$ be the radius given by the Zassenhaus lemma
(i.e. Lemma 7.1). In the positive characteristic case let
$R=\sigma^{-1}\left|\nicefrac{{\mathcal{O}}}{{\mathfrak{m}}}\right|^{-1}$
where the constant $\sigma$ is as given in Equation (7.2). Denote

(7.3) $V=\exp\mathrm{B}_{\mathfrak{g}}(R).$

Namely $V$ is a Zassenhaus neighborhood in the Archimedean case and is an
identity neighborhood contained in the subgroup $G(\mathfrak{m})$ in the non-
Archimedean case.

Let $0<\rho<R$ be a sufficiently small radius such that the identity
neighborhood

(7.4) $V_{0}=\exp\mathrm{B}_{\mathfrak{g}}(\rho)$

satisfies

(7.5) $V_{0}\subset V\cap V^{s}\cap V^{s^{-1}}$

where $s=s(G)\in T$ is the particular semisimple element fixed in the final
paragraph of §6, see Equation (6.22).

We introduce a real-valued _discreteness radius_ function $\mathcal{I}_{G}$ on
the Chabauty space $\mathrm{Sub}_{\text{d}}\left(G\right)$ of discrete
subgroups of the group $G$. Given a discrete subgroup $\Gamma$ of $G$ define

(7.6)
$\displaystyle\begin{split}\mathcal{I}_{G}(\Gamma)&=\begin{cases}\min_{\gamma\in(\Gamma\cap
V_{0})\setminus\\{e\\}}\|\log\gamma\|&\Gamma\cap V_{0}\neq\\{e\\}\\\
\rho&\Gamma\cap V_{0}=\\{e\\}\end{cases}\\\
&=\sup\\{0<r<\rho\>:\>\Gamma\cap\exp\left(\mathrm{B}_{\mathfrak{g}}(r)\right)=\\{e\\}\\}\end{split}$

The function $\mathcal{I}_{G}$ takes values in the interval
$\left(0,\rho\right]$. In the non-Archimedean case $\log$ denotes the Springer
isomorphism $\Sigma$ and Equation (7.6) depends on the fact that $\Gamma\cap
V_{0}$ is unipotent combined with Theorems 6.1 and 7.2.

The logarithm map (and the Springer isomorphism) are $G$-equivariant, namely

(7.7) $\log(x^{g})=\mathrm{Ad}(g)\log(x)$

for every element $x\in V\cap V^{g^{-1}}$. As the norm $\|\cdot\|$ on the Lie
algebra $\mathfrak{g}$ is $\mathrm{Ad}(K)$-invariant, the function
$\mathcal{I}_{G}$ is $K$-invariant with respect to the compact subgroup $K$,
namely

(7.8) $\mathcal{I}_{G}(\Gamma^{k})=\mathcal{I}_{G}(\Gamma)\quad\forall k\in
K.$

### Contraction on the average

Let $\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)$ be a discrete subgroup of
$G$. We show that conjugation by the semisimple element $s$ expands the
function $\mathcal{I}_{G}$ by a definite amount, when it is evaluated at
_most_ conjugates of $\Gamma$ by an element of the compact subgroup $K$.

Let $\eta_{K}$ denote the Haar measure of the compact group $K$ normalized to
be a probability measure so that $\eta_{K}(K)=1$. The precise statement is as
follows.

######  Proposition 7.3.

If the discrete subgroup $\Gamma$ satisfies
$\mathcal{I}_{G}(\Gamma)\leq\frac{\rho}{2}$ then

(7.9) $\eta_{K}(\\{g\in K\>:\>\mathcal{I}_{G}(\Gamma^{gs})\geq
2\mathcal{I}_{G}(\Gamma^{g})\\})\geq p(G)$

where the parameter $p(G)>0$ is as given in Equation (6.19).

We postpone the proof of Proposition 7.3 until the end of the current §7.

######  Proposition 7.4.

For every discrete group $\Gamma$ we have

(7.10)
$\mathcal{I}_{G}(\Gamma^{s})\geq\frac{\mathcal{I}_{G}(\Gamma)}{\|\mathrm{Ad}(s^{-1})\|}.$

###### Proof.

As $\|\mathrm{Ad}(s^{-1})\|>1$ and $0<\mathcal{I}_{G}(\Gamma)\leq\rho$ the
result clearly holds if $\mathcal{I}_{G}(\Gamma^{s})=\rho$. Assume therefore
that $\mathcal{I}_{G}(\Gamma^{s})<\rho$. There is a non-trivial element
$\gamma\in\Gamma$ with $\gamma^{s}\in V_{0}$ and
$\mathcal{I}_{G}(\Gamma^{s})=\|\log\gamma^{s}\|$. Since $V_{0}^{s^{-1}}\subset
V$ we have that $\gamma\in V$. The equivariance of the logarithm map (or the
Springer isomorphism) formulated in Equation (7.7) implies that

(7.11)
$\mathcal{I}_{G}(\Gamma^{s})=\|\log\gamma^{s}\|=\frac{\|\log\gamma\|}{\|\mathrm{Ad}(s^{-1})\|}\geq\frac{\mathcal{I}_{G}(\Gamma)}{\|\mathrm{Ad}(s^{-1})\|}$

as required. ∎

######  Proposition 7.5.

Let $\mathfrak{h}\leq\mathfrak{g}$ be any Lie subalgebra with $\log(\Gamma\cap
V)\subset\mathfrak{h}$. If the discrete group $\Gamma$ and the element $s\in
T$ satisfy

(7.12) $\inf_{0\neq X\in\mathfrak{h}}\frac{\|\mathrm{Ad}(s)X\|}{\|X\|}\geq
2\quad\text{and}\quad\mathcal{I}_{G}(\Gamma)\leq\frac{\rho}{2}$

then

(7.13) $\mathcal{I}_{G}(\Gamma^{s})\geq 2\mathcal{I}_{G}(\Gamma).$

###### Proof.

Assume that two conditions given in Equation (7.12) are satisfied. This
implies $\mathcal{I}_{G}(\Gamma)<\rho$ so that $\Gamma\cap V_{0}\neq\\{e\\}$.
Observe that

(7.14) $\|\log\gamma^{s}\|=\|\mathrm{Ad}(s)\log\gamma\|\geq 2\|\log\gamma\|$

for every element $\gamma\in\Gamma\cap V\cap V^{s^{-1}}$. In particular, as
$V_{0}\subset V\cap V^{s^{-1}}$ it follows that Equation (7.14) holds for
every $\gamma\in\Gamma\cap V_{0}$. On the other hand, note that

(7.15) $\Gamma^{s}\cap V_{0}=(\Gamma\cap V\cap V^{s^{-1}})^{s}\cap V_{0}.$

The containment in the non-trivial direction of Equation (7.15) holds true
since $V_{0}\subset V\cap V^{s}$. The required Equation (7.13) follows by
combining Equations (7.12), (7.14) and (7.15) with the definition of the
function $\mathcal{I}_{G}$. ∎

We are ready to complete the proof of the main result of the current §7.

###### Proof of Proposition 7.3.

Assume that the subgroup $\Gamma$ satisfies
$\mathcal{I}_{G}(\Gamma)\leq\frac{\rho}{2}$. In particular
$\mathcal{I}_{G}(\Gamma^{g})\leq\frac{\rho}{2}$ for all elements $g\in K$, see
Equation (7.8).

In the Archimedean case take $N=N(\Gamma)$ relying on the Zassenhaus Lemma and
let $\mathfrak{n}=\mathfrak{n}(\Gamma)=\mathrm{Lie}(N)$ be the corresponding
Lie algebra. In the non-Archimedean case let $U$ be the unipotent radical of
_some_ parabolic subgroup containing $\Gamma\cap V$ relying on Theorem 6.1 and
let $\mathfrak{n}=\mathfrak{u}^{+}$ be the corresponding Lie subalgebra. In
both cases $\log(\Gamma\cap V)\subset\mathfrak{n}$, see Theorem 7.2 for the
non-Archimedean case

Let $p(G)>0$ be the parameter fixed in the last paragraph of §6. We deduce
from Proposition 6.12 that

(7.16) $\eta_{K}\left(\\{g\in K\>:\>\inf_{0\neq
X\in\mathrm{Ad}(g)\mathfrak{n}}\frac{\|\mathrm{Ad}(s)X\|}{\|X\|}\geq
2\\}\right)\geq p(G).$

The desired Equation (7.9) follows by combining the above Equation (7.16) with
Proposition 7.5. ∎

##  8\. The key inequality and its applications

We maintain the notations introduced in the previous §6 and §7 so that $G$ is
a semisimple analytic group, $K$ is a compact subgroup with Haar probability
measure $\eta_{K}$ and $s$ is the particular semisimple element fixed in the
last paragraph of §6.

We will work with the Borel probability $\mu_{s}$ on the group $G$ given by

(8.1) $\mu_{s}=\eta_{K}*\delta_{s}*\eta_{K}.$

The main goal of the current §8 is to prove the key inequality (Theorem 1.5)
and derive some of its applications. For the reader’s convenience we restate
the inequality using the convolution operator $A_{\mu_{s}}$ introduced in
Equation (2.1).

######  Theorem (The key inequality).

Consider the discreteness radius function
$\mathcal{I}_{G}:\mathrm{Sub}_{\text{d}}\left(G\right)\to\left(0,\rho\right]$
given by

(8.2)
$\mathcal{I}_{G}(\Gamma)=\sup\>\\{0<r<\rho\>:\>\Gamma\cap\exp\left(\mathrm{B}_{\mathfrak{g}}(r)\right)=\\{\mathrm{id}_{G}\\}\\}$

for any discrete subgroup $\Gamma$ of $G$. There are constants $0<c<1$ and
$b>0$ such that the function $\mathcal{I}_{G}$ satisfies

(8.3) $A_{\mu_{s}}\mathcal{I}_{G}^{-\delta}\leq c\mathcal{I}_{G}^{-\delta}+b$

where the parameter $\delta=\delta(G)$ is as in Equation (6.20).

###### Proof.

Let the constants $a_{1}=2,0<a_{2}(G)<1$ and $0<p(G)<1$ be as in Equation
(6.19). Moreover denote $\rho_{0}=\frac{\rho}{2}$. Observe that every discrete
subgroup $\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)$ satisfies

  1. (1)

if $\mathcal{I}_{G}(\Gamma)\leq\rho_{0}$ then $\mu_{s}(\\{g\in
G\>:\>\mathcal{I}_{G}(\Gamma^{g})\geq a_{1}\mathcal{I}_{G}(\Gamma)\\})\geq
p(G)$ and

  2. (2)

$\mathcal{I}_{G}(\Gamma^{g})\geq a_{2}(G)\mathcal{I}_{G}(\Gamma)$ holds every
element $g\in KsK=\mathrm{supp}(\mu_{s})$.

Statements (1) and (2) follow from Propositions 7.3 and 7.4 respectively,
combined with Equation (7.8) to take into account conjugation by elements of
the compact subgroup $K$. Putting all this together and applying Proposition
2.3 we immediately deduce the desired Equation (8.3). ∎

### Effective weak uniform discreteness

The two main Theorems 1.2 and 1.4 appearing in the introduction follow
immediately from Theorem 8.1, which we are now in a position to state and
prove.

Let $\mu$ be any bi-$K$-invariant probability measure on the group $G$ whose
support $\mathrm{supp}(G)$ generates the group.

######  Theorem 8.1.

There are constants $\beta,\delta>0$ such that every $\mu$-stationary
probability $G$-space $(Z,\nu)$ with $\nu$-almost everywhere discrete
stabilizers satisfies

(8.4) $\nu(\\{z\in
Z\>:\>\mathcal{I}_{G}(G_{z})<\varepsilon\\})<\beta\varepsilon^{\delta}$

for all $\varepsilon>0$. The constant $\delta$ satisfies a lower bound as in
Equation (6.21).

Prior to proving Theorem 8.1 we observe that any $\mu$-stationary $G$-space
$(Z,\nu)$ is also $\mu^{\prime}$-stationary for any other probability measure
$\mu^{\prime}$ as above. In other words, the statements of Theorems 1.2, 1.4
and 8.1 are independent of the choice of $\mu$.

Indeed, the Poisson boundary of the pair $(G,\mu)$ can be identified with the
homogoenous space $(G/B,\eta_{B})$ where $B$ is a minimal $k$-parabolic
subgroup and $\nu_{B}$ is the unique $K$-invariant probability
measure141414The uniqueness of $\eta_{B}$ follows from the Iwasawa
decomposition, i.e. the transitivity of the $K$-action on the homogenous space
$G/B$. on $G/B$. From the universal property of the Poisson boundary, a
measure $\nu$ on a $G$-space $Z$ is $\mu$-stationary if and only if $\nu$ is
the $\eta_{B}$-barycentre of some measurable map from $G/B$ to the space
$\text{Prob}(Z)$ of probability measures on $Z$. This condition depends only
on $\eta_{B}$ and not on the specific choice of the measure $\mu$.

###### Proof of Theorem 8.1.

Let $(Z,\nu)$ be any $\mu$-stationary probability $G$-space with $\nu$-almost
surely discrete stabilizers. By the preceding discussion we may assume that
$\nu$ is $\mu_{s}$-stationary where $\mu_{s}=\eta_{K}*\delta_{s}*\eta_{K}$ as
in Equation (8.1).

Consider the probability measure $\nu_{*}$ on the Chabauty space of discrete
subgroups $\mathrm{Sub}_{\text{d}}\left(G\right)$ obtained as the push-
forward151515Varadarajan’s compact model theorem [Zim13, 2.1.19] implies that
the stabilizer map is $\nu$-measurable. of the measure $\nu$ via the
stabilizer map $Z\to\text{Sub}(G),~{}z\mapsto G_{z}$. As the stabilizer map is
$G$-equivariant, the measure $\nu_{*}$ is $\mu$-stationary (as well as
$\mu_{s}$-stationary).

To conclude the proof we apply Lemma 2.1 with respect to the function
$\mathcal{I}_{G}^{-\delta}$ on the Chabauty space of discrete subgroups
$\mathrm{Sub}_{\text{d}}\left(G\right)$. Observe that the discreteness radius
function $\mathcal{I}_{G}$ is continuous on
$\mathrm{Sub}_{\text{d}}\left(G\right)$ with respect to the Chabauty topology
[dlH08]. The key equality coincides with Equation (2.2). We obtain the
estimate

(8.5)
$\nu_{*}(\\{\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)\>:\>\mathcal{I}_{G}^{-\delta}(\Gamma)\geq
M\\})\leq\beta M^{-1}$

for all $M>0$ and with $\beta=\frac{b}{1-c}$. Taking $\varepsilon=\frac{1}{M}$
and rearranging gives

(8.6)
$\nu_{*}(\\{\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)\>:\>\mathcal{I}_{G}(\Gamma)\geq\varepsilon\\})\leq\beta\varepsilon^{\delta}$

for all $\varepsilon>0$. The desired conclusion follows since $\nu_{*}$ is the
push-forward of $\nu$. ∎

In the remainder of the current §8 we complete the proofs of other results
stated in the introduction.

### The real Lie group case

Let $G$ be a real semisimple Lie group without compact factors. For the
purpose of proving the real case of our main result (Theorem 1.2) we may
replace the Lie group $G$ by the group $G^{0}/(G^{0}\cap Z(G))$. In other
words, we may assume without loss of generality that the Lie group $G$ is
connected and center-free. There exists a connected $\mathbb{R}$-linear
semisimple algebraic group $\mathbb{G}$ such that
$G=\mathbb{G}(\mathbb{R})^{0}$. The algebraic group $\mathbb{G}$ is in fact
defined over $\mathbb{Q}$ [Zim13, 3.1.6]. The semisimple algebraic group
$\mathbb{G}$ has no $\mathbb{R}$-anisotropic factors since the Lie group $G$
has no compact factors.

The norm on $B_{\mathfrak{g}}(R)$ given by $g\mapsto\|\log g\|$ is bi-
Lipschitz equivalent to any fixed left-invariant Riemannian (or Finsler)
metric on the Lie group $G$. However note that the value of the function
$\mathcal{I}_{G}$ for a discrete subgroup $\Gamma$ can only be used to
determine whether $\Gamma\cap\mathrm{B}_{\varepsilon}\neq\\{e\\}$ provided
that $\varepsilon<\rho$ where $\rho$ is as in §7.

We conclude that Theorem 1.2 of the introduction follows immediately from the
real case of Theorem 8.1.

### The positive characteristic case

Theorem 1.4 of the introduction is essentially stated in terms of the norm
$\|\cdot\|_{\mathfrak{m}}$ on the $k$-analytic group $G$ defined in Equation
(7.1). The Springer isomorphism $\Sigma$ is bi-Lipschitz with constant
$\sigma>1$ according to Equation $(\ref{eq:sigma})$. Therefore Theorem 1.4
follows from the non-Archimedean case of Theorem 8.1. Note that the value of
the function $\mathcal{I}_{G}$ for a discrete subgroup $\Gamma$ can only be
used to determine whether $\Gamma\cap\mathbb{G}(\mathfrak{m}^{i})\neq\\{e\\}$
provided that
$\left|\nicefrac{{\mathcal{O}}}{{\mathfrak{m}}}\right|^{-i}<\rho$.

### Discrete $\mu$-stationary random subgroups

Let $\mu$ be a probability measure on the group $G$ such that
$\mathrm{supp}(\mu)$ generates $G$. Let $\mathrm{DRS}_{\mu}(G)$ denote the
space of all discrete $\mu$-stationary random subgroups of the group $G$, i.e.
all $\mu$-stationary probability measures $\nu$ on
$\mathrm{Sub}\left(G\right)$ satisfying
$\nu(\mathrm{Sub}_{\text{d}}\left(G\right))=1$. We show that the space
$\mathrm{DRS}_{\mu}(G)$ is compact in the weak-$*$ topology of probability
measures.

###### Proof of Corollary 1.6.

Any weak-$*$ limit of $\mu$-stationary random subgroups (i.e. probability
measures on $\mathrm{Sub}\left(G\right))$ is also a $\mu$-stationary random
subgroup. Assume towards contradiction that $\nu_{n}$ is a sequence of
discrete $\mu$-stationary random subgroups satisfying $\nu_{n}\to\nu_{0}$
where $\nu_{0}(\mathrm{Sub}_{\text{d}}\left(G\right))<1$. In other words

(8.7) $\liminf_{r\to
0}\nu_{0}(\\{H\in\mathrm{Sub}\left(G\right)\>:\>H\cap\mathrm{B}_{r}\neq\\{\mathrm{id}_{G}\\}\\})>0.$

Since the condition of intersecting $\mathrm{B}_{r}$ non-trivially is Chabauty
open, it follows from the Portmanteau theorem that Equation (8.7) contradicts
Theorem 8.1. ∎

### Evanescence for locally symmetric manifolds

Recall that a Riemannian manifold of non-positive sectional curvature is
called _evanescent_ if its injectivity radius function vanishes at infinity.
We now prove that evanescence implies finite volume for locally symmetric
spaces.

###### Proof of Theorem 1.7.

Let $M=K\backslash G/\Gamma$ be a locally symmetric space for some semisimple
Lie group $G$ with maximal compact subgroup $K$ and a discrete torsion-free
subgroup $\Gamma$ of $G$. The homogenous space $G/\Gamma$ admits a natural
quotient map

$\pi:G/\Gamma\to M,\quad\pi:g\Gamma\to Kg\Gamma\in M.$

Furthermore the natural $G$-invariant measure $\sigma$ on $G/\Gamma$ pushes
forward to give a Riemannian volume $\pi_{*}\sigma$ on the manifold $M$. Note
that the locally symmetric space $M$ is evanescent if and only if the
continuous function $g\Gamma\to\mathcal{I}_{G}(g\Gamma g^{-1})^{-\delta}$ is
proper on the homogenous space $G/\Gamma$. The desired conclusion follows from
the key inequality (Theorem 1.5) combined with the argument of [Mar04,
Corollary 1.5]. ∎

##  9\. Weakly cocompact lattices

We briefly discuss the notion weakly cocompact lattices in general. We then
apply our main result Theorem 1.4 towards the study of this notion.

### Weakly cocompact lattices

Let $H$ be a locally compact group and $\Gamma$ be a lattice in the group $H$.
The lattice $\Gamma$ is _weakly co-compact_ if the unitary quasiregular
representation of the group $H$ on the Hilbert space $L^{2}_{0}(H/\Gamma)$
admits no almost invariant vectors.

######  Lemma 9.1.

Let $\mu$ be a compactly supported probability measure on the group $H$. Let
$D\subset H/\Gamma$ be a compact subset. Let $\mathrm{P}:L^{2}(H/\Gamma)\to
L^{2}((H/\Gamma)\backslash D)$ be the orthogonal projection operator. If

(9.1) $\|\mathrm{P}A_{\mu}\mathrm{P}_{|L^{2}(H/\Gamma\backslash D)}\|<1$

then the lattice $\Gamma$ is weakly cocompact.

###### Proof.

Denote $\hat{D}=\mathrm{supp}(\mu)D$. Let $\hat{\mathrm{P}}$ be the orthogonal
projection operator

$\hat{\mathrm{P}}:L^{2}(H/\Gamma)\to L^{2}((H/\Gamma)\backslash\hat{D}).$

It is clear that $\mathrm{P}A_{\mu}\hat{\mathrm{P}}=A_{\mu}\hat{\mathrm{P}}$
and that $\|\hat{\mathrm{P}}f\|\leq\|\mathrm{P}f\|$ for every vector $f\in
L^{2}(H/\Gamma)$.

Assume towards contradiction that the lattice $\Gamma$ is not weakly cocompact
in the group $H$. According to [Mar91, Lemma 1.9, §III.1] there exists a
sequence $v_{n}\in L^{2}(H/F)$ of functions with
$\|v_{n}\|=1,\|A_{\mu}v_{n}\|\to 1$ and
$d_{n}=\|\hat{\mathrm{P}}v_{n}-v_{n}\|\to 0$. Since $A_{\mu}$ are $\mathrm{P}$
are both contracting operators we obtain that

$\displaystyle\|A_{\mu}v_{n}\|$
$\displaystyle\leq\|A_{\mu}\hat{\mathrm{P}}v_{n}\|+\|A_{\mu}(\hat{\mathrm{P}}v_{n}-v_{n})\|\leq\|\mathrm{P}A_{\mu}\hat{\mathrm{P}}v_{n}\|+d_{n}\leq$
$\displaystyle\leq\|\mathrm{P}A_{\mu}\mathrm{P}v_{n}\|+d_{n}\leq\|\mathrm{P}A_{\mu}\mathrm{P}_{|L^{2}(H/\Gamma\backslash
D)}\|+d_{n}$

for all $n\in\mathbb{N}$. This is a contradiction to the assumption stated in
Equation (9.1). ∎

### The thin part

Let us return to our main case of interest. Namely, let $k$ be either the
field $\mathbb{R}$ or a non-Archimedean local field. Let $\mathbb{G}$ be a
connected simply-connected linear $k$-algebraic semisimple group with
$k$-anisotropic factors. Denote $G=\mathbb{G}(k)$ so that $G$ is a
$k$-analytic group.

Let $\mu$ be any bi-$K$-invariant probability measure on the group $G$ whose
support $\mathrm{supp}(\mu)$ generates the group. Let $\nu$ be any
$\mu$-stationary probability measure on the Chabauty space
$\mathrm{Sub}_{\text{d}}\left(G\right)$ of discrete subgroups.

The following elementary integrability result is an immediate consequence of
our main result Theorem 8.1. Its proof is general and can be applied to any
function satisfying the conclusion of Lemma 2.1.

######  Proposition 9.2.

The function $\mathcal{I}_{G}^{-\frac{\delta}{2}}$ is $\nu$-integrable.

###### Proof.

Let $D:\left[0,1\right]\to\mathbb{R}_{\geq 0}$ be the inverse cumulative
distribution function corresponding to $\mathcal{I}_{G}^{-\frac{\delta}{2}}$.
It is defined as

(9.2) $D(t)=\inf\\{x\in\mathbb{R}_{\geq
0}\>:\>\nu(\\{\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)\>:\>\mathcal{I}_{G}^{-\frac{\delta}{2}}(\Gamma)\leq
x\\})\geq t\\}.$

In other words, the function $D$ satisfies

(9.3)
$\nu(\\{\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)\>:\>\mathcal{I}_{G}^{-\frac{\delta}{2}}(\Gamma)\leq
D(t)\\})=t\quad\forall t\in\left[0,1\right].$

Theorem 8.1 implies

(9.4)
$\nu(\\{\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)\>:\>\mathcal{I}^{-\frac{\delta}{2}}_{G}(\Gamma)\leq
M^{-\frac{1}{2}}\\})\geq 1-\beta M$

for some constant $\beta>0$ and all sufficiently small $M>0$. It follows that

(9.5) $D(1-\beta t)\leq t^{-\frac{1}{2}}$

for all sufficiently small values of $t>0$. It follows from Fubini’s theorem
applied to the “area under the graph” of the function
$\mathcal{I}_{G}^{-\frac{\delta}{2}}$ regarded as a subset of
$\mathrm{Sub}_{\text{d}}\left(G\right)\times\mathbb{R}_{\geq 0}$ that

(9.6)
$\int_{Z}\mathcal{I}_{G}^{-\frac{\delta}{2}}(\Gamma)\;\mathrm{d}\nu(\Gamma)=\int_{\left[0,1\right]}D(t)\;\mathrm{d}\lambda(t)$

where $\lambda$ is the Lebesgue measure. As the function $D$ is monotone non-
decreasing and the integral $\int_{0}^{1}x^{-\frac{1}{2}}\mathrm{d}x$
converges, we conclude from Equation (9.6) that the function
$\mathcal{I}_{G}^{-\frac{\delta}{2}}$ is indeed $\nu$-integrable. ∎

A careful examination of §6 and §7 shows that all of our arguments and proofs
apply if the semisimple element $s=s(G)$ is replaced by its inverse
$s(G)^{-1}$. Consider therefore the Borel probability measure

(9.7)
$\hat{\mu}=\eta_{K}*\frac{1}{2}(\delta_{s(G)}+\delta_{s(G)^{-1}})*\eta_{K}=\frac{1}{2}(\mu_{s}+\mu_{s}^{*})$

on the $k$-analytic group $G$. Clearly $\hat{\mu}=\hat{\mu}^{*}$ so that the
averaging operator $A_{\hat{\mu}}$ acting on the Hilbert space $L^{2}(Z,\nu)$
is self-adjoint.

To ease our notations let $Z=\mathrm{Sub}_{\text{d}}\left(G\right)$ so that
$\nu$ is a $\hat{\mu}$-stationary measure on the space $Z$. Recall that
$\mathcal{I}_{G}^{-\frac{\delta}{2}}\in L^{1}(Z,\nu)$ according to Proposition
9.2. This implies that

(9.8) $F_{G}=\mathcal{I}_{G}^{-\frac{\delta}{4}}\in L^{2}(Z,\nu).$

Since $0<\frac{\delta}{4}<\delta=\delta(G)$ it is possible to apply
Proposition 2.3 in a similar way to what was done in the proof of Theorem 8.1
but with respect to the smaller parameter $\frac{\delta}{4}$ and obtain an
analogue of the key inequality Equation (1.8) with a “worse” constant
$0<c<\hat{c}<1$. A careful examination shows that there is a constant
$\rho_{1}>0$ such that the sharper multiplicative inequality

(9.9) $A_{\hat{\mu}}F_{G}\leq\hat{c}F_{G}$

holds for $\nu$-almost every subgroup $\Gamma\in Z$ with
$\mathcal{I}_{G}(\Gamma)<\rho_{1}$. This constant $\rho_{1}$ depends only on
the $k$-analytic group $G$ and is independent of the measure $\nu$.

######  Theorem 9.3.

Let $Z_{<\rho_{1}}$ be the _$\rho_{1}$ -thin part_ of the Chabauty space
$\mathrm{Sub}_{\text{d}}\left(G\right)$, namely

$Z_{<\rho_{1}}=\\{\Gamma\in\mathrm{Sub}_{\text{d}}\left(G\right)\>:\>\mathcal{I}_{G}(\Gamma)<\rho_{1}\\}.$

Let $\mathrm{P}:L^{2}(Z,\nu)\to L^{2}(Z_{<\rho_{1}},\nu)$ be the orthogonal
projection to $Z_{<\rho_{1}}$. Then

(9.10)
$\|\mathrm{P}A_{\hat{\mu}}\mathrm{P}_{|L^{2}(Z_{<\rho_{0}},\nu)}\|\leq\hat{c}$

for some constant $0<\hat{c}<1$.

We emphasize that both constants $\rho_{1}$ and $\hat{c}$ depend _only_ on the
group $G$.

###### Proof of Theorem 9.3.

Consider the spectrum $S$ of the self-adjoint operator
$\mathrm{P}A_{\hat{\mu}}\mathrm{P}$ restricted to the Hilbert subspace
$L^{2}(Z_{<\rho_{1}},\nu)$. In particular $S$ is a compact subset of the real
interval $\left[-1,1\right]$. Assume towards contradiction that the spectrum
$S$ admits a point $\lambda\in S$ with $|\lambda|>\hat{c}$.

The spectral theorem for self-adjoint operators shows that
$\mathrm{P}A_{\hat{\mu}}\mathrm{P}$ is conjugate to a multiplication operator.
Therefore the Hilbert space $L^{2}(Z_{<\rho_{1}},\nu)$ has approximate
eigenvfunctions for the value $\lambda$. Namely, there are vectors $v_{n}\in
L^{2}(Z_{<\rho_{1}},\nu)$ with $\|v_{n}\|=1$ that satisfy

(9.11)
$\|(\mathrm{P}A_{\hat{\mu}}\mathrm{P}-\lambda)v_{n}\|\xrightarrow{n\to\infty}0.$

Consider the inner-products of the vector
$F_{G}=\mathcal{I}_{G}^{-\frac{\delta}{4}}$ with the vectors $|v_{n}|$ in the
Hilbert space $L^{2}(Z,\nu)$. The fact that the operator $A_{\hat{\mu}}$ is
self-adjoint gives

$\displaystyle\hat{c}\left<F_{G},|v_{n}|\right>$
$\displaystyle\geq\left<\mathrm{P}A_{\hat{\mu}}\mathrm{P}F_{G},|v_{n}|\right>=\left<F_{G},\mathrm{P}A_{\hat{\mu}}\mathrm{P}|v_{n}|\right>\geq$
$\displaystyle\geq\left<F_{G},|\mathrm{P}A_{\hat{\mu}}\mathrm{P}v_{n}|\right>=\left<F_{G},|\lambda
v_{n}+(\mathrm{P}A_{\hat{\mu}}\mathrm{P}-\lambda)v_{n}|\right>\geq$
$\displaystyle\geq|\lambda|\left<F_{G},|v_{n}|\right>-\left<F_{G},|(\mathrm{P}A_{\hat{\mu}}\mathrm{P}-\lambda)v_{n}|\right>\geq$
$\displaystyle=|\lambda|\left<F_{G},|v_{n}|\right>-\|F_{G}\|\|(\mathrm{P}A_{\hat{\mu}}\mathrm{P}-\lambda)v_{n}\|.$

This stands in a contradiction to Equation (9.11). ∎

We are ready to conclude the proof of Theorem 1.8 stated in the introduction.

######  Corollary 9.4.

Every lattice $\Gamma$ in the group $G$ is weakly cocompact, i.e.
$L^{2}(G/\Gamma)$ has spectral gap.

###### Proof.

Let $\Gamma$ be a lattice in the $k$-analytic group $G$. Let
$D=(G/\Gamma)_{\geq\rho}$ be the $\rho$-thick part of the homogenous $G$-space
$G/\Gamma$. In particular $D$ is a compact subset. It follows from Theorem 9.3
that

$\|\mathrm{P}A_{\hat{\mu}}\mathrm{P}_{|L^{2}((G/\Gamma)_{<\rho})}\|\
\leq\hat{c}<1.$

Therefore the lattice $\Gamma$ is weakly cocompact by Lemma 9.1. ∎

##  Appendix A Maximally compact Cartan subalgebras

Let $\mathfrak{g}_{0}$ be a real semisimple Lie algebra with Cartan involution
$\theta$. Let $\mathfrak{g}_{0}=\mathfrak{l}_{0}\oplus\mathfrak{p}_{0}$ be the
corresponding Cartan decomposition. Let $\mathfrak{t}_{0}$ be a maximal
abelian Lie subalgebra of $\mathfrak{l}_{0}$. Then
$\mathfrak{h}_{0}=Z_{\mathfrak{g}_{0}}(\mathfrak{t}_{0})$ is a maximally
compact $\theta$-stable Cartan subalgebra of $\mathfrak{g}_{0}$. It satisfies