# Hilbert’s irreducibility theorem via random walks

Lior Bary-Soroker  Lior Bary-Soroker, Raymond and Beverly Sackler School of
Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
<EMAIL_ADDRESS>and  Daniele Garzoni  Daniele Garzoni, Raymond and
Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv
69978, Israel<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a connected linear algebraic group over a number field $K$, let
$\Gamma$ be a finitely generated Zariski dense subgroup of $G(K)$ and let
$Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if
$G/\mathrm{R}_{u}(G)$ is semisimple and $Z$ satisfies certain necessary
conditions, then a long random walk on a Cayley graph of $\Gamma$ hits
elements of $Z$ with negligible probability. We deduce corollaries to Galois
covers, characteristic polynomials, and fixed points in group actions. We also
prove analogous results in the case where $K$ is a global function field.

##  1\. Introduction

###  1.1. Hilbert’s irreducibility theorem

This paper concerns generalizations of the celebrated Hilbert’s Irreducibility
Theorem (HIT). In polynomial terms, HIT asserts that, for every $n\geqslant
1$, if $f(t_{1},\ldots,t_{n},X)\in\mathbf{Q}(t_{1},...,t_{n})[X]$ is
irreducible, then there exists an _irreducible specialization_ , that is to
say, $(\underline{t_{1}},\ldots,\underline{t_{n}})\in\mathbf{Q}^{n}$ such that
$f(\underline{t_{1}},\ldots,\underline{t_{n}},X)\in\mathbf{Q}[X]$ remains
irreducible. Moreover, the Galois group of
$f(\underline{t_{1}},\ldots,\underline{t_{n}},X)$ over $\mathbf{Q}$ is
isomorphic to the Galois group of $f(t_{1},\ldots,t_{n},X)$ over
$\mathbf{Q}(t_{1},\ldots,t_{n})$. Hilbert’s original motivation came from the
inverse Galois problem: In order to realize a finite group as a Galois group
over $\mathbf{Q}$, given HIT it is sufficient to realize it over
$\mathbf{Q}(t_{1},\ldots,t_{n})$, where one has more tools.

HIT found numerous applications in number theory, algebraic geometry, and
algebra, see [Ser89, Ser16, FJ08, MM18].

HIT has an equivalent formulation in terms of rational points, an approach
with traces going back at least to Hilbert’s works on the inverse Galois
problem. It uses the notion of thin sets, introduced by Serre [Ser89] and
Colliot-Thélène–Sansuc [CTS87]. In this paper, a $K$-variety is a separated
$K$-scheme of finite type.

######  Definition.

  1. (1)

Let $K$ be a field and $V$ be an irreducible $K$-variety. A subset $Z$ of
$V(K)$ is called thin if

$Z\subseteq C(K)\cup\bigcup_{i=1}^{t}\pi_{i}(V_{i}(K)),$ (1)

where $C$ is a proper closed subset of $V$, $t\in\mathbf{Z}_{\geqslant 0}$,
and for every $i=1,\ldots,t$, $V_{i}$ is an irreducible $K$-variety,
$\mathrm{dim}(V_{i})=\mathrm{dim}(V)$, and $\pi_{i}\colon V_{i}\to V$ is a
dominant separable morphism with $\deg(\pi_{i})\geqslant 2$.

  2. (2)

We say that $V$ has the Hilbert property (HP) if $V(K)$ is not thin.

HIT is equivalent to the statement that $\mathbb{A}^{n}_{\mathbf{Q}}$ has the
HP for every $n\geqslant 1$. Varieties having the HP satisfy an irreducibility
specialization property, see [Ser16, Proposition 3.3.5]. Every connected
linear algebraic group over a number field has the HP [San81, CTS87, BSFP14].
See [CZ17], where the HP is proven for a geometrically non-rational variety
for the first time, and see its extensions in [Dem20].

For many $V$ satisfying the HP, not only $V(K)$ is not thin, but a thin set
$Z$ is small in $V(K)$ in much stronger senses. For example, if
$V=\mathbb{A}^{n}_{\mathbf{Q}}$, then $H=V(\mathbf{Q})\smallsetminus Z$ is
dense in the real topology, it contains arithmetic progressions, and $H$ has
asymptotic density $1$, with respect to counting either integral or rational
points by height.

For a connected linear algebraic group $V=G$ over a number field $K$, Liu
[Liu21] and Corvaja [Cor07], based on Zannier [Zan10] and Ferretti–Zannier
[FZ07], proved that thin sets $Z$ are small with respect to the group
structure. More precisely, let $\Gamma$ be a Zariski dense subgroup of $G(K)$
and let $Z$ be a thin set of $G(K)$ as in (1). For every $i$, assume that
$V_{i}$ is normal and $\pi_{i}\colon V_{i}\to G$ is finite and ramified. Then,
$\Gamma\nsubseteq Z$.

The ramification assumption is essential for the conclusion. Indeed, if
$\pi\colon G_{1}\to G$ is a nontrivial isogeny of algebraic groups; in
particular, $\pi$ is unramified, then $\Gamma:=\pi(G_{1}(K))$ is a Zariski
dense subgroup of $G(K)$, and also a thin set. We remark that the connection
between the HP and ramification is made explicit in [CZ17].

The objective of this paper is to obtain a quantitative version of this
result. Loosely speaking, we will show that, under suitable conditions on $G$,
“most” points of $\Gamma$ do not belong to $Z$. We will measure this by
performing long random walks on a Cayley graph of $\Gamma$. This direction is
inspired by the works of Rivin [Riv08], Jouve–Kowalski–Zywina [JKZ13],
Lubotzky–Meiri [LM12], and Lubotzky–Rosenzweig [LR14].

###  1.2. Random walks

We fix notation for this subsection.

######  Notation list 1.

  * $\diamond$

$K$ is a number field.

  * $\diamond$

$G$ is a connected linear algebraic $K$-group such that $G/\text{R}_{u}(G)$ is
either trivial or semisimple.

  * $\diamond$

$\Gamma$ is a finitely generated Zariski dense subgroup of $G(K)$.

  * $\diamond$

$A$ is a finite symmetric generating multiset of $\Gamma$, with $1\in A$.

  * $\diamond$

$\omega_{n}$ is the $n$-th step of a random walk on the Cayley graph
$\mathrm{Cay}(\Gamma,A)$, starting at the identity.

  * $\diamond$

For every $G$, we define a function
$\Xi\colon\mathbf{R}_{>0}\times\mathbf{Z}_{\geqslant 1}\to\mathbf{R}_{>0}$, as
follows:

$\Xi(C,n)=\begin{cases}e^{-n/C}&\mbox{if $G$ is semisimple}\\\
Cn^{-1/(10\,\mathrm{dim}\,G)}&\mbox{otherwise.}\end{cases}$

Let us clarify the notation: By $\mathrm{R}_{u}(G)$, we denote the unipotent
radical of $G$. By $A$ symmetric, we mean that $g$ and $g^{-1}$ appear in $A$
with the same multiplicity, for every $g\in\Gamma$. The Cayley graph
$\mathrm{Cay}(\Gamma,A)$ attached to $\Gamma$ and $A$ has $\Gamma$ as the set
of vertices and $\\{(g,ga):a\in A,g\in\Gamma\\}$ as the multiset of edges.
Then, $\omega_{n}$ is the $n$-th step of the uniform random walk on
$\mathrm{Cay}(\Gamma,A)$, starting at the identity.

Our main theorem asserts that, under our assumptions on $G$, the random walk
$\omega_{n}$ is asymptotically almost never in a thin set:

######  Theorem 1.1.

Let $Z$ be a thin subset of $G(K)$ as in (1) with the additional assumptions
that $V_{i}$ is normal and $\pi_{i}\colon V_{i}\to G$ is finite and ramified,
for each $i$. Then, there exists $C>0$ such that

$\mathbf{P}(\omega_{n}\in Z)\leqslant\Xi(C,n).$

In particular, $\displaystyle\lim_{n\to\infty}\mathbf{P}(\omega_{n}\in Z)=0$.

If $G$ is semisimple, Theorem 1.1 gives exponential decay. When $G$ is not
semisimple, one cannot get exponential decay. Indeed, take $G=\mathbb{A}^{1}$,
$\Gamma=\mathbf{Z}$, $A=\\{0,1,-1\\}$, and consider the thin set $Z=\\{0\\}$.
Then, $\omega_{n}$ is a classical one dimensional random walk, hence

$\mathbf{P}(\omega_{n}\in Z)=\mathbf{P}(\omega_{n}=0)\geqslant cn^{-1/2}.$ (2)

for some $c>0$.

If $G$ is not semisimple, then $C$ in Theorem 1.1 depends on $G$, $K$,
$\Gamma$, $|A|$, and on $Z$. In fact, the dependence on $Z$ is via the
_complexity_ , see Definition 2.1. If $G$ is semisimple, $C$ depends also on
the spectral gap of the Cayley graphs induced on the congruent quotients of
$\Gamma$; see § 2 for further details. In all the theorems of the
introduction, $C$ will depend on these parameters.

As in Liu’s result, the assumption that the morphisms $\pi_{i}$ are ramified
is crucial: If $\pi\colon G_{1}\to G$ is a nontrivial isogeny of algebraic
groups, then $\mathbf{P}(\omega_{n}\in\pi(G_{1}(K)))\geqslant c>0$. See Remark
3.2 below for further details.

The assumption that the varieties $V_{i}$ are normal and that the morphisms
are finite is technical and can be removed. See Remark 3.3 and Lemma 3.1.

It seems that Theorem 1.1 was known to experts in the case where $G$ is
semisimple and simply connected ([Kow10, § 6.2]). We note that the estimates
in _loc. cit._ are not valid for semisimple non-simply connected groups, as
the example of a nontrivial isogeny shows.

We apply Theorem 1.1 to get irreducibility, in the spirit of the classical HIT
formulation.

######  Corollary 1.2.

Let $V$ be a normal irreducible $K$-variety, and $\pi\colon V\to G$ be a
finite surjective morphism. Assume that every non-scalar subcover of $\pi$ is
ramified. Then, there exists $C>0$ such that

$\mathbf{P}(\pi^{-1}(\omega_{n})\mbox{ is not integral\,})\leqslant\Xi(C,n).$

The original motivation of Hilbert came from Galois theory, where he used HIT
to specialize Galois groups from $\mathbf{Q}(t)$ to $\mathbf{Q}$. The next
application is in this spirit. If $\pi\colon V\to G$ is a generically Galois
cover of normal irreducible $K$-varieties with group
$\mathcal{G}=\mathrm{Gal}(K(V)/K(G))$, then every $g\in G(K)$ comes with the
specialized Galois group, denoted $\mathcal{G}_{g}$, which is the automorphism
group of the residue field extension at a point $x\in V$ lying above $g$; this
group is isomorphic to the decomposition group modulo the inertia group.
Hence, the specialized Galois group is always a sub-quotient of $\mathcal{G}$.
We show that typically it equals $\mathcal{G}$:

######  Corollary 1.3.

Let $V$ be a normal irreducible $K$-variety, and $\pi\colon V\to G$ be a
finite surjective morphism, which is generically Galois with group
$\mathcal{G}$. Assume that every nontrivial geometrically irreducible subcover
of $\pi$ is ramified. Then, there exists $C>0$ such that

$\mathbf{P}(\mathcal{G}_{\omega_{n}}\neq\mathcal{G})\leqslant\Xi(C,n).$

A different sense for smallness of a thin set $Z$ in an algebraic group $G$ is
the following: Take $\Gamma\subseteq G(K)$ a finitely generated Zariski dense
subgroup and prove that there exists a finite index coset that is disjoint
from $Z$. As in our case, a ramification condition is necessary. Schinzel
[Sch65] proved it for the additive group $\mathbb{G}_{a}$, Zannier [Zan10] for
a torus or a power of a non-CM elliptic curve. Corvaja, Demeio, Javanpeykar,
Lombardo, and Zannier [CDJ+20] generalized the latter to any abelian variety,
and Liu [Liu21] and Corvaja [Cor07] generalized it to any connected linear
algebraic group.

###  1.3. Characteristic polynomial

We are still under Notation List 1. The typical Galois group of the
characteristic polynomial of $\omega_{n}$ is well studied in the literature,
see [Riv08, JKZ13, LR14].

Our objective in this subsection is to show how these type of results follow
from the general result on generic Galois groups (Corollary 1.3). The
advantage of our approach is that we do not use anything on the characteristic
polynomial, except for the computation of the generic Galois group by Prasad
and Rapinchuk. The original proofs use more specific information, see for
instance [JKZ13, § 4 and Proposition 4.1].

Fix an embedding $\rho\colon G\hookrightarrow\mathrm{GL}_{N}$. Then, we have
the generic characteristic polynomial $\chi_{G,\rho}\in K(G)[X]$ such that
$\chi_{G,\rho}|_{g}=\mathrm{det}(X-g^{\rho})$ for every $g\in G(K)$. Let $V$
be the normalization of $G$ in the splitting field $F$ of $\chi_{G,\rho}$ over
$K(G)$, and let $\pi\colon V\to G$ be the corresponding finite morphism, which
is generically Galois.

It turns out that $F$ is independent of $\rho$. It follows from a result of
Prasad and Rapinchuk [PR03] that $\mathrm{Gal}(F/K(G))$ is isomorphic to a
certain extension of the Weyl group $W(G)$ of $G$, that we denote by $\Pi(G)$.
For instance, if $G$ is split, then $\Pi(G)=W(G)$. See § 4.1 for the precise
description of $\Pi(G)$.

It is easy to see that, in this problem, one can replace $G$ by
$G/\mathrm{R}_{u}(G)$. We will deduce from Corollary 1.3 the following:

######  Corollary 1.4.

Assume that $G/\mathrm{R}_{u}(G)$ is either trivial or semisimple, and put
$\chi_{n}=\mathrm{det}(X-\omega_{n}^{\rho})$. Then, there exists $C>0$ such
that

$\mathbf{P}(\mathrm{Gal}(\chi_{n}/K)\not\cong\Pi(G))\leqslant e^{-n/C}.$

A more general result was proved by Lubotzky and Rosenzweig [LR14], who also
considered non-connected groups.

In the case of arithmetic subgroups of simple simply connected groups over a
number field, Gorodnik and Nevo [GN11] considered the typical characteristic
polynomial of elements by counting in archimedean balls, instead of using
random walks. This is a substantially different problem. We remark that the
methods in [GN11] can be used to deal with thin sets in the archimedean model.

###  1.4. Fixed points

We are still under Notation list 1. In this subsection, we deal with actions
of $G$ on varieties $X$. We show that typically $\omega_{n}$ has no rational
fixed points, unless generically every element has a fixed point. Our result
is the random walk analogue of a theorem of Corvaja [Cor07].

Let us be more precise. Assume that $G$ acts, as a group scheme, on a
$K$-variety $X$. We can then consider the variety of fixed points

$V=\\{(x,g)\mid xg=g\\}\subseteq X\times G.$

It is equipped with the natural projection morphism $\pi\colon V\to G$. Let
$\accentset{\rule{4.89998pt}{0.8pt}}{K}$ be an algebraic closure of $K$. In
order to apply Theorem 1.1, we need the existence of a Zariski dense subset of
$G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$ consisting of elements fixing
finitely many points on $X(\accentset{\rule{4.89998pt}{0.8pt}}{K})$. For
technical reasons, we will also require that, letting $p\colon\widetilde{G}\to
G$ be the universal cover of $G$ (which exists since $G/\mathrm{R}_{u}(G)$ is
semisimple), we have that $p^{-1}(\Gamma)\leqslant\widetilde{G}(K)$. In
applications, this extra assumption can be obtained by replacing $K$ by a
finite extension. We write $\theta\colon U\dashrightarrow W$ for rational
maps, i.e., defined on an open dense $U_{0}\subseteq U$.

######  Corollary 1.5.

Assume that, with notation as above,
$p^{-1}(\Gamma)\leqslant\widetilde{G}(K)$, and assume that there exists a
Zariski dense subset of $G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$ consisting
of elements fixing finitely many points on
$X(\accentset{\rule{4.89998pt}{0.8pt}}{K})$. Then one of the following holds:

  1. (i)

There exists a rational map $\theta\colon G\dashrightarrow X$ such that
$\theta(g)g=\theta(g)$ for every $g$ in the domain of $\theta$. Moreover, if
$X$ is projective, every element of $G(K)$ fixes a point of $X(K)$.

  2. (ii)

There exists $C>0$ such that

$\mathbf{P}(\omega_{n}\mbox{ has a fixed point in $X(K)$})\leqslant\Xi(C,n).$

We have, therefore, a dichotomy: Either generically every element of $G$ fixes
a point on $X$, or almost surely a long random walk will hit elements with no
rational fixed points.

To have an example, one may take $X$ to be a flag variety (Corollary 4.3). For
more examples, we refer the reader to [Cor07].

###  1.5. Function fields

The goal of this subsection is to extend our results to global fields of
positive characteristic. The approach we take over number fields works for
general global fields. However, the proof of Theorem 1.1 is based on the
“strong approximation theorem”, which is more delicate in positive
characteristic (see, e.g., [LS12, pp. 416–417]). To overcome this obstacle, we
add assumptions on $G$ and $\Gamma$, in such a way that the setting remains
quite general.

We start by modifying Notation list 1: $K$ denotes any global function field.
We assume that $G$ is simply connected and semisimple. (See Remark 2.10 for a
more general setting to which our results apply.) In this case,
$G=\prod_{i=1}^{t}G_{i}$, where $G_{i}$ is simply connected and $K$-simple for
each $i=1,\ldots,t$.

We consider more specific $\Gamma$. Let $S$ be a finite set of places of $K$,
let $\mathcal{O}_{S}$ be the ring of $S$-integers of $K$ and, for $\nu\in S$,
let $K_{\nu}$ be the completion of $K$ with respect to $\nu$. Assume that
$\prod_{\nu\in S}G_{i}(K_{\nu})$ is non-compact for every $i=1\ldots,t$; in
particular, $S$ is non-empty. We then take

$\Gamma=G(\mathcal{O}_{S})$

to be the corresponding $S$-arithmetic subgroup, and we assume that $\Gamma$
is finitely generated. To have a concrete example in mind, we note that for
the group $G=\mathrm{SL}_{N}$ with $N\geqslant 3$, all these assumptions are
satisfied for every non-empty $S$ and for every $K$.

######  Theorem 1.6.

Let $Z$ be a thin set of $G(K)$. Then, there exists $C>0$ such that

$\mathbf{P}(\omega_{n}\in Z)\leqslant Cn^{-1/(10\,\mathrm{dim}\,G)}.$

We extend Corollary 1.3 to function fields.

######  Corollary 1.7.

Let $V$ be a normal irreducible $K$-variety, and $\pi\colon V\to G$ be a
finite surjective morphism, which is generically Galois with group
$\mathcal{G}$. Then, there exists $C>0$ such that

$\mathbf{P}(\mathcal{G}_{\omega_{n}}\neq\mathcal{G})\leqslant
Cn^{-1/(10\,\mathrm{dim}\,G)}.$

We also obtain a result on characteristic polynomials over function fields.

######  Corollary 1.8.

Fix an embedding $\rho\colon G\hookrightarrow\mathrm{GL}_{N}$. Let $\Pi(G)$ be
the extension of the Weyl group of $G$, as explained in § 1.3, and put
$\chi_{n}=\mathrm{det}(X-\omega_{n}^{\rho})$. Then, there exists $C>0$ such
that

$\mathbf{P}(\chi_{n}\mbox{ has an inseparable irreducible factor or
}\mathrm{Gal}(\chi_{n}/K)\not\cong\Pi(G))\leqslant
Cn^{-1/(10\,\mathrm{dim}\,G)}.$

We remark that there exists a finite place $\nu$ such that, if $\nu\in S$,
then Theorem 1.6, Corollary 1.7 and Corollary 1.8 hold with an exponential
bound $e^{-n/C}$. See Remark 2.8.

###  1.6. Acknowledgements

The authors are grateful to Arno Fehm and Sebastian Petersen for stimulating
discussions. They also thank Alex Lubotzky and Alexei Entin for an explanation
on strong approximation and on Bertini theorem, respectively. Finally, they
thank Pietro Corvaja for pointing out Fei Liu’s paper.

Both authors were partially supported by a grant of the Israel Science
Foundation No. 702/19, and the second named author has received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 850956).

##  2\. Simply connected groups: Large sieve

In this section we will prove Theorem 1.1 in the case where $G$ is simply
connected, and we will prove Theorem 1.6. We introduce the following technical
notion.

######  Definition 2.1.

Let $K$ be a field, $V$ be an irreducible $K$-variety, and $Z$ be a thin set
of $V(K)$. We define the complexity of $Z$ to be the minimum of

$\max\\{t,\deg(V_{i}),\deg(\pi_{i}),N\mid i=0,\ldots,t\\},$

running over all $t\geqslant 0$, $N\geqslant 1$, subvarieties
$V_{0},\ldots,V_{t}\subseteq\mathbb{P}_{K}^{N}$, and morphisms $\pi_{i}\colon
V_{i}\to V$ subject to the following conditions:
$\mathrm{dim}(V_{0})<\mathrm{dim}(V)$; for every $i\geqslant 1$, $V_{i}$ is
irreducible, $\mathrm{dim}(V_{i})=\mathrm{dim}(V)$, $\pi_{i}$ is dominant
separable with $\deg(\pi_{i})\geqslant 2$;
$Z\subseteq\bigcup_{i=0}^{t}\pi_{i}(V_{i}(K))$.

Let $K$ be a global field and let $G$ be a smooth connected linear algebraic
$K$-group (any further requirement will be made explicit). Let $\Gamma$ be a
finitely generated Zariski dense subgroup of $G(K)$, and $A$ be a finite
symmetric generating multiset of $\Gamma$, with $1\in A$.

If $\mathfrak{p}$ is a nonzero prime of $\mathcal{O}_{K}$ (the ring of
integers of $K$) of sufficiently large norm, the equations defining $G$ can be
reduced modulo $\mathfrak{p}$, to give a smooth connected linear algebraic
group over $K(\mathfrak{p}):=\mathcal{O}_{K}/\mathfrak{p}$, that we still
denote by $G$. Since $\Gamma$ is finitely generated, if $\mathfrak{p}$ is
sufficiently large we get a well defined map $\Gamma\to G(K(\mathfrak{p}))$.

If $Q$ is a finite quotient of $\Gamma$, we will consider the Cayley graph of
$Q$ with respect to the multiset induced by $A$. We will say that $Q$ is a
one-sided $\epsilon$-expander with respect to $A$ if the second eigenvalue of
the normalized adjacency operator of the corresponding Cayley graph is at most
$1-\epsilon$.

######  Theorem 2.2.

With notation as above, assume that there exists a set $\mathscr{P}$ of primes
of $K$ of density one such that, for every distinct
$\mathfrak{p},\mathfrak{q}\in\mathscr{P}$, the morphism $\Gamma\rightarrow
G(k(\mathfrak{p}))\times G(k(\mathfrak{q}))$ is well defined and surjective.
Let $Z$ be a thin set of $G(K)$ of complexity $M$.

  1. (1)

There exists $C_{1}>0$ such that

$\mathbf{P}(\omega_{n}\in Z)\leqslant C_{1}n^{-1/(10\,\mathrm{dim}\,G)},$

where $C_{1}$ depends on $G$, $K$, $|A|$, $M$.

  2. (2)

Assume furthermore that, for every distinct
$\mathfrak{p},\mathfrak{q}\in\mathscr{P}$, the finite groups
$G(k(\mathfrak{p}))\times G(k(\mathfrak{q}))$ are one-sided
$\epsilon$-expanders with respect to $A$. Then, there exists $C_{2}>0$ such
that

$\mathbf{P}(\omega_{n}\in Z)\leqslant e^{-n/C_{2}},$

where $C_{2}$ depends on $G$, $K$, $|A|$, $M$, $\epsilon$.

###  2.1. Group large sieve method

The proof of Theorem 2.2 is based on the “group large sieve method”,
introduced by Kowalski [Kow08, Chapter 7]; revised and reformulated in a neat
way by Lubotzky–Meiri [LM12].

We first recall the main ideas of this method. Let $\Gamma$ be a finitely
generated group, $A$ be a finite symmetric generating multiset of $\Gamma$,
with $1\in A$, and $Z$ be a subset of $\Gamma$. The group sieve method
provides conditions for which a random walk on $\mathrm{Cay}(\Gamma,A)$ hits
elements of $Z$ with probability tending to zero exponentially fast.

The method is based on looking at finite quotients of $\Gamma$. In order to
get exponentially fast decay, the crux is that the Cayley graphs induced on
these quotients should be uniform expanders, so that the corresponding random
walks gets equidistributed in logarithmic time.

If the quotients are not expanders, we do not have equidistribution in
logarithmic time, but still, we have equidistribution in polynomial time. This
is sufficient in order to show that a random walk on $\mathrm{Cay}(\Gamma,A)$
hits elements of $Z$ with probability decaying as $n^{-c}$. We make this
precise in Lemma 2.5, below, which is the “polynomial” analogue of [LM12,
Theorem B]; see also [Bre15, Lemma 7.5]. We first need a lemma.

######  Lemma 2.3.

Let $G$ be a finite group, and let $A$ be a finite symmetric generating
multiset of $G$, with $1\in A$. Let $\omega_{n}$ denote the $n$-th step of the
uniform random walk on $\mathrm{Cay}(G,A)$, starting at the identity. Then,
for every $g\in G$,

$\left|\mathbf{P}(\omega_{n}=g)-\frac{1}{|G|}\right|\leqslant|G|^{1/2}\cdot\left(1-\frac{1}{|A||G|^{2}}\right)^{n}.$

###### Proof.

Let $1=\pi_{0}\geqslant\pi_{1}\geqslant\cdots\geqslant\pi_{\ell-1}\geqslant-1$
be the eigenvalues of the normalized adjacency operator associated to
$\mathrm{Cay}(G,A)$. Set $\pi_{*}:=\max\\{\pi_{1},|\pi_{\ell-1}|\\}$. By
[DSC93, Lemma 2], we have that, for every $g\in G$,

$\left|\mathbf{P}(\omega_{n}=g)-\frac{1}{|G|}\right|\leqslant|G|^{1/2}\cdot\pi_{*}^{n}.$

Now, [DSC93, Lemma 1 and Corollary 1] show that

$\pi_{*}\leqslant 1-\frac{1}{|A||G|^{2}}.$

This concludes the proof. ∎

We recall another lemma, which is an application of Chebyshev’s inequality.

######  Lemma 2.4.

Let $A_{i}$, $i=1,\ldots,t$ be events in a probability space. Assume that

  1. (1)

$\mathbf{P}(A_{i})\leqslant 1-\beta$ for every $i$.

  2. (2)

$\mathbf{P}(A_{i}\cap
A_{j})\leqslant\mathbf{P}(A_{i})\mathbf{P}(A_{j})+\delta$ for every $i\neq j$.

Then,

$\mathbf{P}\left(\bigcap_{i=1}^{t}A_{i}\right)\leqslant\frac{1}{\beta^{2}}\left(\delta+\frac{\beta}{t}\right).$

###### Proof.

See [LM12, Lemma 3.1]. (In fact, that lemma is stated with $1/t$ replacing
$\beta/t$ in the conclusion, but the proof gives the above, slightly stronger,
statement.) ∎

######  Lemma 2.5.

Let $\Gamma$ be a finitely generated group, let $A$ be a finite symmetric
generating multiset of $\Gamma$, with $1\in A$, and let $Z$ be a subset of
$\Gamma$. Let $N_{1},\ldots,N_{t}$ be finite index normal subgroups of
$\Gamma$. For every $i$, denote by $\pi_{i}:\Gamma\rightarrow\Gamma/N_{i}$ the
natural projection. Assume that there exist positive constants $C$, $D$ and
$0<\alpha<1$ such that the following conditions are satisfied.

  1. (1)

(CRT condition) $\Gamma/(N_{i}\cap
N_{j})\cong\Gamma/{N_{i}}\times\Gamma/{N_{j}}$ for every $i\neq j$.

  2. (2)

(Quotients grow sub-polynomially) $|\pi_{i}(\Gamma)|\leqslant C\cdot t^{D}$
for every $i$.

  3. (3)

(Large sieve assumption) $|\pi_{i}(Z)|\leqslant(1-\alpha)|\pi_{i}(\Gamma)|$
for every $i$.

Then, for every $n\geqslant 10|A|C^{5}t^{5D}/\alpha$,

$\mathbf{P}(\omega_{n}\in Z)\leqslant\frac{3}{\alpha t}.$

In particular, if $t\asymp n^{1/(5D)}$, we get $\mathbf{P}(\omega_{n}\in Z)\ll
n^{-1/(5D)}$, where the implied constant depends polynomially on
$1/\alpha,|A|,C$.

###### Proof.

For every $i\in\\{1,\ldots,t\\}$ and for every $y\in\pi_{i}(\Gamma)$, by Lemma
2.3 and by assumption (2) we have

$\displaystyle\left|\mathbf{P}(\pi_{i}(\omega_{n})=y)-\frac{1}{|\pi_{i}(\Gamma)|}\right|$
$\displaystyle\leqslant|\pi_{i}(\Gamma)|^{1/2}\cdot\left(1-\frac{1}{|A||\pi_{i}(\Gamma)|^{2}}\right)^{n}$
(3) $\displaystyle\leqslant
C^{1/2}t^{D/2}\cdot\left(1-\frac{1}{|A|C^{2}t^{2D}}\right)^{n}.$

By the same reasoning and by assumption (1), for every $i\neq j$ and
$y_{i}\in\pi_{i}(\Gamma)$, $y_{j}\in\pi_{j}(\Gamma)$, we have

$\displaystyle\left|\mathbf{P}(\pi_{i}(\omega_{n})=y_{i},\pi_{j}(\omega_{n})=y_{j})-\frac{1}{|\pi_{i}(\Gamma)||\pi_{j}(\Gamma)|}\right|$
(4)
$\displaystyle\qquad\leqslant(|\pi_{i}(\Gamma)||\pi_{j}(\Gamma)|)^{1/2}\cdot\left(1-\frac{1}{|A|(|\pi_{i}(\Gamma)||\pi_{j}(\Gamma)|)^{2}}\right)^{n}$
$\displaystyle\qquad\leqslant
Ct^{D}\cdot\left(1-\frac{1}{|A|C^{4}t^{4D}}\right)^{n}.$

Now, let $A_{i}$ denote the event “$\pi_{i}(\omega_{n})\in\pi_{i}(Z)$”.
Applying a union bound and three times the triangle inequality, we deduce from
(3) and (4) that

$\mathbf{P}(A_{i}\cap
A_{j})\leqslant\mathbf{P}(A_{i})\mathbf{P}(A_{j})+3C^{3}t^{3D}\cdot\left(1-\frac{1}{|A|C^{4}t^{4D}}\right)^{n}.$

Set now

$\displaystyle\delta$
$\displaystyle:=3C^{3}t^{3D}\cdot\left(1-\frac{1}{|A|C^{4}t^{4D}}\right)^{n},$
$\displaystyle\beta$ $\displaystyle:=\alpha/2.$

For every $n\geqslant 10|A|C^{5}t^{5D}/\alpha$, we have that
$4\delta/\alpha^{2}\leqslant 1/\alpha t$. Applying Lemma 2.4 to the events
$A_{1},\ldots,A_{t}$, we get that

$\mathbf{P}(\omega_{n}\in
Z)\leqslant\mathbf{P}\left(\bigcap_{i=1}^{t}A_{i}\right)\leqslant\frac{4}{\alpha^{2}}\left(\delta+\frac{\alpha}{2t}\right)\leqslant\frac{3}{\alpha
t}.$

This concludes the proof. ∎

We will now prove Theorem 2.2.

###### Proof of Theorem 2.2.

We will first prove item (1). Since the constant $C_{1}$ in the statement
depends on the complexity $M$ of the thin set $Z$, we may assume that $Z=C(K)$
where $C$ is a proper closed subvariety of $G$, or $Z=\pi(V(K))$ for some
separable dominant morphism $\pi\colon V\to G$, with $V$ quasi-projective,
$\mathrm{dim}(V)=\mathrm{dim}(G)$, and $\deg(\pi)\geqslant 2$. The case where
$Z=C(K)$ will be dealt in Lemma 2.6, below, under weaker assumptions.
Therefore, we focus here on the case $Z=\pi(V(K))$. We may assume that $V$ is
geometrically irreducible, for otherwise $V(K)$ is contained in a proper
closed subset of $V$, hence $Z$ is contained in a proper closed subset of $G$
and we are in the previous case.

Let $K_{\pi}$ be the algebraic closure of $K$ in the Galois closure of
$K(V)/K(G)$. Let $\mathscr{P}$ be the set of primes given in the statement of
Theorem 2.2. Let $\mathscr{P}^{\prime}$ be the set of primes $\mathfrak{p}$ of
$\mathscr{P}$ having the following property:

  * $\diamond$

$\mathfrak{p}$ splits completely in $K_{\pi}$, and $\text{N}(\mathfrak{p})$ is
sufficiently large, depending on $G$ and $M$, in such a way that
$|\pi(V(K(\mathfrak{p})))|\leqslant(1-\frac{1}{2M!})\cdot|G(K(\mathfrak{p}))|$.
(Here, we still use the letters $G$ and $V$ to denote the base change to
$K(\mathfrak{p})$ of models over a suitable ring of $S$-integers of $K$.)

We can ensure this condition in view of the Lang–Weyl estimate [LW54] and in
view of [Ser16, Theorem 3.6.2]. The latter is stated over number fields, but
the proof works also over function fields if $\pi$ is separable.

We order the primes in $\mathscr{P}^{\prime}$, in such a way that the norm of
the primes is non-decreasing, and we get a sequence
$(\mathfrak{p}_{i})_{i\in\mathbf{N}}$. Let $n$ be a positive integer; our aim
is to bound $\mathbf{P}(\omega_{n}\in Z)$. We want to apply Lemma 2.5 to the
finite quotients $G(K(\mathfrak{p}_{i}))$, $i=1,\ldots,t$, of $\Gamma$, for
some $t$ that will be chosen later. We now check that conditions (1), (2) and
(3) of that lemma are satisfied. Condition (1) is the main assumption of
Theorem 2.2, so it is satisfied.

By Chebotarev Density Theorem, the density of primes of $K$ splitting
completely in $K_{\pi}$ is $1/|K_{\pi}:K|\geqslant 1/M!$, hence so is the
density of primes in $\mathscr{P}^{\prime}$. Combining this with Landau Prime
Ideal Theorem, we get $\text{N}(\mathfrak{p}_{i})\ll i\log i\ll i^{2}$, where
the implied constant depends on $G$, $K$, $M$. Hence, by the Lang–Weyl
estimate, we get $|G(K(\mathfrak{p}_{i}))|\ll i^{2\,\mathrm{dim}(G)}$. In
particular, condition (2) of Lemma 2.5 is satisfied, with
$D=2\cdot\mathrm{dim}(G)$. Finally, condition (3) holds by the definition of
$\mathscr{P}^{\prime}$. At this point, we can choose any $t\asymp
n^{1/(10\,\mathrm{dim}\,G)}$, and Theorem 2.2(1) follows from Lemma 2.5.

The proof of Theorem 2.2(2) is similar, but instead of using Lemma 2.5, we use
[LM12, Theorem B]. ∎

###  2.2. Proof of Theorems 1.1 and 1.6 for $G$ simply connected

###### Proof of Theorem 1.1 for $G$ simply connected.

Let $G$ be a connected simply connected linear algebraic $K$-group, where $K$
is a number field, and let $\Gamma$ be a finitely generated Zariski dense
subgroup of $G(K)$. Let $\mathscr{P}$ be the set of nonzero primes of
$\mathcal{O}_{K}$ having inertia degree one over $\mathbf{Q}$, and of
sufficiently large norm so that $G$ can be reduced modulo $\mathfrak{p}$ and
we have a map $\Gamma\to G(K(\mathfrak{p}))$ for every
$\mathfrak{p}\in\mathscr{P}$. Note that $\mathscr{P}$ has density one among
all primes. We recall two facts.

(i) The “strong approximation theorem” (see Nori [Nor87, § 5]) implies that,
possibly removing finitely many primes from $\mathscr{P}$, the morphism
$\Gamma\to G(K(\mathfrak{p}))\times G(K(\mathfrak{q}))$ is surjective for
every $\mathfrak{p}\neq\mathfrak{q}\in\mathscr{P}$. (We refer to [LS12, pp.
399–417] and [PR94, Chapter 7] for thorough discussions on strong
approximation.)

(ii) Assume, furthermore, that $G$ is semisimple. The “superstrong
approximation theorem”, proved by Salehi Golsefidy and Varjú [SGV12, Corollary
6], asserts that, possibly removing finitely many primes from $\mathscr{P}$,
the quotients $G(k(\mathfrak{p}))\times G(k(\mathfrak{q}))$ of $\Gamma$ are
$\epsilon$-expanders for every $\mathfrak{p}\neq\mathfrak{q}\in\mathscr{P}$
and for some $\epsilon>0$ independent of $\mathfrak{p}$.

Now Theorem 1.1 for $G$ simply connected follows from (i), (ii), and Theorem
2.2. ∎

###### Proof of Theorem 1.6.

Let $K$ be a global function field, let $G$ be connected and simply connected
and let $\Gamma=G(\mathcal{O}_{S})$ be an $S$-arithmetic group, as in § 1.5.
Prasad [Pra77] proved the strong approximation theorem for function fields.
(The theorem is stated for simply connected $K$-simple groups, but a simply
connected semisimple group is a direct product of such factors, and strong
approximation behaves well with respect to direct products; see, e.g., [PR94,
Proposition 7.1].) This implies that the assumptions of Theorem 2.2 are
satisfied, with $\mathscr{P}$ denoting the set of all primes of sufficiently
large norm (in terms of $G$ and $\Gamma$). Then, Theorem 1.6 follows from
Theorem 2.2(1). ∎

Recall that, in the proof of Theorem 2.2, we only dealt with thin sets of the
form $Z=\pi(V(K))$. We now close the gap, by proving Theorems 1.1 and 1.6 in
the case $Z=C(K)$ where $C$ is a proper closed subset of $G$. In particular,
over number fields we do not require that $G$ is simply connected. The case
$Z=C(K)$ is easier: It is sufficient to reduce modulo a single prime, rather
than modulo pairs of primes, as in Lemma 2.5. We decided to postpone the proof
here because we need the results on strong and superstrong approximation,
which we recalled only in this subsection.

######  Lemma 2.6.

Theorems 1.1 and 1.6 hold when $Z=C(K)$ and $C$ is a proper closed subset of
$G$.

###### Proof.

Let $\mathfrak{p}$ be a prime of sufficiently large norm, to be determined
later, and of inertia degree one over $\mathbf{Q}$ in the case where $K$ is a
number field. Denote by $\pi_{\mathfrak{p}}\colon\Gamma\to G(K(\mathfrak{p}))$
the reduction morphism. We note that the index of $\pi_{\mathfrak{p}}(\Gamma)$
in $G(K(\mathfrak{p}))$ is bounded, independently of $\mathfrak{p}$. Indeed,
for function fields, this follows from strong approximation [Pra77]; for
number fields (where $G$ is not necessarily simply connected), see [Nor87, §
5]. Then, by the Lang–Weyl estimate,
$|\pi_{\mathfrak{p}}(\Gamma)|\asymp\text{N}(\mathfrak{p})^{\mathrm{dim}(G)}$,
and
$|\pi_{\mathfrak{p}}(Z\cap\Gamma)|\ll\text{N}(\mathfrak{p})^{\mathrm{dim}(G)-1}$,
so
$|\pi_{\mathfrak{p}}(Z\cap\Gamma)|\ll|\pi_{p}(\Gamma)|/\text{N}(\mathfrak{p})$.
Now, reasoning as in (3) in the proof of Lemma 2.5, we get

$\displaystyle\mathbf{P}(\omega_{n}\in Z)$
$\displaystyle\leqslant\mathbf{P}(\pi_{\mathfrak{p}}(\omega_{n})\in\pi_{\mathfrak{p}}(Z\cap\Gamma))$
$\displaystyle\leqslant\frac{|\pi_{\mathfrak{p}}(Z\cap\Gamma)|}{|\pi_{\mathfrak{p}}(\Gamma)|}+|\pi_{\mathfrak{p}}(Z\cap\Gamma)||\pi_{\mathfrak{p}}(\Gamma)|^{1/2}\cdot\left(1-\frac{1}{|A||\pi_{\mathfrak{p}}(\Gamma)|^{2}}\right)^{n}.$

If we choose $\mathfrak{p}$ of norm $\asymp n^{1/(3\,\mathrm{dim}\,G)}$, we
get $\mathbf{P}(\omega_{n}\in Z)\ll n^{-1/(3\,\mathrm{dim}\,G)}$, which proves
Theorem 1.1 for $G/\mathrm{R}_{u}(G)$ semisimple, as well as Theorem 1.6.
Assume now that $G$ is semisimple. Then, the quotients
$\pi_{\mathfrak{p}}(\Gamma)$ are uniform expanders [SGV12], hence (in the
notation of the proof of Lemma 2.3) $\pi_{*}$ is bounded away from one, and we
can choose $\mathfrak{p}$ of norm $\asymp e^{n/C}$, so that
$\mathbf{P}(\omega_{n}\in Z)\ll e^{-n/C}$. This concludes the proof of Theorem
1.1. (The statement for $G$ semisimple and $Z=C(K)$ is known and can be found
for instance in [LM12, Proposition 2.7].) ∎

Even in the case $Z=C(K)$, in general we cannot get exponential decay, as (2)
demonstrates. We conclude this section with some remarks on Theorem 1.1 and
Theorem 1.6.

######  Remark 2.7.

In the proof of Theorem 1.1 for $G$ simply connected, we have not used any
assumption on $Z$, beyond the fact that it is a thin set. Therefore, for
simply connected groups $G$, in Theorem 1.1 it is not necessary to assume that
the varieties are normal and the morphisms are finite and ramified. (We recall
that, in fact, $G$ does not admit absolutely irreducible nontrivial unramified
covers.) The same remark applies to Corollary 1.2 and Corollary 1.3, which
will be proved in § 3.

######  Remark 2.8.

Let $K$ be a global function field and let $G$ and $\Gamma$ be as in Theorem
1.6. There exists a finite place $\nu$ of $K$ such that, if $\nu\in S$, then
the quotients $G(K(\mathfrak{p}))\times G(K(\mathfrak{q}))$ of
$\Gamma=G(\mathcal{O}_{S})$ are uniform expanders, for
$\mathfrak{p}\neq\mathfrak{q}$ of sufficiently large norm (see Lubotzky and
Zuk [LZ, Theorem 4.4]). If such $\nu$ belongs to $S$ then, using Theorem
2.2(2), we deduce that we can get exponential decay in Theorem 1.6:

$\mathbf{P}(\omega_{n}\in Z)\leqslant e^{-n/C}.$

In fact, conjecturally Theorem 4.4 of [LZ] should hold for every $S$, see [LZ,
remarks after Theorem 4.4], so that in Theorem 1.6 one would get exponential
decay for every $S$. The same remark applies to Corollaries 1.7 and 1.8, which
will be proved in § 3 and § 4.1.

######  Remark 2.9.

In Theorem 1.6, we require that $\Gamma=G(\mathcal{O}_{S})$ is finitely
generated (which, of course, is a necessary assumption for our method). We
note that this holds in many cases. For instance, if $G$ is $K$-simple, then
$G(\mathcal{O}_{S})$ is finitely generated unless $\sum_{\nu\in
S}\text{rank}(G(K_{\nu}))=1$ (see [LW93, Remark 2.2]).

######  Remark 2.10.

The proof of Theorem 1.6 works under more general assumptions. For instance,
let $S$ be a finite set of places, and $L$ be simply connected, semisimple,
such that $\prod_{\nu\in S}L_{i}(K_{\nu})$ is non-compact for every $K$-simple
factor $L_{i}$ of $L$. Let $U$ be unipotent and $K$-split (i.e., it admits a
composition series with factors isomorphic to $\mathbb{A}_{K}^{1}$). Assume
that $L$ acts on $U$, and set $G=U\rtimes L$ and $\Gamma=G(\mathcal{O}_{S})$.
Assume that $\Gamma$ is finitely generated.

Then, Theorem 1.6 holds for $G$ and $\Gamma$. Indeed, $U$ is isomorphic to
some $\mathbb{A}_{K}^{m}$ as a $K$-variety (see [KMT74, Remark A.3]), and so
satisfies strong approximation with respect to $S$. Now, as a $K$-variety
$G\cong\mathbb{A}_{K}^{m}\times L$, therefore $G$ satisfies strong
approximation with respect to $S$ and the claim follows from Theorem 2.2(2).

##  3\. General case: a pullback argument

In this section we will prove Theorem 1.1 and Corollaries 1.2, 1.3 and 1.7.

We assume first that $K$ is a number field, and $G$ is a connected linear
algebraic $K$-group, such that $G/\mathrm{R}_{u}(G)$ is either trivial or
semisimple. In the first case, $G$ is unipotent, hence simply connected. This
case was considered in § 2.2, hence we may assume that $G/\mathrm{R}_{u}(G)$
is semisimple. Then, $G$ admits a universal cover ([Mil17, Theorem 18.25 and
Remark 18.27]), that is, a connected simply connected linear algebraic
$K$-group $\widetilde{G}$, equipped with an isogeny (i.e., surjective
homomorphism with finite central kernel) $p\colon\widetilde{G}\to G$. We fix
this and other notation, as follows.

######  Notation list 2.

  * $\diamond$

Notation list 1 holds.

  * $\diamond$

$\accentset{\rule{4.89998pt}{0.8pt}}{K}$ is an algebraic closure of $K$.

  * $\diamond$

$G$ is such that $G/\mathrm{R}_{u}(G)$ is semisimple.

  * $\diamond$

$\widetilde{G}$ is the universal cover of $G$, with isogeny
$p\colon\widetilde{G}\to G$.

  * $\diamond$

$E$ is a finite extension of $K$ such that
$p^{-1}(\Gamma)\subseteq\widetilde{G}(E)$.

  * $\diamond$

$\widetilde{\Gamma}:=p^{-1}(\Gamma)$.

  * $\diamond$

$\widetilde{A}:=p^{-1}(A)$, a finite symmetric generating multiset of the
Zariski dense subgroup $\widetilde{\Gamma}$ of $\widetilde{G}_{E}(E)$.

  * $\diamond$

$\widetilde{\omega_{n}}$ is the $n$-th step of the random walk on
$\mathrm{Cay}(\widetilde{\Gamma},\widetilde{A})$, starting at the identity.

(In the definition of $\widetilde{\Gamma}$ and $\widetilde{A}$, $p$ denotes
the morphism $\widetilde{G}(\accentset{\rule{4.89998pt}{0.8pt}}{K})\to
G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$ induced on
$\accentset{\rule{4.89998pt}{0.8pt}}{K}$-points.) Now, let $V$ be a
geometrically irreducible $K$-variety, with $\mathrm{dim}(V)=\mathrm{dim}(G)$,
and let $\pi:V\rightarrow G$ be a dominant morphism. Base change to $E$, and
get a pullback diagram

${\widetilde{V}:=V_{E}\times_{G_{E}}\widetilde{G}_{E}}$${\widetilde{G}_{E}}$${V_{E}}$${G_{E}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p}$$\scriptstyle{\pi}$
(5)

######  Lemma 3.1.

Let $Z=\pi(V(K))$. Under Notation list 2, one of the following holds:

  * (i)

There exists $C>0$ such that $\mathbf{P}(\omega_{n}\in Z)\leqslant\Xi(C,n)$.

  * (ii)

There exists an irreducible component $W$ of $\widetilde{V}$ such that
$\pi^{\prime}|_{W}\colon W\to\widetilde{G}_{E}$ has degree one. In particular,
there exists a rational map $\theta\colon\widetilde{G}_{E}\dashrightarrow
V_{E}$ such that $\pi\circ\theta=p$.

###### Proof.

In this proof, we will denote by $p,p^{\prime},\pi,\pi^{\prime}$ the morphisms
induced on $\accentset{\rule{4.89998pt}{0.8pt}}{K}$-points. Note, first, that
$\mathbf{P}(\omega_{n}\in Z)=\mathbf{P}(\omega_{n}\in
Z\cap\Gamma)=\mathbf{P}(\widetilde{\omega_{n}}\in p^{-1}(Z\cap\Gamma))$. Set
now $Y:=V(K)\cap\pi^{-1}(\Gamma)$. It follows from (5) that
$\pi^{\prime}(p^{\prime-1}(Y))=p^{-1}(Z\cap\Gamma)$, from which
$\mathbf{P}(\omega_{n}\in
Z)=\mathbf{P}(\widetilde{\omega_{n}}\in\pi^{\prime}(p^{\prime-1}(Y)))$.
Moreover, $p^{\prime-1}(Y)$ is a subset of $\widetilde{V}(E)$. Assume that
(ii) does not hold; then $\pi^{\prime}(p^{\prime-1}(Y))$ is a thin set of
$\widetilde{G}_{E}(E)$. Since the complexity of
$\pi^{\prime}(p^{\prime-1}(Y))$ is bounded in terms of $G$ and the complexity
of $Z$, (i) follows from Theorem 1.1 for simply connected groups, proved in §
2.2 (see also Remark 2.7). This concludes the proof of the lemma. ∎

At this point, we prove Theorem 1.1.

###### Proof of Theorem 1.1.

Since the constant $C$ in the statement depends on the complexity of $Z$, we
may assume that either $Z=X(K)$ where $X$ is a proper closed subvariety of
$G$, or $Z=\pi(V(K))$ for some $\pi\colon V\to G$, where $V$ is a normal
geometrically irreducible $K$-variety, and $\pi$ is finite, surjective and
ramified. The case $Z=X(K)$ follows from Lemma 2.6, hence we assume
$Z=\pi(V(K))$.

If conclusion (i) of Lemma 3.1 holds, we are done. Assume then that conclusion
(ii) holds, seeking for a contradiction. Since the finiteness of a morphism is
stable under base change and restriction to closed subvarieties, we have that
$\pi^{\prime}|_{W}\colon W\to\widetilde{G}_{E}$ is finite and surjective.
Since $\widetilde{G}_{E}$ is normal, Zariski’s main theorem implies that
$\pi^{\prime}|_{W}$ is an isomorphism. Set
$f:=p^{\prime}\circ(\pi^{\prime}|_{W})^{-1}\colon\widetilde{G}_{E}\to V_{E}$.
Then $f$ is a finite morphism, and $\pi\circ f=p$ is unramified. Since $V_{E}$
is normal, it follows that $\pi$ is unramified. This contradicts the choice of
$\pi$, and the proof is concluded. ∎

######  Remark 3.2.

(i) In Theorem 1.1, the assumption that the morphisms are ramified is
necessary. For instance, assume that $\pi\colon G_{1}\to G$ is an isogeny of
algebraic $K$-groups, of degree at least two, and let $\Gamma\leqslant G(K)$
be finitely generated and Zariski dense. Then, still denoting by $\pi$ the map
on $\accentset{\rule{4.89998pt}{0.8pt}}{K}$-points
$G_{1}(\accentset{\rule{4.89998pt}{0.8pt}}{K})\to
G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$, we have that
$Z:=\pi(\pi^{-1}(\Gamma)\cap G_{1}(K))$ is a finite index subgroup of $\Gamma$
(see, e.g., [LS12, Lemma 16.4.14]). It is easy, then, to see that a random
walk on a Cayley graph of $\Gamma$ hits elements of $Z$ with positive
probability, so that Theorem 1.1 fails.

(ii) The examples given in (i) cover essentially all the cases excluded in
Theorem 1.1. Indeed, in Theorem 1.1, we can rephrase the assumption that the
morphism $\pi_{i}\colon V_{i}\to G$ is ramified, by requiring that the
morphism $\pi_{i}\colon:(V_{i})_{E}\to G_{E}$ is not an isogeny of algebraic
groups; that is, there does not exist an isomorphism of $E$-varieties
$\phi_{i}\colon(V_{i})_{E}\to G_{i}$, where $G_{i}$ is an algebraic $E$-group,
and an isogeny $f_{i}\colon G_{i}\to G_{E}$ such that
$f_{i}\circ\phi_{i}=\pi_{i}$.

######  Remark 3.3.

In Theorem 1.1, the assumption that the varieties are normal is not crucial.
Some modification in the statement, however, is needed, because ramification
can be resolved by the normalization map. See Lemma 3.1 for a reformulation.

We now deduce Corollaries 1.3 and 1.7 on generically Galois covers, and
Corollary 1.2 on irreducibility of fibres.

###### Proof of Corollaries 1.3 and 1.7.

We first prove Corollary 1.3, so that $K$ is a number field. Let $V$ be a
normal irreducible $K$-variety, and $\pi\colon V\to G$ be finite, surjective
and generically Galois, with group $\mathcal{G}$. We may assume
$\deg(\pi)\geqslant 2$, otherwise the statement is trivial. Let $\Omega$ be
the set of maximal subgroups of $\mathcal{G}$. For $\mathcal{H}\in\Omega$, set
$V_{\mathcal{H}}:=V/\mathcal{H}$, a normal variety, and denote by
$\pi_{\mathcal{H}}\colon V_{\mathcal{H}}\to G$ the induced finite morphism.
Set $Z:=\cup_{\mathcal{H}\in\Omega}\pi_{\mathcal{H}}(V_{\mathcal{H}}(K))$, and
let $U$ be an open dense subset of $G$ such that the cover $\pi^{-1}(U)\to U$
is étale. The set of elements $g$ of $U(K)$ whose specialized Galois group
$\mathcal{G}_{g}$ is not generic is contained in $Z$ (see, e.g., [Ser16,
Proposition 3.3.1]). In view of Lemma 2.6, it is then sufficient to bound
$\mathbf{P}(\omega_{n}\in Z)$.

Note, now, that if $V_{\mathcal{H}}$ is not geometrically irreducible, then
$V_{\mathcal{H}}(K)=\varnothing$, and so can be removed from the definition of
$Z$. On the other hand, if $V_{\mathcal{H}}$ if geometrically irreducible,
then $\pi_{\mathcal{H}}$ is ramified, by the assumptions of Corollary 1.3. In
particular, $Z$ is a thin subset of $G(K)$ that satisfies the assumptions of
Theorem 1.1; so Theorem 1.1 holds for $Z$ and Corollary 1.3 follows.

The proof of Corollary 1.7 is identical, using Theorems 1.6 (which has no
ramification assumption) instead of Theorem 1.1. ∎

###### Proof of Corollary 1.2.

We are under Notation list 2. Let $V$ be a normal irreducible $K$-variety, and
$\pi\colon V\to G$ be a finite surjective morphism. Let
$L:=\accentset{\rule{4.89998pt}{0.8pt}}{K}\cap K(V)$, so that $K(G)\subseteq
LK(G)\subseteq K(V)$ and the morphism $\pi$ factors as $\pi=f\circ\phi$, where
$\phi\colon V\to G_{L}$ and $f\colon G_{L}\to G$ is the base change morphism.
Then, we may regard $V$ as an absolutely irreducible $L$-variety. Up to
replacing $E$ by $EL$, we may assume that $E$ contains $L$. Set
$V_{E}:=V\times_{L}E$. Still denoting by $\phi$ and $f$ the induced morphisms
$V_{E}\to G_{E}$ and $G_{E}\to G$, respectively, and by $\pi$ their
composition, we have a diagram

${\widetilde{V}:=V_{E}\times_{G_{E}}\widetilde{G}_{E}}$${\widetilde{G}_{E}}$${V_{E}}$${G_{E}}$${G}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p}$$\scriptstyle{\pi}$$\scriptstyle{\phi}$$\scriptstyle{f}$

By assumption and by the choice of $L$, every nontrivial subcover of $\phi$ is
ramified. Since $p$ is unramified, it follows that $\widetilde{V}$ is
irreducible. Let $\widehat{V}$ be the normalization of $\widetilde{G}_{E}$ in
the Galois closure of $E(\widetilde{V})/E(\widetilde{G}_{E})$. We get a
generically Galois morphism $\psi\colon\widehat{V}\to\widetilde{G}_{E}$, whose
group we denote by $\mathcal{G}$. Then

$\displaystyle\mathbf{P}(\pi^{-1}(\omega_{n})\mbox{ is not integral})$
$\displaystyle\qquad=\mathbf{P}(\phi^{-1}(\omega_{n})\mbox{ is not integral})$
$\displaystyle\qquad\leqslant\mathbf{P}(\phi^{\prime-1}(\widetilde{\omega}_{n})\mbox{
is not integral})$
$\displaystyle\qquad\leqslant\mathbf{P}(\mathcal{G}_{\widetilde{\omega}_{n}}\neq\mathcal{G})$
$\displaystyle\qquad\leqslant\Xi(C,n),$

where the last inequality follows from Corollary 1.3 (see also Remark 2.7).
This concludes the proof. ∎

##  4\. Applications

In this section we prove Corollaries 1.4, 1.8 and 1.5.

###  4.1. Corollaries 1.4 and 1.8: Characteristic polynomial

Let $K$ be a global field, and let $G$ be as in Corollary 1.4 or 1.8.

We will use that $G$ has the HP over $K$, which, of course, follows from
Theorems 1.1 and 1.6. (In fact, this is known for more general $G$ and $K$,
see [BSFP14, CTS87].)

Fix an embedding $\rho\colon G\hookrightarrow\mathrm{GL}_{N}$. Let $F$ be the
largest separable extension of $K(G)$ contained in the splitting field of
$\chi_{G,\rho}$ over $K(G)$. Let $V$ be the normalization of $G$ in $F$, and
let $\pi\colon V\to G$ be the corresponding finite surjective morphism, which
is generically Galois with group $\mathcal{G}:=\mathrm{Gal}(F/K(G))$.

We now recall some preliminary facts from [JKZ13, § 2] or [PR03]. For an
element $g\in G(K)$, let $K_{g}$ the splitting field of
$\mathrm{det}(X-g^{\rho})$ over $K$. If $g$ is semisimple or $K$ is a number
field, then $K_{g}$ is a Galois extension of $K$, which is independent of the
choice of the embedding $\rho$. For every $g\in G(K)$, let $\mathcal{G}_{g}$
be the specialized group of $\pi$ at $g$. The following holds:

  * $(\star)$

There exists an open dense subset $U_{0}$ of $G$ such that, for every $g\in
U_{0}(K)$, $K_{g}/K$ is Galois and
$\mathcal{G}_{g}\cong\mathrm{Gal}(K_{g}/K)$.

Now, if $K$ is a number field, $K_{g}=K_{g^{\phi}}$, where $\phi\colon G\to
G/\mathrm{R}_{u}(G)$ (see [JKZ13, Lemma 2.3]). In particular, if $K$ is a
number field we may replace $G$ by $G^{\phi}$, $\Gamma$ by $\Gamma^{\phi}$,
and assume that $G$ is semisimple. Note that this holds by assumption if $K$
is a function field.

  * $(\star\star)$

From now on, therefore, $G$ is a semisimple group.

Let $K^{s}$ be a fixed separable closure of $K$. For a maximal $K$-torus $T$
of $G$, we let $X(T):=\mathrm{Hom}(T_{K^{s}},(\mathbb{G}_{m})_{K^{s}})$, the
character group of $T$, and let $K_{T}$ be the splitting field of $T$, a
Galois extension of $K$ ([Mil17, Theorem 12.18]). Let $W(G,T)$ denote the
$K^{s}$-points of the group scheme $\text{N}_{G}(T)/T$. Then $W(G,T)$ is a
finite group, called the absolute Weyl group of $G$ with respect to $T$. Now,
$W(G,T)$ and $\mathrm{Gal}(K_{T}/K)$ can be viewed as subgroups of
$\text{Aut}(X(T))$; define, then, $\Pi(G,T):=\langle
W(G,T),\mathrm{Gal}(K_{T}/K)\rangle=W(G,T)\mathrm{Gal}(K_{T}/K)\leqslant\text{Aut}(X(T))$.
This is a finite group and, if $T$ is $K$-split, $\Pi(G,T)=W(G,T)$. It turns
out that the isomorphism type of this group is independent of the choice of
$T$, so that we can define $\Pi(G):=\Pi(G,T)$ for any choice of $T$.

It follows from a result of Prasad and Rapinchuk [PR03], and from the fact
that $G$ has the HP over $K$, that $\Pi(G)$ is the generic Galois group of
$\pi$:

######  Lemma 4.1.

$\mathcal{G}\cong\Pi(G)$.

###### Proof.

We divide the proof of the lemma in two parts.

(i) We show that $\mathcal{G}$ is isomorphic to a subgroup of $\Pi(G)$.

There exists an open dense subset $U$ of $G$ such that, for every $g\in U(K)$,
$g$ is regular semisimple and, denoting by $T_{g}$ the maximal torus of $g$,
$K_{g}=K_{T_{g}}$, so that $\mathrm{Gal}(K_{g}/K)$ is isomorphic to a subgroup
of $\Pi(G)$ (see, e.g., [JKZ13, Lemma 2.4], which is stated over number field
but does not use this). Let $U_{0}$ be as in $(\star)$. By the HP, we know
that $\mathcal{G}_{g}\cong\mathcal{G}$ for at least one $g\in U\cap U_{0}(K)$,
and in particular, $\mathcal{G}$ is isomorphic to a subgroup of $\Pi(G)$.

(ii) We show that $\mathcal{G}\cong\Pi(G)$.

In [PR03, Theorem 3 and Remark 6] (see also [PR14, p. 242]), it was shown
that, if $G$ is semisimple, there exists a Zariski dense subset $U^{\prime}$
of $G(K)$ such that, for every $g\in U^{\prime}$, $g$ is regular semisimple
and, with notation as in (i), $\mathrm{Gal}(K_{T_{g}}/K)=\Pi(G,T)$. In
particular, taking $g\in U^{\prime}\cap U\cap U_{0}$, we get that
$\mathcal{G}_{g}\cong\mathrm{Gal}(K_{g}/K)=\mathrm{Gal}(K_{T_{g}}/K)=\Pi(G,T)\cong\Pi(G)$.
Hence $\Pi(G)$, as the Galois group of a specialization, is a sub-quotient of
$\mathcal{G}$. From (i), we deduce that $\mathcal{G}\cong\Pi(G)$. ∎

######  Remark 4.2.

In fact, we used only a weak form of [PR03], namely, the existence of one
element $g\in G(K)$ such that $\mathrm{Gal}(K_{g}/K)\cong\Pi(G)$. This already
implies that $\Pi(G)$ is the generic Galois group of $\pi$. In particular, by
the HP, it follows automatically that the set of elements $g\in G(K)$ such
that $\mathrm{Gal}(K_{g}/K)\cong\Pi(G)$ is Zariski dense in $G$.

###### Proof of Corollaries 1.4 and 1.8.

By $(\star\star)$, we may assume that $G$ is semisimple. Let $Z$ be the set of
elements $g$ of $G(K)$ such that $\mathrm{det}(X-g^{\rho})$ has an inseparable
irreducible factor or $\mathrm{Gal}(\mathrm{det}(X-g^{\rho})/K)$ is not
isomorphic to $\Pi(G)$. We want to show that $\mathbf{P}(\omega_{n}\in
Z)\leqslant e^{-n/C}$. By Lemma 2.6, it is sufficient to show that
$\mathbf{P}(\omega_{n}\in Z^{\prime})\leqslant e^{-n/C}$, where
$Z^{\prime}:=Z\cap U\cap U_{0}$, and $U_{0}$ and $U$ were defined in $(\star)$
and in the proof of Lemma 4.1, respectively. In particular, for $g\in
Z^{\prime}$, $g$ is regular semisimple, with maximal torus $T_{g}$,
$K_{g}=K_{T_{g}}$, and $\mathcal{G}_{g}\cong\mathrm{Gal}(K_{g}/K)$.

Assume first that $G$ is simply connected. Then the statement follows from
Corollary 1.3 (if $K$ is a number field) or Corollary 1.7 (if $K$ is a
function field), applied to the cover $\pi\colon V\to G$. In particular, from
now on we may assume that $G$ is not simply connected; hence, $K$ is a number
field. We use now Notation list 2.

We have that $\mathbf{P}(\omega_{n}\in
Z^{\prime})=\mathbf{P}(\widetilde{\omega_{n}}\in
p^{-1}(Z^{\prime}\cap\Gamma)$), where
$p^{-1}(Z^{\prime}\cap\Gamma)\leqslant\widetilde{G}_{E}(E)$. Now let $g\in
Z^{\prime}$ and, for ease of notation, set $T:=T_{g}$. Then
$\mathrm{Gal}(K_{T}/K)\nsupseteq W(G,T)$. Let $\widetilde{g}\in
p^{-1}(Z^{\prime}\cap\Gamma)$; note that $\widetilde{g}$ is regular
semisimple, and denote by $\widetilde{T}$ the maximal $E$-torus of
$\widetilde{G}_{E}$ containing $\widetilde{g}$. Then $p(\widetilde{T})=T_{E}$
and $E_{\widetilde{T}}=EK_{T}$, where $E_{\widetilde{T}}$ denotes the
splitting field of the torus $\widetilde{T}$. We see that
$\mathrm{Gal}(E_{\widetilde{T}}/E)\nsupseteq
W(\widetilde{G}_{E},\widetilde{T})\cong W(G,T)$, from which
$\mathrm{Gal}(E_{\widetilde{g}}/E)\not\cong\Pi(\widetilde{G}_{E})$. Then, by
the simply connected case, we deduce that
$\mathbf{P}(\widetilde{\omega_{n}}\in p^{-1}(Z^{\prime}\cap\Gamma))\leqslant
e^{-n/C}$, which concludes the proof. ∎

###  4.2. Corollary 1.5: Fixed points

###### Proof of Corollary 1.5.

We use Notation list 2. By assumption,
$p^{-1}(\Gamma)\leqslant\widetilde{G}(K)$, hence we may take $E=K$.

Note, first, that the set of elements of $G(K)$ fixing a point on $X(K)$ is
$\pi(V(K))$. Let $W$ be an irreducible component of $V$. By assumption, there
exists a Zariski dense subset of $G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$
consisting of elements fixing finitely many points on
$X(\accentset{\rule{4.89998pt}{0.8pt}}{K})$. This implies that, if $\pi|_{W}$
is dominant, then $\pi|_{W}$ has finite degree. We can, therefore, apply Lemma
2.6 or Lemma 3.1 to each irreducible component $W$ and to the set
$Z=\pi(W(K))$.

We deduce that, if conclusion (ii) of Corollary 1.5 does not hold, then there
exists a rational map $\theta^{\prime}:\widetilde{G}\dashrightarrow V$ such
that $\pi\circ\theta^{\prime}=p$.

At this point, [Cor07, Proposition 5.4] shows that $\theta^{\prime}$ is
constant on the fibres of $p$. In particular, there exists a rational map
$\theta^{\prime\prime}\colon G\dashrightarrow V$ such that
$\pi\circ\theta^{\prime\prime}=\text{id}$. Setting
$\theta:=\pi_{X}\circ\theta^{\prime\prime}$, where $\pi_{X}\colon V\to X$ is
the natural projection, proves the first part of item (i) in the statement.
Assume now that $X$ is projective; let $U$ be the domain of $\theta$ and let
$g\in G(K)\setminus U$. Let $C$ be a smooth irreducible $K$-curve passing
through $g$ and having non-empty intersection with $U$. Since $X$ is
projective, the rational map $\theta|_{C}\colon C\dashrightarrow X$ can be
extended to a morphism $\accentset{\rule{4.89998pt}{0.8pt}}{\theta}\colon C\to
X$. Then
$\accentset{\rule{4.89998pt}{0.8pt}}{\theta}(g)g=\accentset{\rule{4.89998pt}{0.8pt}}{\theta}(g)$,
which proves the last part of item (i). (This argument can be found in [Cor07,
p. 594].) This concludes the proof. ∎

An interesting special case of Corollary 1.5 comes from the action of $G$ on
flag varieties. Fix a morphism $\rho\colon G\to\mathrm{GL}_{N}$, and let $V$
be an $N$-dimensional $K$-vector space. Let $\mathbf{a}=(a_{1},\ldots,a_{t})$
be a sequence of integers, with $t\geqslant 1$ and $0<a_{1}<\cdots<a_{t}<N$.
Then $G(K)$ acts naturally on the set of $\mathbf{a}$-flags of $V$

$0<W_{1}<\cdots<W_{t}<V,$

where $W_{i}$ is a $K$-subspace of $V$ of dimension $a_{i}$. Similarly,
$G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$ acts on the set of
$\mathbf{a}$-flags of
$\accentset{\rule{4.89998pt}{0.8pt}}{V}:=V\otimes_{K}\accentset{\rule{4.89998pt}{0.8pt}}{K}$.

######  Corollary 4.3.

Assume that, with notation as above,
$p^{-1}(\Gamma)\leqslant\widetilde{G}(K)$, and assume that there exists a
Zariski dense subset of $G(\accentset{\rule{4.89998pt}{0.8pt}}{K})$ consisting
of elements fixing finitely many $\mathbf{a}$-flags of
$\accentset{\rule{4.89998pt}{0.8pt}}{V}$. Then one of the following holds:

  1. (i)

Every element of $G(K)$ fixes an $\mathbf{a}$-flag of $V$.

  2. (ii)

There exists $C>0$ such that

$\mathbf{P}(\mbox{$\omega_{n}$ fixes an $\mathbf{a}$-flag of
$V$})\leqslant\Xi(C,n).$

###### Proof.

The set of $\mathbf{a}$-flags of $V$ are the $K$-points of a projective
$K$-variety $X$ which goes under the name of flag variety (see [Mil17, p.
146]). The morphism $\rho\colon G\to\mathrm{GL}_{N}$ equips $G$ with an action
on $X$. The statement, then, follows from Corollary 1.5. ∎

## References

  * [Bre15] E. Breuillard.  Approximate subgroups and super-strong approximation.  In Groups St Andrews 2013. Selected papers of the conference, St. Andrews, UK, August 3–11, 2013., pages 1–50. Cambridge: Cambridge University Press, 2015.
  * [BSFP14] L. Bary-Soroker, A. Fehm, and S. Petersen.  On varieties of Hilbert type.  Ann. Inst. Fourier (Grenoble), 64(5):1893–1901, 2014.
  * [CDJ+20] P. Corvaja, J.L. Demeio, A. Javanpeykar, D. Lombardo, and U. Zannier.  On the distribution of rational points on ramified covers of abelian varieties.  arXiv preprint arXiv:2011.12840, 2020.
  * [Cor07] P. Corvaja.  Rational fixed points for linear group actions.  Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 6(4):561–597, 2007.
  * [CTS87] J.L. Colliot-Thélène and J.J. Sansuc.  Principal homogeneous spaces under flasque tori: applications.  Journal of algebra, 106(1):148–205, 1987.
  * [CZ17] P. Corvaja and U. Zannier.  On the Hilbert property and the fundamental group of algebraic varieties.  Mathematische Zeitschrift, 286(1):579–602, 2017.
  * [Dem20] J.L. Demeio.  Non-rational varieties with the Hilbert property.  International Journal of Number Theory, 16(04):803–822, 2020.
  * [DSC93] P. Diaconis and L. Saloff-Coste.  Comparison techniques for random walk on finite groups.  The Annals of Probability, pages 2131–2156, 1993.
  * [FJ08] M. D. Fried and M. Jarden.  Field arithmetic, volume 11 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, third edition, 2008.  Revised by Jarden.
  * [FZ07] A. Ferretti and U. Zannier.  Equations in the Hadamard ring of rational functions.  Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 6(3):457–475, 2007.
  * [GN11] A. Gorodnik and A. Nevo.  Splitting fields of elements in arithmetic groups.  arXiv preprint arXiv:1105.0858, 2011.
  * [JKZ13] F. Jouve, E. Kowalski, and D. Zywina.  Splitting fields of characteristic polynomials of random elements in arithmetic groups.  Israel Journal of Mathematics, 193(1):263–307, 2013.
  * [KMT74] T. Kambayashi, M. Miyanishi, and M. Takeuchi.  Unipotent algebraic groups, volume 414.  Springer, Cham, 1974.
  * [Kow08] E. Kowalski.  The large sieve and its applications: arithmetic geometry, random walks and discrete groups, volume 175.  Cambridge University Press, 2008.
  * [Kow10] E. Kowalski.  Sieve in expansion.  Séminaire Bourbaki, page 63, 2010.
  * [Liu21] F. Liu.  On Hilbert’s irreducibility theorem for linear algebraic groups.  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, in press:33, 2021.
  * [LM12] A. Lubotzky and C. Meiri.  Sieve methods in group theory I: Powers in linear groups.  Journal of the American Mathematical Society, 25(4):1119–1148, 2012\.
  * [LR14] A. Lubotzky and L. Rosenzweig.  The Galois group of random elements of linear groups.  American Journal of Mathematics, 136(5):1347–1383, 2014.
  * [LS12] A. Lubotzky and D. Segal.  Subgroup growth, volume 212.  Birkhäuser, 2012.
  * [LW54] S. Lang and A. Weil.  Number of points of varieties in finite fields.  American Journal of Mathematics, 76(4):819–827, 1954.
  * [LW93] A. Lubotzky and B. Weiss.  Groups and expanders.  In Expanding graphs. Proceedings of the DIMACS workshop on expander graphs, May 11-14, 1992, Princeton University, NJ (USA), pages 95–109. Providence, RI: American Mathematical Society, 1993.
  * [LZ] A. Lubotzky and A. Zuk.  On property ($\tau$).  Available at http://www.ma.huji.ac.il/~alexlub/.
  * [Mil17] J.S. Milne.  Algebraic groups: the theory of group schemes of finite type over a field, volume 170.  Cambridge University Press, 2017.
  * [MM18] G. Malle and B.H. Matzat.  Inverse Galois theory.  Berlin: Springer, 2018.
  * [Nor87] M.V. Nori.  On subgroups of ${GL}_{n}(\mathbf{F}_{p})$.  Inventiones mathematicae, 88(2):257–275, 1987.
  * [PR94] V.P. Platonov and A.S. Rapinchuk.  Algebraic groups and number theory, volume 139 of Pure and Applied Mathematics.  Academic Press, Inc., Boston, MA, 1994.  Translated from the 1991 Russian original by Rachel Rowen.
  * [PR03] G. Prasad and A.S. Rapinchuk.  Existence of irreducible $\mathbb{R}$-regular elements in Zariski-dense subgroups.  Mathematical research letters, 10:21–32, 2003.
  * [PR14] G. Prasad and A.S. Rapinchuk.  Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces.  Thin groups and superstrong approximation, 61:211–252, 2014.
  * [Pra77] G. Prasad.  Strong approximation for semi-simple groups over function fields.  Annals of Mathematics, 105(3):553–572, 1977.
  * [Riv08] I. Rivin.  Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms.  Duke Mathematical Journal, 142(2):353–379, 2008.
  * [San81] J.J. Sansuc.  Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.  J. Reine Angew. Math., 327:12–80, 1981.
  * [Sch65] A. Schinzel.  On Hilbert’s irreducibility theorem.  Ann. Polon. Math., 16:333–340, 1965.
  * [Ser89] J.P. Serre.  Lectures on the Mordell-Weil theorem. Transl. and ed. by Martin Brown from notes by Michel Waldschmidt, volume E15.  Braunschweig etc.: Friedr. Vieweg &— Sohn, 1989.
  * [Ser16] J.P. Serre.  Topics in Galois theory.  CRC Press, 2016.
  * [SGV12] A. Salehi Golsefidy and P.P. Varjú.  Expansion in perfect groups.  Geometric and Functional Analysis, 22(6):1832–1891, 2012.
  * [Zan10] U. Zannier.  Hilbert irreducibility above algebraic groups.  Duke Mathematical Journal, 153(2):397–425, 2010.