# Automorphisms of algebraic varieties and
infinite transitivity

Ivan Arzhantsev  Faculty of Computer Science, HSE University, Pokrovsky
Boulevard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>
###### Abstract.

We survey recent results on multiple transitivity of automorphism groups of
affine algebraic varieties. We consider the property of infinite transitivity
of the special automorphism group, which is equivalent to flexibility of the
corresponding affine variety. These properties have important algebraic and
geometric consequences. At the same time they are fulfilled for wide classes
of varieties. Also we study situations where infinite transitivity takes place
for automorphism groups generated by finitely many one-parameter subgroups. In
the appendices to the paper, the results on infinitely transitive actions in
complex analysis and in combinatorial group theory are discussed.

###### Key words and phrases:

Algebraic variety, automorphism, group action, multiple transitivity, locally
nilpotent derivation, toric variety, affine cone

###### 2010 Mathematics Subject Classification:

Primary 14L30, 14R10; Secondary 13E10, 14M25, 20M32

Supported by Russian Foundation for Basic Research, grant 20-11-50106

###### Contents

  1. 1 Basic facts on infinitely transitive actions and flexible varieties
    1. 1.1 Infinite transitivity on an affine space
    2. 1.2 Algebraically generated automorphism groups
    3. 1.3 Locally nilpotent derivations and replicas of ${\mathbb{G}}_{a}$-subgroups
    4. 1.4 Flexibility and infinite transitivity
  2. 2 Properties of flexible varieties
    1. 2.1 The Makar-Limanov invariant and rationality properties
    2. 2.2 Collective transitivity
    3. 2.3 ${\mathbb{A}}^{1}$-richness and the Gromov-Winkelmann Theorem
    4. 2.4 Transitivity on jets
    5. 2.5 Multiple transitivity of the automorphism group
  3. 3 Examples of flexible varieties
    1. 3.1 Suspensions
    2. 3.2 Affine toric varieties
    3. 3.3 Homogeneous and almost homogeneous varieties
      1. 3.3.1 The smooth case.
      2. 3.3.2 ${\rm SL}_{2}$-embeddings.
      3. 3.3.3 Horospherical and spherical varieties
    4. 3.4 Vector bundles
    5. 3.5 Affine cones over projective varieties
    6. 3.6 Universal torsors
    7. 3.7 Gizatullin surfaces
    8. 3.8 Calogero-Moser spaces
  4. 4 Infinite transitivity and finite generation
    1. 4.1 The case of an affine toric variety
    2. 4.2 The case of an affine space
    3. 4.3 Groups generated by root subgroups and the Tits alternative
  5. 5 Appendix A. Infinitely transitive actions in complex analysis
  6. 6 Appendix B. Infinitely transitive actions of finitely generated groups

## Introduction

In this survey we study multiply transitive actions of automorphism groups of
algebraic varieties and the closely related concept of a flexible affine
algebraic variety. Let us start by reminding basic definitions.

Let us fix a positive integer $m$. We say that an action of a group $G$ on a
set $X$ is _$m$ -transitive_ if for any two tuples of $m$ pairwise distinct
points of the set $X$ there exists an element of the group $G$ that sends the
first tuple to the second one. The action of $G$ on $X$ is called _infinitely
transitive_ if it is $m$-transitive for any positive integer $m$. Infinite
transitivity is equivalent to the fact that the pointwise stabilizer of any
finite set of points acts transitively on the complement to this set.

In the case of finite groups, it is easy to see that the symmetric group
$S_{n}$ acts on a set with $n$ elements $n$-transitively. The same action
restricted to the subgroup of even permutations $A_{n}$ becomes
$(n-2)$-transitive. The surprising fact is that all other finite permutation
groups have the transitivity degree not greater than $5$ [30, p. 2.1].
Moreover, only the Mathieu groups $M_{12}$ and $M_{24}$ have the transitivity
degree $5$, and the transitivity degree equals $4$ only for the Mathieu groups
$M_{11}$ and $M_{23}$ [30, Chapter 6]. At the same time, there are infinitely
many finite groups of the transitivity degree $3$. For example, we may
consider the action of the group ${\rm PGL}_{2}({\mathbb{F}}_{q})$ on the
projective line ${\mathbb{P}}^{1}({\mathbb{F}}_{q})$ over a field with $q$
elements. With different values of $q$ we obtain different $3$-transitive
actions.

Let us move to the case of infinite groups. First of all, we are interested in
automorphism groups of algebraic varieties. It is well known that the
connected component of the automorphism group of a complete (in particular,
projective) algebraic variety is a (finite-dimensional) algebraic group [68].
In particular, such a group can not act on a variety infinitely transitively.

More precisely, it is known that the maximum of transitivity degree of the
action of an algebraic group is $3$ and this value is achieved only for the
action of the group ${\rm PGL}_{2}$ on the projective line ${\mathbb{P}}^{1}$
[57, Corollary 2]. All $2$-transitive actions of algebraic groups over an
arbitrary algebraically closed field are classified in [57]. It turned out
that in addition to the action of the group ${\rm PGL}_{n+1}$ on
${\mathbb{P}}^{n}$, such actions are only the actions of the semidirect
product $G\rightthreetimes V$ on $V$, where $G$ is a linear algebraic group
and $V$ is a rational $G$-module such that $G$ acts on $V\setminus\\{0\\}$
transitively. All such actions can be explicitly listed. Multiply transitive
actions of Lie groups were studied in [19, 60].

In the case of affine varieties, the situation changes essentially. It is not
difficult to show that for $n\geq 2$ the automorphism group of the affine
space ${\mathbb{A}}^{n}$ over an infinite field acts on ${\mathbb{A}}^{n}$
infinitely transitively. Constructing other examples of infinitely transitive
actions on algebraic varieties is a nontrivial task that requires to develop a
special technique. One of the first works in this direction is the paper [54].
Here the infinite transitivity is proved for the action of the automorphism
group on the smooth locus of certain affine hypersurfaces. Such hypersurfaces
are called suspensions. In [13], the concept of a flexible variety is defined,
and the properties of flexibility and infinite transitivity were proved for
affine cones over varieties of generalized flags, non-degenerate affine toric
varieties and suspensions over flexible varieties.

One interpretation of the infinite transitivity of the action of the
automorphism group is that any two embeddings of a given finite set into the
smooth locus of a flexible variety are equivalent, that is, they can be
translated to each other by a suitable automorphism of the variety. This
statement is consonant with the Abyankar-Moh-Suzuki Theorem on the equivalence
of embeddings of a line into a plane.

Unless otherwise stated, we assume the ground field ${\mathbb{K}}$ to be an
algebraically closed field of characteristic zero. Consider the additive group
${\mathbb{G}}_{a}$ of the field ${\mathbb{K}}$ with the natural structure of a
one-dimensional linear algebraic group. For each non-identical regular action
${\mathbb{G}}_{a}\times X\to X$, the image of ${\mathbb{G}}_{a}$ in the
automorphism group ${\rm Aut}(X)$ is called a _${\mathbb{G}}_{a}$ -subgroup_.
Denote by ${\rm SAut}(X)$ the subgroup of the group ${\rm Aut}(X)$ generated
by all ${\mathbb{G}}_{a}$-subgroups. Since a subgroup conjugated to a
${\mathbb{G}}_{a}$-subgroup by an arbitrary automorphism is again a
${\mathbb{G}}_{a}$-subgroup, the subgroup ${\rm SAut}(X)$ is normal in ${\rm
Aut}(X)$.

We say that a smooth point $x$ of a variety $X$ is _flexible_ if the tangent
space $T_{x}(X)$ is generated by tangent vectors to the orbits of
${\mathbb{G}}_{a}$-actions on $X$ passing through $x$. An algebraic variety
$X$ is called _flexible_ if every its smooth point is flexible. For example,
this condition is satisfied if there is at least one flexible point on $X$ and
the group ${\rm Aut}(X)$ acts transitively on the smooth locus.

We are ready to formulate the main result of the paper [6]. This is one of the
central results for this survey.

######  Theorem 1.

Let $X$ be an irreducible affine algebraic variety of dimension $\geq 2$. Then
the following conditions are equivalent.

  1. (i)

The group ${\rm SAut}(X)$ acts on the smooth locus $X^{{\rm reg}}$ of the
variety $X$ transitively.

  2. (ii)

The group ${\rm SAut}(X)$ acts on the smooth locus $X^{{\rm reg}}$ infinitely
transitively.

  3. (iii)

The variety $X$ is flexible.

In other words, the transitivity of the group ${\rm SAut}(X)$ implies infinite
transitivity, and these conditions are equivalent to the flexibility condition
formulated in infinitesimal terms. Note that in [11, Theorem 2] and [33,
Theorem 1.11] Theorem 1 is generalized to the case of irreducible quasi-affine
varieties of dimension $\geq 2$.

Let us take a closer look at the structure of this paper. In the first
section, following [6], we define algebraically generated subgroups of the
automorphism group of an algebraic variety as subgroups generated by a family
of algebraic subgroups. Note that the group ${\rm SAut}(X)$ defined above is
algebraically generated. Actions of such subgroups have a number of properties
in common with the actions of finite-dimensional groups. Such properties
include the local closedness of orbits, as well as an analogue of Rosenlicht’s
theorem on the separation of orbits by rational invariants and an analogue of
Kleiman’s theorem on transversality. After that we proceed to the discussion
on locally nilpotent derivations of affine algebras. The technique of locally
nilpotent derivations and the construction of replicas of
${\mathbb{G}}_{a}$-subgroups are among the main technical tools of this
theory. The discussion of this technique ends with the proof of equivalence of
conditions (i) and (iii) in Theorem 1. Finally, in the last part of this
section we discuss results which lead to implication (i) $\\!\Rightarrow$
(ii) in Theorem 1. In fact, we deal with more general setup: infinite
transitivity is considered on an open orbit, which need not coincide with the
smooth locus of the variety, and the acting group need not be ${\rm SAut}(X)$,
but any subgroup of this group generated by a saturated family of
${\mathbb{G}}_{a}$-subgroups.

Section 2 is devoted to the study of properties of flexible varieties
established in [6, 33] and subsequent papers. We show that the existence of a
flexible point on an affine variety is equivalent to the triviality of the
Makar-Limanov field invariant and prove that every such variety is
unirational. Bogomolov’s conjecture [17] is also discussed here. It offers a
characterization of unirational varieties in terms of stably infinitely
transitive birational models. We consider collectively infinitely transitive
actions, the ${\mathbb{A}}^{1}$-richness property, and a generalization of the
Gromov-Winkelmann Theorem to arbitrary flexible varieties obtained in the work
of Flenner, Kaliman and Zaidenberg [33]. The theorem claims that flexibility
is preserved when passing to a quasi-affine variety obtained from a flexible
affine variety by removing a subvariety of codimension $\geq 2$. We also
consider the question of when the automorphism group of an affine variety $X$
acts on the smooth locus $2$-transitively. In the case when some affine
algebraic group of positive dimension acts non-identically on $X$, this
condition is equivalent to the flexibility of the variety [4].

In Section 3 we consider examples of flexible varieties. The presentation
begins with the already mentioned construction of suspension. Then non-
degenerate affine toric varieties are considered. To prove flexibility, the
root ${\mathbb{G}}_{a}$-subgroups and related Demazure roots are used. A
generalization of this result is the theorem on flexibility of non-degenerate
affine horospherical varieties proved in recent papers by Gaifullin and
Shafarevich [84, 40]. Flexibility of affine cones over projective varieties is
studied, in particuar over del Pezzo surfaces [72, 74] and higher dimensional
Fano varieties [70, 80]. Here the proof of flexibility uses the construction
of a cylindrical subset on a projective variety proposed by Kishimoto,
Prokhorov and Zaidenberg, see e.g. [21]. We discuss the flexibility property
for a universal torsor and for a total coordinate space over varieties covered
by open charts isomorphic to an affine space [11]. These objects arise in the
theory of Cox rings. The flexibility of Gizatullin surfaces and Calogero-Moser
varieties is also considered.

In Section 4 we discuss a recently discovered effect — infinite transitivity
can occur for subgroups of the automorphism group generated by a finite number
of ${\mathbb{G}}_{a}$-subgroups. In the case of affine spaces, this result can
be extracted from the theorems of Derksen and Bodnarchuk. In [9], we formulate
a conjecture that for any flexible affine variety there exists such a finite
set of one-dimensional unipotent subgroups that the group generated by them
acts infinitely transitively on its open orbit. This conjecture is proved in
[9] for non-degenerate affine toric varieties of dimension $\geq 2$, which are
nonsingular in codimension $2$. It is known that in the case of an affine
space, three generating ${\mathbb{G}}_{a}$-subgroups are sufficient for
infinite transitivity. The ind-structure on the automorphism group ${\rm
Aut}(X)$ and the passage from the subgroup $G$ to its closure $\overline{G}$
in the ind-topology play an important role here. In the last part, we follow
the paper [12] and discuss the Tits alternative for groups generated by a
finite number of root subgroups in the automorphism group of an affine toric
variety.

The last part of the survey consists of two appendices. Appendix A is devoted
to holomorphic flexibility and infinite transitivity for groups of
automorphisms of complex analytic manifolds, as well as to the explanation of
the connection of concepts and constructions introduced above with the
Andersen-Lempert theory, Gromov sprays, Oka manifolds and Oka maps.

In Appendix B we prove several results on infinite transitivity of actions of
abstract groups. To the reader who has not worked with infinitely transitive
actions before, we recommend to read this appendix before reading the main
part of the text. Here we prove the infinite transitivity for a normal
subgroup of an infinitely transitive group. Following [69], we construct an
example of an infinitely transitive action of a $2$-generated group and give a
brief overview of the known facts on existence of infinitely transitive
effective actions of finitely generated groups. Such actions are implemented
both on abstract sets and on infinite graphs, metric spaces and other natural
objects. Recently, this direction has been actively developing, but this
happens without any connection with the results on infinite transitivity of
automorphism groups of algebraic varieties. It is natural to expect that in
the future the two theories will enrich each other.

Note also that the survey contains some original results and observations. For
example, Proposition 6 seems to be new.

In our opinion, the theory of flexible varieties is currently at its zenith.
On the one hand, this area has proved its right to exist: a number of general
structural theorems have been obtained, interesting applications and
consequences of the results on flexibility have been found, it has been shown
that wide classes of affine varieties have the flexibility property,
nontrivial connections with other areas of mathematics have been established.
On the other hand, a number of important open questions and conjectures are
connected with this topic, which seem to be accessible, including within the
framework of already developed techniques, and which are interesting to work
on. We are confident that in the coming years the theory of flexible varieties
will be supplemented with new results, and we hope that this survey will
contribute to the further development of this area.

The author would like to take this opportunity to thank his co-authors and
colleagues with whom he works together in this field or discusses related
issues. He is sincerely grateful to Mikhail Zaidenberg, Jürgen Hausen, Shulim
Kaliman, Frank Kutzschebauch, Hubert Flenner, Hendrik Süss, Gene Freudenburg,
Alvaro Liendo, Yuri Prokhorov, Dmitry Timashev, Karine Kuyumzhiyan, Sergey
Gaifullin, Alexander Perepechko and many others. Special thanks are due to the
referee for a careful reading of the text and many valuable comments.

##  1\. Basic facts on infinitely transitive actions and flexible varieties

###  1.1. Infinite transitivity on an affine space

We start this section by proving a well-known fact: for $n\geq 2$, the group
${\rm Aut}({\mathbb{A}}^{n})$ acts on ${\mathbb{A}}^{n}$ infinitely
transitively. This result hardly makes sense to associate with a specific
work, it has a folklore nature and can be proved by a junior math student who
is familiar with the concept of an interpolation polynomial. It is also an
obvious consequence of Theorem 1: the space ${\mathbb{A}}^{n}$ is flexible,
since the tangent space at each point is generated by tangent vectors to the
orbits of one-parameter subgroups of parallel translations along the
coordinate axes. Nevertheless, we give here a direct proof of the infinite
transitivity of the action of the automorphism group, since it illustrates a
number of ideas and methods underlying this theory. An affine space is
considered here over an arbitrary infinite field.

######  Proposition 1.

For $n\geq 2$, the group ${\rm Aut}({\mathbb{A}}^{n})$ acts on
${\mathbb{A}}^{n}$ infinitely transitively.

###### Proof.

We fix a positive integer $m$ and choose $m$ pairwise distinct points

$(x_{11},\ldots,x_{n1}),\ldots,(x_{1m},\ldots,x_{nm}).$

It suffices to prove that these points can be moved to a standard position by
a suitable automorphism of the space ${\mathbb{A}}^{n}$. Over a field of
characteristic zero, we can assume that a standard position is the set

$(1,\ldots,1),\ldots,(m,\ldots,m),$

and over fields of positive characteristic, obvious changes need to be made in
the arguments given below.

Step $1$. We are going to ensure that $x_{1i}\neq x_{1j}$ for all $i\neq j$.
We consistently increase the number of such inequalities. Let us suppose that
$x_{11}=x_{12}$. Then we can assume that $x_{21}\neq x_{22}$. Let us find a
polynomial $f(x)$ such that $f(x_{21})=a$ and $f(x_{2i})=0$ for all
$x_{2i}\neq x_{21}$, where $a\neq x_{1i}-x_{1j}$ for all $i,j=1,\ldots,m$.
Applying the automorphism $(x_{1}+f(x_{2}),x_{2},\ldots,x_{n})$ of
${\mathbb{A}}^{n}$ to our set, we achieve that $x_{11}\neq x_{12}$ and the
inequalities of the first coordinates of the points that took place before
will not turn to equalities after such a transformation. Thus, after a finite
number of such transformations, we obtain the required.

Step $2$. Now we assume that $x_{1i}\neq x_{1j}$ for all $i\neq j$. Let us
take such polynomials $f_{2}(x),\ldots,f_{n}(x)$ that $x_{si}+f_{s}(x_{1i})=i$
for all $s=2,\ldots,n$ and ${i=1,\ldots,m}$. After applying the automorphism

$(x_{1},x_{2}+f_{2}(x_{1}),\ldots,x_{n}+f_{n}(x_{1}))$

we obtain a set of points

$(x_{11},1,\ldots,1),\ldots,(x_{1m},m,\ldots,m).$

Step $3$. Take a polynomial $h(x)$ such that

$x_{1i}+h(i)=i$

for all $i=1,\ldots,m$. Then the automorphism
$(x_{1}+h(x_{2}),x_{2},\ldots,x_{n})$ moves our set to the standard position.
∎

######  Remark 1.

For $n=1$, the group ${\rm Aut}({\mathbb{A}}^{1})$ is isomorphic to the
semidirect product ${{\mathbb{G}}_{m}\rightthreetimes{\mathbb{G}}_{a}}$, where
the torus ${\mathbb{G}}_{m}$ acts by scalar multiplications and the group
${\mathbb{G}}_{a}$ acts by parallel translations. This implies that ${\rm
Aut}({\mathbb{A}}^{1})$ acts on ${\mathbb{A}}^{1}$ $2$-transitively, but not
$3$-transitively.

###  1.2. Algebraically generated automorphism groups

Let $X$ be an algebraic variety over the ground field ${\mathbb{K}}$. An
_algebraic subgroup_ of the automorphism group ${\rm Aut}(X)$ is a subgroup
$H\subseteq{\rm Aut}(X)$ with a structure of an algebraic group such that the
action map $H\times X\to X$ is a morphism of algebraic varieties. We say that
a subgroup $G$ of the group ${\rm Aut}(X)$ is _algebraically generated_ if it
is generated as an abstract group by a family $S$ of connected algebraic
subgroups of the group ${\rm Aut}(X)$.

######  Proposition 2 ([6, Propositions 1.3 and 1.5]).

Consider an algebraically generated subgroup $G\subseteq{\rm Aut}(X)$. Then

  1. (a)

for any point $x\in X$, the orbit is $G.x$ is locally closed in $X$;

  2. (b)

there is such a finite set of not necessarily pairwise distinct subgroups
$H_{1},\ldots,H_{s}\in S$ that

$G.x=H_{1}.(H_{2}.\cdots(H_{s}.x))$

for any point $x\in X$.

######  Proposition 3 ([6, Proposition 1.8]).

Suppose that a generating family $S$ of connected algebraic subgroups is
invariant under conjugation by elements of the group $G$. Then there is such a
sequence $(H_{1},\ldots,H_{s})$ of subgroups in $S$ that for all points $x\in
X$ the tangent space $T_{x}(G.x)$ to the orbit $G.x$ is generated by tangent
spaces

$T_{x}(H_{1}.x),\ldots,T_{x}(H_{s}.x).$

We consider an algebraically generated subgroup $G\subseteq{\rm Aut}(X)$ with
a generating family $S$ of connected algebraic subgroups that is invariant
under conjugation by elements of the group $G$. We say that a point $x\in
X^{{\rm reg}}$ is _$G$ -flexible_ if the tangent space $T_{x}X$ is generated
by subspaces $T_{x}(H.x)$, where $H\in S$. Then [6, Corollary 1.11] states
that a point $x\in X^{{\rm reg}}$ is $G$-flexible if and only if the orbit
$G.x$ is open in $X$. Further, if such an open orbit exists, then it is unique
and consists exactly of all $G$-flexible points in $X^{\rm reg}$. Note also
that the definition of a $G$-flexible point does not depend on the choice of a
generating family $S$ of the group $G$.

Using the semi-continuity theorem for the dimension of fibers of a morphism,
it is not difficult to prove [6, Corollary 1.12] that for an algebraically
generated group $G\subseteq{\rm SAut}(X)$, the function
$X\to{\mathbb{Z}}_{\geq 0}$, $x\mapsto\dim G.x$ is lower semi-continuous. In
particular, the union of orbits of maximum dimension in $X$ is a Zariski-open
subset.

The following result is an analogue of the well-known Rosenlicht theorem for
algebraic groups [79, Theorem 2.3].

######  Theorem 2 ([6, Theorem 1.13]).

Let $G\subseteq{\rm Aut}(X)$ be an algebraically generated subgroup. Then
there is a finite set of rational $G$-invariant functions which separate
$G$-orbits in general position on $X$.

The proof of this result repeats almost verbatim the proof of the original
Rosenlicht theorem given in [79].

######  Corollary 1 ([6, Corollary 1.14]).

Let an algebraically generated group $G$ act on an irreducible variety $X$.
Then

${\rm tr.deg}({\mathbb{K}}(X)^{G}:{\mathbb{K}})=\min_{x\in X}\,\\{{{\rm
codim}}_{X}G.x\\}\,.$

In particular, there is an open $G$-orbit on $X$ if and only if
${\mathbb{K}}(X)^{G}={\mathbb{K}}$.

If an algebraic group $G$ acts transitively on an algebraic variety $X$ and
$Y,Z$ are smooth subvarieties in $X$, Kleiman’s Transversality theorem claims
that a typical translate $g.Z$ ($g\in G$) intersects $Y$ transversally. The
following generalization of this result takes place in the case of
algebraically generated subgroups.

######  Theorem 3 ([6, Theorem 1.15]).

Let an algebraically generated subgroup $G\subseteq{\rm Aut}(X)$ be generated
by a family $S$ of connected algebraic subgroups that is invariant under
conjugation by elements of $G$. Suppose that $G$ acts with the open orbit
$O\subseteq X$. Then there are such subgroups $H_{1},\ldots,H_{s}\in S$ that
for any locally closed reduced subschemes $Y$ and $Z$ in $O$ one can find such
a Zariski-open subset $U=U(Y,Z)\subseteq H_{1}\times\ldots\times H_{s},$ that
any elements $(h_{1},\ldots,h_{s})\in U$ satisfy the following conditions:

  1. (a)

the translate $(h_{1}\cdot\ldots\cdot h_{s}).Z^{\rm reg}$ intersects $Y^{\rm
reg}$ transversely;

  2. (b)

$\dim(Y\cap(h_{1}\cdot\ldots\cdot h_{s}).Z)\leq\dim Y+\dim Z-\dim X$.

In particular, if $\dim Y+\dim Z<\dim X,$ then $Y\cap(h_{1}\cdot\ldots\cdot
h_{s}).Z=\varnothing$.

Note that in the work [77] results on algebraically generated subgroups have
been further generalized. A number of properties of actions including local
closureness of orbits and an analogue of Rosenlicht’s theorem on rational
invariants, are proved here for any connected subgroup of the automorphism
group of an arbitrary irreducible algebraic variety.

###  1.3. Locally nilpotent derivations and replicas of
${\mathbb{G}}_{a}$-subgroups

For our further purposes it is important to know how to work with
infinitesimal generators of ${\mathbb{G}}_{a}$-subgroups of the automorphism
group ${\rm Aut}(X)$ of an affine variety $X$. Below we formulate the related
results.

(1) If the group ${\mathbb{G}}_{a}$ acts on an affine variety $X={\rm
Spec\,}A$, then the derivation $\partial$ of the algebra $A$ associated with
this action is _locally nilpotent_ , that is, for every $f\in A$ there is such
an $n\in{\mathbb{N}}$ that $\partial^{n}(f)=0$. This result is explained by
the fact that each function $f$ lies in a finite-dimensional
${\mathbb{G}}_{a}$-invariant subspace in $A$, and, by the Lie-Kolchin Theorem,
in a suitable basis of this subspace the group ${\mathbb{G}}_{a}$ acts by
unitriangular matrices. Hence, the differential of such an action is given in
this subspace by a nilpotent operator.

(2) Conversely, given a locally nilpotent ${\mathbb{K}}$-linear derivation
$\partial:A\to A$ and $t\in{\mathbb{K}}$, the mapping ${\rm
exp}(t\partial):A\to A$ defines an automorphism of the algebra $A$ and,
consequently, of the variety $X$. Thus, each locally nilpotent derivation
$\partial\neq 0$ is associated with the ${\mathbb{G}}_{a}$-subgroup $H={\rm
exp}({\mathbb{K}}\partial)$ in ${\rm Aut}(X)$; for more details see [37]. If
we consider $\partial$ as a vector field on $X$, the action of $H$ on $X$ is
exactly the phase flow associated with this field.

(3) The algebra of invariants ${\mathbb{K}}[X]^{H}=\ker\partial$ has the
transcendence degree ${\dim X-1}$. Further, each $H$-invariant function
$f\in{\mathbb{K}}[X]^{H}$ defines a new ${\mathbb{G}}_{a}$-subgroup
$H_{f}:={\rm exp}({\mathbb{K}}f\partial)$. We call such subgroups _replicas_
of the subgroup $H$. Bellow such subgroups play a significant role.

The subgroup $H_{f}$ acts in the same direction as the subgroup $H$, but with
a different velocity along the orbits. In particular, those $H$-orbits, where
the function $f$ vanishes, consist of $H_{f}$-fixed points. The mapping
$f\mapsto f\partial$ embeds $\ker\partial$ as an abelian subalgebra into the
Lie algebra of all regular vector fields on $X$.

For a subgroup $G\subseteq{\rm SAut}(X)$, we denote by ${\rm LND}(G)$ the set
of locally nilpotent vector fields on $X$ that generate
${\mathbb{G}}_{a}$-subgroups contained in $G$. This set is invariant under
conjugation by elements of $G$ and is a cone, that is, ${\mathbb{K}}\cdot{\rm
LND}(G)\subseteq{\rm LND}(G)$. In the following, we consider subsets ${\rm
N}\subseteq{\rm LND}(G)$ of locally nilpotent vector fields for which the
associated subgroups $\\{{\rm exp}({\mathbb{K}}\partial)\\}$, $\partial\in{\rm
N}$ generate the group $G$. For brevity, we say that ${\rm N}$ generates the
group $G$ and write $G=\langle{\rm N}\rangle$.

The following result can be deduced from Proposition 3.

######  Corollary 2 ([6, Corollary 1.21]).

For a subgroup $G\\!=\\!\langle{\rm N}\rangle\\!$ of the group ${\rm
Aut}(X),$ where ${\rm N}\subseteq{\rm LND}(G)$ is invariant under conjugation
in $G$, there are locally nilpotent vector fields
$\partial_{1},\ldots,\partial_{s}\in{\rm N}$ that generate the tangent space
$T_{x}(G.x)$ at each point $x\in X$.

For a point $x\in X$, let ${\rm LND}_{x}(G)\subseteq T_{x}X$ denote the cone
of all tangent vectors $\partial(x)$, where $\partial$ runs through ${\rm
LND}(G)$. According to Corollary 2, we have $T_{x}(G.x)=\langle{\rm
LND}_{x}(G)\rangle$. Thus, a point $x\in X^{\rm reg}$ is $G$-flexible if and
only if the cone ${\rm LND}_{x}(G)$ generates the tangent space $T_{x}X$.

Applying Corollary 2 to the group $G={\rm SAut}(X)$, we obtain equivalence
(i)$\Leftrightarrow$(iii) in Theorem 1 from the Introduction.

######  Corollary 3.

For an irreducible affine variety $X$, the action of the group ${\rm SAut}(X)$
on $X^{\rm reg}$ is transitive if and only if $X$ is flexible.

######  Example 1.

We illustrate the notion of a replica and a special automorphism group ${\rm
SAut}(X)$ in the case of the affine space $X={\mathbb{A}}^{n}$. The group
${\rm SAut}({\mathbb{A}}^{n})$ contains ${\mathbb{G}}_{a}$-subgroups of
parallel translations in all directions. The infinitesimal generator of such a
subgroup coincides with the partial derivative in this direction, which is
considered as a locally nilpotent derivation of the polynomial ring in $n$
variables. For example, in the case of the parallel translation along the
first coordinate axis $(x_{1}+t,x_{2},\ldots,x_{n})$, $t\in{\mathbb{K}}$, the
corresponding derivation $\partial$ is $\frac{\partial}{\partial x_{1}}$.
Replicas of such a subgroup are translations of the form
$(x_{1}+f,x_{2},\ldots,x_{n})$ for an arbitrary polynomial
$f\in{\mathbb{K}}[x_{2},\ldots,x_{n}]$.

As one more example, we consider the locally nilpotent derivation
$\partial=x_{1}\frac{\partial}{\partial x_{2}}+x_{2}\frac{\partial}{\partial
x_{3}}$ of the polynomial ring ${\mathbb{K}}[x_{1},x_{2},x_{3}]$ and the
invariant function ${f=x_{2}^{2}-2x_{1}x_{3}\in\ker\partial}$. The
corresponding replica $H_{f}$ contains the well-known automorphism of Nagata

$H_{f}(1)={\rm exp}(f\cdot\partial)\in{\rm SAut}({\mathbb{A}}^{3}).$

Note that any automorphism from ${\rm SAut}({\mathbb{A}}^{n})$ preserves the
volume form on ${\mathbb{A}}^{n}$. So we have

${\rm SAut}({\mathbb{A}}^{n})\subseteq J_{n},$

where $J_{n}$ denotes the subgroup in ${\rm Aut}({\mathbb{A}}^{n})$ consisting
of all automorphisms such that the determinant of the Jacobi matrix is $1$.

Let us recall that the subgroup of _tame_ automorphisms ${\rm
Tame}_{n}\subseteq{\rm Aut}({\mathbb{A}}^{n})$ is the subgroup generated by
all elementary automorphisms $\sigma\in{\rm Aut}({\mathbb{A}}^{n})$ of the
form

$\displaystyle\sigma$
$\displaystyle=\sigma_{i,\lambda,f}\colon(x_{1},\ldots,x_{i-1},x_{i},x_{i+1},\ldots,x_{n})$
$\displaystyle\longmapsto(x_{1},\ldots,x_{i-1},\lambda
x_{i}+f,x_{i+1}\ldots,x_{n}),$

where $f\in{\mathbb{K}}[x_{1},\ldots,x_{i-1},x_{i+1}\ldots,x_{n}]$ and
$\lambda\in{\mathbb{G}}_{m}$. An elementary automorphism $\sigma$ with
$\lambda=1$ is contained in the ${\mathbb{G}}_{a}$-subgroup
$\\{\sigma_{i,1,tf}\\}_{t\in{\mathbb{K}}}$. Since for any
$\lambda\in{\mathbb{G}}_{m}$ the conjugated element
$h_{\lambda}\circ\sigma\circ h_{\lambda}^{-1}$, where $h_{\lambda}(x)=\lambda
x$, is again elementary, any tame automorphism $\beta$ with Jacobi matrix
having determinant $1$ can be written as a product of elementary automorphisms
with Jacobi matrix having determinant $1$. Thus ${{\rm Tame}_{n}\cap
J_{n}\subseteq{\rm SAut}({\mathbb{A}}^{n})\subseteq J_{n}.}$

For $n=3$ the first inclusion is strict. In fact, the Shestakov-Umirbaev
Theorem states that the Nagata automorphism $H_{f}(1)$ is wild or, in other
words,

$H_{f}(1)\in{\rm SAut}({\mathbb{A}}^{3})\setminus({\rm Tame}_{3}\cap J_{3}).$

For $n=2$, the Jung–van der Kulk Theorem states that

${\rm Aut}({\mathbb{A}}^{2})={\rm Tame}_{2}$

and so ${\rm SAut}({\mathbb{A}}^{2})=J_{2}$. It is also known that the group
$J_{2}$ is perfect and its derived subgroup coincides with the derived
subgroup of the group ${\rm Aut}({\mathbb{A}}^{2})$. The question of whether
the equality ${\rm SAut}({\mathbb{A}}^{n})=J_{n}$ holds for $n\geq 3$ remains
open, see [38, Question 15.5.4]. Moreover, it is not known whether the group
$J_{n}$ coincides with the closure of the subgroup ${\rm
SAut}({\mathbb{A}}^{n})$ with respect to the ind-topology on the group ${\rm
Aut}({\mathbb{A}}^{n})$.

###  1.4. Flexibility and infinite transitivity

Let $X$ be an affine algebraic variety and ${G\subseteq{\rm SAut}(X)}$ be a
subgroup generated by a set ${\rm N}$ of locally nilpotent vector fields
satisfying the following conditions:

  1. (a)

${\rm N}$ is invariant under the action of the group $G$ by conjugation;

  2. (b)

${\rm N}$ is closed under taking replicas.

We call such a generating set saturated. Note that a more essential condition
here is condition (b), since starting from an arbitrary set of locally
nilpotent vector fields ${\rm N}_{0}$ generating $G$ and satisfying condition
(b) we can add all conjugates by elements of $G$ to elements of ${\rm N}_{0}$
and obtain a saturated set ${\rm N}$ generating the same group $G$.

The following theorem proves (i)$\Rightarrow$(ii) in Theorem 1 from the
Introduction.

######  Theorem 4 ([6, Theorem 2.2]).

Let $X$ be an affine variety of dimension $\geq 2$ and ${G\subseteq{\rm
SAut}(X)}$ be a subgroup generated by a saturated set of locally nilpotent
vector fields. Suppose that $G$ acts with the open orbit $O\subseteq X$. Then
$G$ acts on $O$ infinitely transitively.

For a subset $Z\subseteq X$, we denote by ${\rm N}_{Z}$ the set of vector
fields from ${\rm N}$ that vanish on $Z$. Let

$G_{{\rm N},Z}=\langle H={\rm exp}({\mathbb{K}}\partial):\partial\in{\rm
N}_{Z}\rangle$

be the subgroup in $G$ generated by exponents of all elements in ${\rm
N}_{Z}$. It is clear that automorphisms from $G_{{\rm N},Z}$ preserve the
subset $Z$ pointwise. It is easy to see that ${\rm N}_{Z}$ is a saturated
generating set of the group $G_{{\rm N},Z}$.

One of the main technical results of the work [6] can be formulated as
follows.

######  Theorem 5 ([6, Theorem 2.5]).

Let $X$ be an affine variety of dimension $\geq 2$ and ${G\subseteq{\rm
SAut}(X)}$ be a subgroup generated by a saturated set $N$ of locally nilpotent
vector fields. Suppose that $G$ acts with the open orbit $O\subseteq X$. Then
for each finite subset $Z\subseteq O$ the group $G_{N,Z}$ acts transitively on
$O\setminus Z$.

Theorem 5 implies Theorem 4. In turn, the proof of Theorem 5 is based on the
orbit separation property [6, Definition 2.6]. This proof requires
considerable technical efforts.

Finally, let us mention one more result on this subject, which is of
independent interest.

######  Theorem 6 ([6, Theorem 2.15]).

Let $X$ be an affine variety of dimension $\geq 2$. Suppose that a group
$G\subseteq{\rm SAut}(X)$ generated by a saturated set $N$ of locally
nilpotent vector fields acts on $X$ with an open orbit $O$ and $Y=X\setminus
O$. Then the group $G_{N,Y}$ acts on $O$ infinitely transitively.

##  2\. Properties of flexible varieties

In this section we consider several algebraic and geometric results related to
flexibility of an algebraic variety.

###  2.1. The Makar-Limanov invariant and rationality properties

Recall [37] that the _Makar-Limanov invariant_ $\text{ML}(X)$ of an affine
algebraic variety $X$ is the intersection of kernels of locally nilpotent
derivations on ${\mathbb{K}}[X]$. In other words, $\text{ML}(X)$ is the
subalgebra of ${\rm SAut}(X)$-invariant regular functions on $X$.

In [67], a close concept is defined: the _Makar-Limanov field invariant_
$\text{FML}(X)$ is the intersection of kernels of derivations obtained by
extension of locally nilpotent derivations of the algebra ${\mathbb{K}}[X]$ to
the field of rational functions ${\mathbb{K}}(X)$. This is a subfield of the
field ${\mathbb{K}}(X)$ consisting of rational functions invariant with
respect to the group ${\rm SAut}(X)$. If this invariant is trivial, i.e.
$\text{FML}(X)={\mathbb{K}}$, then the invariant $\text{ML}(X)$ is also
trivial, but the converse statement is generally not true. The triviality of
$\text{FML}(X)$ is equivalent to the existence of an open ${\rm
SAut}(X)$-orbit in $X$ (Corollary 1) or, equivalently, the existence of a
flexible point on $X$.

It is natural to assume that the triviality of such invariants should be
connected with one or another version of rationality properties of the variety
$X$. Recall that an irreducible algebraic variety $X$ is _rational_ if the
field of rational functions ${\mathbb{K}}(X)$ is generated over the field
${\mathbb{K}}$ by a finite set of algebraically independent elements.
Equivalently, there is an open subset in $X$ isomorphic to an open subset of
an affine space. An irreducible variety $X$ is _stably rational_ if the
variety $X\times{\mathbb{A}}^{k}$ is rational for some positive integer $k$.
Finally, an irreducible variety $X$ is _unirational_ if the field of rational
functions ${\mathbb{K}}(X)$ can be embedded as a subfield in the field of
rational fractions ${\mathbb{K}}(x_{1},\ldots,x_{m})$. Geometrically, this
means that there exists a rational dominant morphism to $X$ from an affine
space. It is known that if $X$ is a curve or a surface then the rationality of
$X$ is equivalent to unirationality. At the same time, there are unirational
but not rational varieties of dimension $3$.

Examples of non-unirational three-dimensional affine varieties $X$ with
$\text{ML}(X)={\mathbb{K}}$ are known; in these examples, the varieties are
birationally equivalent to the direct product $C\times{\mathbb{A}}^{2}$, where
$C$ is a smooth curve of genus $g\geq 1$; see [66, Example 4.2]. For such
varieties, typical ${\rm SAut}(X)$-orbits have dimension two, that is, the
field invariant $\text{FML}(X)$ is nontrivial.

The following proposition confirms, in particular, Conjecture 5.3 from [67].

######  Proposition 4 ([6, Proposition 5.1]).

Let $X$ be an irreducible affine variety. If ${\text{FML}(X)={\mathbb{K}},}$
then the variety $X$ is unirational.

The proof follows from the observation that the condition
$\text{FML}(X)={\mathbb{K}}$ means that there is an open ${\rm SAut}(X)$-orbit
$O$ in $X$. Proposition 2 guarantees that there are
${\mathbb{G}}_{a}$-subgroups $H_{1},\ldots,H_{s}$ in ${\rm SAut}(X)$ and a
point $x\in X$ such that the image of the map

$H_{1}\times\ldots\times H_{s}\to
X,\quad(h_{1},\ldots,h_{s})\mapsto(h_{1}\ldots h_{s}).x$

equals $O$. Since the product $H_{1}\times\ldots\times H_{s}$ is isomorphic
(as a variety) to the affine space ${\mathbb{A}}^{s}$, this implies
unirationality of $X$. Moreover, every two points on $O$ are connected by the
image of a morphism ${\mathbb{A}}^{1}\to O$. In other words, the orbit $O$ is
_${\mathbb{A}}^{1}$ -connected_.

In general, flexibility entails neither rationality nor stable rationality.
For example, Popov [76, Example 1.22] pointed out examples of finite subgroups
$F\subseteq{\rm SL}_{n}$ for some $n$ such that the smooth flexible
unirational affine homogeneous space $X={\rm SL}_{n}/F$ is not stably
rational.

The paper [17] is devoted to a conjecture that characterizes unirationality in
terms of infinitely transitive actions. To formulate the conjecture, we need
several definitions.

We say that an irreducible variety $X$ is _stably birationally infinitely
transitive_ if for some positive integer $m$ the purely transcendental
extension ${\mathbb{K}}(X)(y_{1},\ldots,y_{m})$ of the field ${\mathbb{K}}(X)$
is the field of rational functions on some flexible affine variety of
dimension $\geq 2$. If this condition holds for $m=0$, the variety $X$ is
called _birationally infinitely transitive_.

It is clear that the affine space ${\mathbb{A}}^{n}$ is stably birationally
infinitely transitive, and for $n\geq 2$ it is birationally infinitely
transitive. Thus, the same properties hold for any rational algebraic variety.

######  Conjecture 1 ([17, Conjecture 1.4]).

Any unirational variety is stably birationally infinitely transitive.

In [17, Example 1.6] it is shown that the stability requirement is essential
for the conjecture. In fact, the authors show that any unirational three-
dimensional variety admitting an infinitely transitive birational model is
rational. Indeed, for a typical ${\mathbb{G}}_{a}$-orbit $C$ on a given model,
this model is birationally isomorphic to $C\times Y$ for some surface $Y$. By
assumptions, the surface $Y$ is unirational, and therefore rational. At the
same time, there are examples of unirational but not rational three-
dimensional varieties.

In [17, Theorems 2.1–2.2], sufficient conditions for stable birational
infinite transitivity of a variety are given. These conditions allow to prove
Conjecture 1 for wide classes of varieties, see [17, Section 3].

###  2.2. Collective transitivity

By _collective infinite transitivity_ we mean the possibility to translate
simultaneously, that is, by the same automorphism, arbitrary finite sets of
points to arbitrary finite sets of points of the same cardinality in the same
orbits. The general results on collective infinite transitivity obtained in
[6, Section 3] are illustrated here by examples from matrix algebra, see [81,
82].

Let $X={\rm Mat}(n,m)$ be the space of matrices of size $n\times m$ over the
ground field ${\mathbb{K}}$. It is well known that a locally closed subset
$X_{r}\subseteq X$ of matrices of rank $r$ has dimension $mn-(m-r)(n-r)$.
Further we assume that this dimension is at least $2$. The group ${\rm
SL}_{n}\times{\rm SL}_{m}$ acts on $X$ by means of left/right multiplications
and preserves the subsets $X_{r}$. For any $k\neq l$, let $E_{kl}$ and
$E^{kl}$ be matrix units, that is, $x_{kl}=1$ and all other matrix elements
are zero. Let $H_{kl}=I_{n}+{\mathbb{K}}E_{kl}\subseteq{\rm SL}_{n}$ and
$H^{kl}=I_{m}+{\mathbb{K}}E^{kl}\subseteq{\rm SL}_{m}$ denote the
corresponding ${\mathbb{G}}_{a}$-subgroups. They preserve the stratification
$X=\bigcup_{r}X_{r}$. Let $\delta_{kl}$ and $\delta^{kl}$ be the corresponding
locally nilpotent vector fields on $X$. These fields are tangent to the
elements of the stratification.

Let us say that the ${\mathbb{G}}_{a}$-subgroups $H_{kl}$, $H^{kl}$ and their
replicas are _elementary_ ${\mathbb{G}}_{a}$-subgroups. Our goal is to
establish collective infinite transitivity on strata for the subgroup $G$ of
the group ${\rm SAut}(X)$ generated by such two-sided elementary subgroups.

We know from linear algebra that the subgroup ${\rm SL}_{n}\times{\rm
SL}_{m}\subseteq G$ acts transitively on every subvariety $X_{r}$ with the
exception of the open subvariety $X_{n}$ in the case $m=n$. In the latter
case, $G$-orbits lying in $X_{n}$ are the level subvarieties of the
determinant.

######  Theorem 7 ([6, Theorem 3.3]).

Let us consider two finite ordered sets $S_{1}$ and $S_{2}$ of the same
cardinality, consisting of such matrices from ${\rm Mat}(n,m)$ that the
sequences of ranks of these matrices coincide. In the case $m=n$, we
additionally require a matching of the sequences of determinants. Then there
is an element of the group $G$ whose diagonal action translates $S_{1}$ to
$S_{2}$.

Similar results in the case of symmetric and skew-symmetric matrices are
obtained in [6, Section 3.3].

###  2.3. ${\mathbb{A}}^{1}$-richness and the Gromov-Winkelmann Theorem

Let $X$ be a flexible affine variety of dimension $\geq 2$. Consider a set of
pairwise distinct points $p_{1},\ldots,p_{k}\in X^{\rm reg}$. Let us fix a
${\mathbb{G}}_{a}$-orbit $C$ on $X^{\rm reg}$ and a set of $k$ pairwise
distinct points ${q_{1},\ldots,q_{k}\in C}$. It follows from infinite
transitivity that there is such an element $g\in{\rm SAut}(X)$ that $g\cdot
q_{1}=p_{1},\ldots,g\cdot q_{k}=p_{k}$. Thus, the shift $g\cdot C$ of the
curve $C$ is a ${\mathbb{G}}_{a}$-orbit on $X$ with respect to the conjugated
action and it passes through the points $p_{1},\ldots,p_{k}$. This elementary
observation can be strengthened as follows.

An algebraic variety $X$ is called _${\mathbb{A}}^{1}$ -rich_ if for any
finite subset $Z$ and any closed subvariety $Y$ of codimension $\geq 2$ that
does not intersect $Z$, there is a curve on $X$ that is isomorphic to the line
${\mathbb{A}}^{1}$, does not intersect $Y$, and passes through all points of
$Z$. The next result follows from the Transversality Theorem (Theorem 3).

######  Proposition 5 ([6, Corollary 4.18]).

Let us consider an affine variety $X$ and assume that the group ${\rm
SAut}(X)$ acts with an open orbit ${O\subseteq X}$. Then for any finite subset
$Z\subseteq O$ and any closed subvariety $Y\subseteq X$ of codimension $\geq
2$ with the condition $Z\cap Y=\varnothing$ there is an orbit
$C\cong{\mathbb{A}}^{1}$ of some ${\mathbb{G}}_{a}$-action on $X$ which does
not intersect $Y$ and passes through all points of the subset $Z$

In the case $X={\mathbb{A}}^{n}$, this result follows from the Gromov-
Winkelmann Theorem [87]. This theorem states that the group ${\rm
SAut}({\mathbb{A}}^{n}\setminus Y)$ acts on ${\mathbb{A}}^{n}\setminus Y$
transitively and, as we know, this entails infinite transitivity of the
action.

A generalization of the Gromov-Winkelmann Theorem is obtained in [33].
Firstly, the authors show that the flexibility of an affine variety $X$ is
equivalent to the flexibility of the smooth quasi-affine variety $X^{\rm
reg}$.

######  Theorem 8 ([33, Theorem 0.1]).

Let $X$ be a smooth flexible quasi-affine variety of dimension $\geq 2$ and
$Y\subseteq X$ be a closed subscheme of codimension $\geq 2$. Then the variety
$X\setminus Y$ is also flexible.

Finally, it is shown in [33, Proposition 1.8] that for a normal quasi-affine
variety $X$ and a closed subset $Y\subseteq X$ of codimension $\geq 2$, each
${\mathbb{G}}_{a}$-action on $X\setminus Y$ can be extended to a
${\mathbb{G}}_{a}$-action on $X$, which stabilizes $Y$.

###  2.4. Transitivity on jets

We limit ourselves in this subsection to the formulation of only one result.
By the volume form on a variety $X$ of dimension $n$, we mean a differential
$n$-form $\omega$ defined on $X^{\rm reg}$ that does not vanish anywhere.

######  Theorem 9 ([6, Theorem 4.14 and Remark 4.16]).

Let $X$ be a flexible affine variety of dimension $n\geq 2$ equipped with a
volume form $\omega$. Then for any $m\geq 0$ and any finite subset $Z\subseteq
X^{\rm reg}$ there is an automorphism $g\in{\rm SAut}(X)$ with prescribed
$m$-jets at points $p\in Z$ provided that these jets preserve the form
$\omega$ and map $Z$ to $X^{{\rm reg}}$ injectively. The same holds without
assuming that there is a global volume form on $X^{{\rm reg}}$ provided that
each point $p\in Z$ corresponds to a jet that fixes the point $p$ and the
linear part of the jet belongs to ${\rm SL}(T_{p}X)$.

Other results on transitivity on jets can be found in [6, Section 4.2].

###  2.5. Multiple transitivity of the automorphism group

It is natural to ask whether the flexibility is the only reason that the
automorphism group ${\rm Aut}(X)$ of an affine variety $X$ acts on the smooth
locus of $X$ multiply transitively. In [4] a positive answer to this question
is obtained under some additional restrictions.

We say that a variety $X$ satisfies _condition $(*)$_ if $X$ admits a non-
identical action of a connected affine algebraic group of positive dimension.
Condition $(*)$ is equivalent to the fact that there is a non-identical action
of either the group ${\mathbb{G}}_{a}$ or the group ${\mathbb{G}}_{m}$ on $X$.

######  Theorem 10 ([4, Theorem 11]).

Let $X$ be an irreducible quasi-affine variety of dimension $\geq 2$
satisfying condition $(*)$. Suppose that the group ${\rm Aut}(X)$ acts on $X$
with an open orbit $O$. Then the following conditions are equivalent.

  1. (1)

The group ${\rm Aut}(X)$ acts on $O$ $2$-transitively.

  2. (2)

The group ${\rm Aut}(X)$ acts on $O$ infinitely transitively.

  3. (3)

The group ${\rm SAut}(X)$ acts on $O$ transitively.

  4. (4)

The group ${\rm SAut}(X)$ acts on $O$ infinitely transitively.

In particular, the group ${\rm Aut}(X)$ acts on the smooth locus
$2$-transitively if and only if the variety $X$ is flexible.

Implications $(4)\Rightarrow(2)$ and $(2)\Rightarrow(1)$ are obvious.
Implication $(3)\Rightarrow(4)$ follows from Theorem 4.

It remains to prove implication $(1)\Rightarrow(3)$. Here we give arguments
that are somewhat different from the arguments in [4] and are based on the
group-theoretic facts set out in the second appendix to this paper. Let us
first assume that there is a non-identical action of the group
${\mathbb{G}}_{a}$ on $X$. This means that the group ${\rm SAut}(X)$ is
nontrivial. Since ${\rm SAut}(X)$ is a normal subgroup of the group ${\rm
Aut}(X)$, condition $(1)$ and Lemma 7 imply condition $(3)$.

Let us recall that an algebraic variety $X$ is called _rigid_ if there is no
non-identical ${\mathbb{G}}_{a}$-action on $X$. For an affine $X$, this
condition is equivalent to the fact that the algebra ${\mathbb{K}}[X]$ does
not admit nonzero locally nilpotent derivation. In this case, there is a
unique maximal torus $T$ in the group ${\rm Aut}(X)$. This result is proved in
[8, Theorem 1] for an affine variety $X$ and is generalized to the quasi-
affine case in [4, Theorem 5]. In particular, the maximal torus $T$ is a
normal subgroup in ${\rm Aut}(X)$. According to condition $(*)$, the torus $T$
is nontrivial. From Lemma 7 we conclude again that $T$ acts transitively on
$O$. Thus, the orbit $O$ is isomorphic to the torus $T$. However, the
automorphism group of the variety $T$ is a semidirect product of $T$ and the
discrete group ${\rm GL}_{n}({\mathbb{Z}})$. Such a group acts transitively
but not $2$-transitively on $T$. Thus, for a rigid variety $X$ with condition
$(*)$ condition $(1)$ does not hold, and implication $(1)\Rightarrow(3)$ is
proved.

The ideas of this reasoning can be used in the following situation.

######  Definition 1.

We call a set of automorphisms of an algebraic variety $X$ _geometric_ if this
set is invariant under conjugation by any automorphism of the variety $X$ and
contains at least one non-identical automorphism.

Informally speaking, geometric sets of automorphisms can be understood as sets
that are defined by certain invariant geometric properties. For example, such
sets form all automorphisms of order two that have a single fixed point, or
all automorphisms that preserve the set of curves that do not intersect, cover
the smooth locus of the variety $X$, and each of these curves is isomorphic to
the affine line.

######  Proposition 6.

Let $X$ be a flexible quasi-affine variety of dimension $\geq 2$. Then for any
geometric set of automorphisms of the variety $X$, the subgroup of the
automorphism group generated by this set acts on the smooth locus $X^{{\rm
reg}}$ infinitely transitively.

The proof follows immediately from the fact that a geometric set of
automorphisms generates a nontrivial normal subgroup in ${\rm Aut}(X)$, and
Theorem 24 shows that this group is infinitely transitive on $X^{{\rm reg}}$.

In [77] the author studies relations between multiple transitivity of the
action of a group of automorphisms and geometric properties of a variety, such
as rationality or unirationality. An action of a group $G$ on a variety $X$ is
said to be _generically $m$-transitive_ if the restriction of this action to
some open invariant subset of the variety $X$ is $m$-transitive. In [77,
Theorem 5] it is proved over an uncountable algebraically closed field that
for any irreducible variety $X$, the condition of generic $2$-transitivity of
the action of the group ${\rm Aut}(X)$ implies that either $X$ is unirational,
or the group ${\rm Aut}(X)$ does not contain nontrivial connected algebraic
subgroups. In particular, if the variety $X$ is complete, the generic
$2$-transitivity of the action of ${\rm Aut}(X)$ on $X$ implies
unirationality.

We also note that flexible varieties form a natural class of varieties where
it makes sense to investigate the problem of continuation of isomorphisms
between subvarieties. Namely, let $X$ be a smooth flexible quasi-affine
variety and $Y_{1},Y_{2}$ be two closed subvarieties in $X$. The extension
problem consists in finding conditions on the variety $X$, the subvarieties
$Y_{1},Y_{2}$, and possibly tangent subbundles $TY_{1},TY_{2}$, under which
any isomorphism $Y_{1}\to Y_{2}$ can be extended to an automorphism of the
variety $X$. A classical result of this type is the Abyankar-Moh-Suzuki
Theorem, which states that the images of any two isomorphic embeddings of the
affine line ${\mathbb{A}}^{1}$ into the affine plane ${\mathbb{A}}^{2}$ can be
sent to each other by an automorphism of the plane. A number of sufficient
conditions for extension of an isomorphism between subvarieties in a flexible
variety can be found in [50, 53] and in the works listed there in the
references.

##  3\. Examples of flexible varieties

Considerable interest in flexible varieties is caused not only because such
varieties have a number of remarkable geometric properties, but also by the
fact that the class of flexible varieties is surprisingly wide. In this
section we give various examples of flexible varieties. We largely follow the
work plan from [7], supplementing the presentation with the results of recent
years.

###  3.1. Suspensions

One of the first papers, where the property of infinite transitivity of the
action of a group of automorphisms of an algebraic variety was studied, is the
work [54].

######  Theorem 11 ([54, Theorem 5.1]).

Consider a hyperplane $X$ in the affine space ${\mathbb{A}}^{n},$ $n\geq 4,$
given by an equation of the form

$x_{1}x_{2}=f(x_{3},\ldots,x_{n}),$

where $f$ is an arbitrary non-constant polynomial in $n-2$ variables. Then the
group ${\rm SAut}(X)$ acts on the smooth locus of the variety $X$ infinitely
transitively.

This result motivates the following definition.

######  Definition 2.

Let $X$ be an irreducible affine variety and $f\in{\mathbb{K}}[X]$ be a non-
constant function. The _suspension_ over the variety $X$ is the affine variety

${\rm Susp}(X,f)=\\{uv-f(x)=0\\}\subseteq{\mathbb{A}}^{2}\times X,$

where $u$ and $v$ are coordinates on the plane ${\mathbb{A}}^{2}$.

In these terms, the varieties from Theorem 11 are exactly suspensions over the
affine space ${\mathbb{A}}^{n-2}$. The following general fact holds.

######  Theorem 12 ([13, Theorem 3.2]).

Let $X$ be a flexible affine variety and $f\in{\mathbb{K}}[X]$ be a non-
constant function. Then the suspension ${\rm Susp}(X,f)$ is also a flexible
variety.

Thus, iterating the suspension construction, we obtain a wide class of
flexible varieties.

Note that in Theorem 12, the case of a one-dimensional variety $X$ is also
allowed. In this case, we have $X={\mathbb{A}}^{1}$ automatically. In fact, a
similar result for the affine line can be obtained over any field of
characteristic zero.

######  Theorem 13 ([13, Theorem 3.1]).

Let ${\mathbb{K}}$ be a field of characteristic zero and $f\in{\mathbb{K}}[x]$
be a polynomial such that $f({\mathbb{K}})={\mathbb{K}}$. Consider a
hypersurface $X\subseteq{\mathbb{A}}^{3}$ given by the equation
$x_{1}x_{2}=f(x_{3})$. Then the group ${\rm SAut}(X)$ acts on the smooth locus
of the variety $X$ infinitely transitively.

The case of the field of real numbers ${\mathbb{K}}={\mathbb{R}}$ was
considered separately. The initial result [13, Theorem 3.3] has been
strengthened to the following general fact.

######  Theorem 14 ([63, Theorem 1]).

Let $X$ be an affine algebraic variety over the field of real numbers and
$f\in{\mathbb{R}}[X]$. Suppose that each connected component $X^{i}$ of the
smooth locus of the variety $X$ has dimension $\geq 2$ and $f$ is non-constant
on $X^{i}$. If the variety $X$ is flexible and the action of the group ${\rm
SAut}(X)$ is infinitely transitive on each component $X^{i}$, then the same
properties hold for the suspension ${\rm Susp}(X,f)$.

Let us return to the case of an algebraically closed field of characteristic
zero and describe several more classes of flexible affine hypersurfaces. We
fix positive integers $n_{0},n_{1},n_{2}$ and put
${n\\!=\\!n_{0}\\!+\\!n_{1}\\!+\\!n_{2}}$. For each $i=0,1,2$, we also fix a
set of positive integers $l_{i}\in{\mathbb{Z}}^{n_{i}}_{>0}$ and define
monomials

$T_{i}^{l_{i}}:=T_{i1}^{l_{i1}}\ldots
T_{in_{i}}^{l_{in_{i}}}\in{\mathbb{K}}[T_{ij}\,;\,i=0,1,2,\
j=1,\ldots,n_{i}].$

A _trinomial_ is a polynomial of the form
$f=T_{0}^{l_{0}}+T_{1}^{l_{1}}+T_{2}^{l_{2}}$, and a _trinomial hypersurface_
$X$ is the set of zeroes of a trinomial $f$ in the affine space
${\mathbb{A}}^{n}$. It is not difficult to verify that $X$ is an irreducible
normal affine variety of dimension $n-1$. Among such hypersurfaces there are
examples of both rigid and flexible varieties. For example, if for some $i$ we
have $n_{i}=l_{i1}=1$, then the trinomial hypersurface is isomorphic to the
affine space ${\mathbb{A}}^{n-1}$, and if $n_{i}=2$ and $l_{i1}=l_{i2}=1$,
then the trinomial hypersurface is a suspension over the affine space
${\mathbb{A}}^{n-2}$. In addition to these examples, five sufficiently broad
classes of flexible trinomial hypersurfaces are described in [39, Theorem 4].
For instance, hypersurfaces corresponding to trinomials of the form
$T_{0}^{l_{0}}+T_{1}^{l_{1}}+T_{21}T_{22}\ldots T_{2n_{2}}$ are flexible. At
the same time, in [39] flexibility and rigidity are investigated both for
trinomial hypersurfaces and for a wider class of affine trinomial varieties.

###  3.2. Affine toric varieties

We recall that a normal algebraic variety $X$ is _toric_ if there exists a
regular effective action of an algebraic torus $T$ on $X$ with an open orbit.
Some affine toric varieties are not flexible. For example, if we put $X=T$,
then the algebra ${\mathbb{K}}[X]$ is generated by invertible functions, and
the group ${\rm SAut}(X)$ is trivial.

Let us call a toric variety _non-degenerate_ if every invertible regular
function on $X$ is constant. This condition is equivalent to the fact that $X$
cannot be represented as a direct product $X^{\prime}\times{\mathbb{G}}_{m}$
for some toric variety $X^{\prime}$.

######  Theorem 15 ([13, Theorem 2.1]).

Evere non-degenerate affine toric variety is flexible.

In order to explain the idea of the proof of this result, we consider
$T$-normalized ${\mathbb{G}}_{a}$-subgroups and describe such subgroups in
terms of so-called Demazure roots.

Let $N$ be the lattice of one-parameter subgroups of the torus $T$ and $M$ be
the dual lattice of characters. Let $\langle\,\cdot\,,\,\cdot\,\rangle\colon
N\times M\to{\mathbb{Z}}$ be the canonical pairing. We let $\chi^{m}$ denote
the character of the torus $T$ corresponding to a point $m\in M$. In this case
we have $\chi^{m}\chi^{m^{\prime}}\,=\,\chi^{m+m^{\prime}}$, so the group
algebra

${\mathbb{K}}[M]:=\bigoplus_{m\in M}{\mathbb{K}}\chi^{m}$

can be identified with the algebra ${\mathbb{K}}[T]$ of regular functions on
the torus $T$. Denote by $T.x_{0}$ the open $T$-orbit on $X$. Since the orbit
map $T\to X$, $t\longmapsto t.x_{0}$ is dominant, we can identify
${\mathbb{K}}[X]$ with a subalgebra in ${\mathbb{K}}[M]$. More precisely,
there is such a convex polyhedral cone $\sigma^{\vee}\subseteq
M_{{\mathbb{Q}}}:=M\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ that ${\mathbb{K}}[X]$
coincides with the semigroup algebra of the semigroup $\sigma^{\vee}\cap M$,
that is

${\mathbb{K}}[X]=\bigoplus_{m\in\sigma^{\vee}\cap M}{\mathbb{K}}\chi^{m}.$

The dual cone $\sigma$ to the cone $\sigma^{\vee}$ is a strictly convex cone
in $N_{{\mathbb{Q}}}$. We assume that the variety $X$ is non-degenerate. This
means that the cone $\sigma$ does not lie in a proper subspace of the space
$N_{{\mathbb{Q}}}$. Let $\Xi=\\{p_{1},\ldots,p_{r}\\}$ be the set of
generators on rays, that is, the set of primitive vectors on one-dimensional
faces of the cone $\sigma$. Each vector $p\in N$ is associated with a one-
parameter subgroup $R_{p}$ of the torus $T$.

There are natural one-to-one correspondences between faces $\delta$ of the
cone $\sigma$, the dual faces $\tau=\delta^{\bot}$ of the cone
$\sigma^{\vee}$, and $T$-orbits $O_{\tau}$ on $X$, where $\dim
O_{\tau}=\dim\tau=\dim\sigma-\dim\delta$. In particular, the only $T$-fixed
point on $X$ corresponds to the vertex of the cone $\sigma^{\vee}$, and the
open $T$-orbit corresponds to the cone $\sigma^{\vee}$ itself. These
correspondences preserve inclusions: a $T$-orbit $O_{\mu}$ is contained in the
closure $\overline{O_{\tau}}$ if and only if $\mu\subseteq\tau$ or,
equivalently, $\mu^{\bot}\supseteq\tau^{\bot}$.

Every face $\tau\subseteq\sigma^{\vee}$ defines a decomposition

${\mathbb{K}}[X]\ ={\mathbb{K}}[\overline{O_{\tau}}]\oplus
I(\overline{O_{\tau}}),$

where

${\mathbb{K}}[\overline{O_{\tau}}]=\bigoplus_{m\in\tau\cap
M}{\mathbb{K}}\chi^{m},\qquad\mbox{and}\qquad
I(\overline{O_{\tau}})=\bigoplus_{m\in(\sigma^{\vee}\setminus\tau)\cap
M}{\mathbb{K}}\chi^{m}$

is the ideal of the subvariety $\overline{O_{\tau}}$ in ${\mathbb{K}}[X]$.

The stabilizer $T_{p}$ of a point $p\in X$ is connected, that is,
$T_{p}\subseteq T$ is a subtorus. Further, we have an inclusion
$T_{p}\subseteq T_{q}$ exactly when $T.q\subseteq\overline{T.p}$, and
${\overline{T.p}=X^{T_{p}}}$, where $X^{G}$ denotes the set of fixed points of
an action of a group $G$ on a set $X$.

######  Definition 3 ([28]).

A _Demazure root_ of a cone $\sigma$ is a vector ${e\in M}$ such that for some
$i$ with $1\leq i\leq r$, where $r$ is the number of elements in $\Xi$, we
have

$\langle p_{i},e\rangle\,=\,-1\quad\text{and}\quad\langle p_{j},e\rangle\geq
0\quad\text{for all}\quad j\neq i.$

Denote by ${\mathcal{R}}(\sigma)$ the set of all Demazure roots of a cone
$\sigma$. There is a one-to-one correspondence $e\longleftrightarrow H_{e}$
between roots of the cone $\sigma$ and ${\mathbb{G}}_{a}$-subgroups in ${\rm
Aut}(X)$ normalized by the acting torus, see [28, 66]. Such subgroups are
called _root subgroups_.

We put $p_{e}:=p_{i}$. A root $e\in{\mathcal{R}}(\sigma)$ defines a locally
nilpotent derivation $\partial_{e}$ on the $M$-graded algebra
${\mathbb{K}}[X]$ given as

$\partial_{e}(\chi^{m})\ =\ \langle p_{e},m\rangle\chi^{m+e}.$

Its kernel is a finitely generated homogeneous subalgebra in
${\mathbb{K}}[X]$:

$\ker\partial_{e}=\bigoplus_{m\in p_{e}^{\bot}\cap M}{\mathbb{K}}\chi^{m},$

where $p_{e}^{\bot}=\\{m\in\sigma^{\vee}\cap M,\,\langle p_{e},m\rangle=0\\}$
is a facet, i.e. a face of codimension one, of the cone $\sigma^{\vee}$
orthogonal to $p_{e}$.

Two roots $e$ and $e^{\prime}$ such that $p_{e}=p_{e^{\prime}}$ are called
equivalent; we write $e\sim e^{\prime}$. The roots $e$ and $e^{\prime}$ are
equivalent if and only if $\ker\partial_{e}=\ker\partial_{e^{\prime}}$.

A numbering of the generators on the rays $\Xi=\\{p_{1},\ldots,p_{r}\\}$
determines the partition

${\mathcal{R}}(\sigma)=\bigcup_{i=1}^{r}{\mathcal{R}}_{i},\quad\mbox{where}\quad{\mathcal{R}}_{i}=\\{e\in{\mathcal{R}}(\sigma)\,|\,p_{e}=p_{i}\\}\,$

are nonempty. Moreover, these subsets are infinite if the dimension of the
cone is $\geq 2$.

As an example, we consider the affine plane $X={\mathbb{A}}^{2}$ with the
standard action of a two-dimensional torus. Here the cones $\sigma$ and
$\sigma^{\vee}$ coincide with the positive quadrants. The set
${\mathcal{R}}(\sigma)$ consists of two equivalence classes

${\mathcal{R}}_{1}=\\{(-1,y)\,|\,y\in{\mathbb{Z}}_{\geq
0}\\}\quad\mbox{and}\quad{\mathcal{R}}_{2}=\\{(x,-1)\,|\,x\in{\mathbb{Z}}_{\geq
0}\\}.$

The derivation $\partial_{e}$ generates a ${\mathbb{G}}_{a}$-subgroup

$H_{e}=\lambda_{e}({\mathbb{K}})\subseteq{\rm Aut}(X),$

where $\lambda_{e}\colon t\longmapsto{\rm exp}(t\partial_{e})$. The algebra of
invariants ${\mathbb{K}}[X]^{H_{e}}$ coincides with $\ker\partial_{e}$. The
embedding ${\mathbb{K}}[X]^{H_{e}}\subseteq{\mathbb{K}}[X]$ induces a morphism
$\pi\colon X\to Z={\rm Spec\,}{\mathbb{K}}[X]^{H_{e}}$ whose typical fibers
are one-dimensional $H_{e}$-orbits isomorphic to the line ${\mathbb{A}}^{1}$.
The torus $T$ normalizes the subgroup $H_{e}$. In particular, $T$ leaves the
set of fixed points $X^{H_{e}}$ invariant.

Let $R_{e}=R_{p_{e}}\subseteq T$ be the one-dimensional subtorus corresponding
to the vector ${p_{e}\in N}$. The action of $R_{e}$ on the graded algebra
${\mathbb{K}}[X]$ with a suitable parametrization
$p_{e}:{\mathbb{G}}_{\text{m}}({\mathbb{K}})\ni t\longmapsto p_{e}(t)\in
R_{e}$ is given by

$t.\chi^{m}\ =\ t^{\langle p_{e},m\rangle}\chi^{m},\quad
t\in{\mathbb{G}}_{\text{m}}({\mathbb{K}}).$

In particular, we have ${\mathbb{K}}[X]^{R_{e}}={\mathbb{K}}[X]^{H_{e}}$.
Therefore, the morphism ${\pi:X\to Z}$ coincides with the quotient morphism
$X\to X/\\!/R_{e}$, and typical $H_{e}$-orbits coincide with the closures of
typical $R_{e}$-orbits. This statement holds for any one-dimensional
$H_{e}$-orbit.

We have a decomposition

${\mathbb{K}}[X]={\mathbb{K}}[X]^{R_{e}}\ \oplus\
\bigoplus_{m\in\sigma^{\vee}\cap M\setminus
p_{e}^{\bot}}{\mathbb{K}}\chi^{m}={\mathbb{K}}[X]^{R_{e}}\oplus I(D_{e}),$

where $D_{e}:=X^{R_{e}}\cong Z$. The divisor $D_{e}$ coincides with the set of
limit points of the action of $R_{e}$ on $X$. Thus, each one-dimensional
$R_{e}$-orbit has a limit point on $D_{e}$.

######  Proposition 7 ([13, Proposition 2.1]).

Let $e\in{\mathcal{R}}(\sigma)$ and $H_{e}\subseteq{\rm SAut}(X)$ be the
associated ${\mathbb{G}}_{a}$-subgroup. Then the following statements hold.

  1. (a)

For any point $x\in X\setminus X^{H_{e}}$ the orbit $H_{e}.x$ intersects
precisely two $T$-orbits ${\mathcal{O}}_{1}$ and ${\mathcal{O}}_{2}$ on $X,$
and $\dim{\mathcal{O}}_{1}=\dim{\mathcal{O}}_{2}+1$.

  2. (b)

The intersection ${\mathcal{O}}_{2}\cap H_{e}.x$ consists of a unique point,
while

${\mathcal{O}}_{1}\cap H_{e}.x=R_{e}.y,\quad y\in{\mathcal{O}}_{1}\cap
H_{e}.x.$

We say that a pair of $T$-orbits $({\mathcal{O}}_{1},{\mathcal{O}}_{2})$ on
$X$ is $H_{e}$-connected if
$H_{e}.x\subseteq{\mathcal{O}}_{1}\cup{\mathcal{O}}_{2}$ for some $x\in
X\setminus X^{H_{e}}$. Proposition 7 shows that for such a pair (up to
permutation) we have ${\mathcal{O}}_{2}\subseteq\overline{{\mathcal{O}}_{1}}$
and $\dim{\mathcal{O}}_{1}=\dim{\mathcal{O}}_{2}+1$. Here we choose the point
$x$ on the orbit ${\mathcal{O}}_{1}$. Since the torus normalizes the subgroup
$H_{e}$, any point on ${\mathcal{O}}_{1}$ can serve as the point $x$ from the
definition of $H_{e}$-connectedness.

Proposition 7 implies the following criterion of $H_{e}$-connectedness.

######  Lemma 1 ([13, Lemma 2.2]).

Let $({\mathcal{O}}_{1},{\mathcal{O}}_{2})$ be a pair of $T$-orbits on $X$
with ${{\mathcal{O}}_{2}\subseteq\overline{{\mathcal{O}}_{1}},}$ where
${\mathcal{O}}_{i}={\mathcal{O}}_{\sigma_{i}^{\bot}}$ for a face $\sigma_{i}$
of the cone $\sigma,$ $i=1,2$. For a given root $e\in{\mathcal{R}}(\sigma)$ a
pair $({\mathcal{O}}_{1},{\mathcal{O}}_{2})$ is $H_{e}$-connected if and only
if we have $e|_{\sigma_{2}}\\!\leq\\!0$ and $\sigma_{1}$ is a facet of the
cone $\sigma_{2}$ given by the equation $\langle v,e\rangle=0$.

The following lemma is a key technical result.

######  Lemma 2 ([13, Lemma 2.3]).

For any point $x\in X^{{\rm reg}}\setminus{\mathcal{O}}_{\sigma^{\vee}}$ there
is a root $e\in{\mathcal{R}}(\sigma)$ such that

$\dim T.y\ >\ \dim T.x$

for a generic point $y\in H_{e}.x$. In particular, the pair $(T.y,T.x)$ is
$H_{e}$-connected.

From this lemma it is not difficult to deduce the transitivity of the action
of the group ${\rm SAut}(X)$ on the smooth locus of a non-degenerate affine
toric variety $X$, which leads to the proof of Theorem 15.

######  Remark 2.

Already in the case of affine toric surfaces, we can give examples of flexible
varieties $X$ such that the smooth locus $X^{{\rm reg}}$ is not a homogeneous
space for any algebraic group, see [13, Example 2.2].

To study actions of ${\mathbb{G}}_{a}$-subgroups, the following generalization
of the notion of a root subgroup on a toric variety is very useful.

######  Definition 4.

Let $X$ be an algebraic variety with an action of an algebraic torus $T$. A
${\mathbb{G}}_{a}$-subgroup in ${\rm Aut}(X)$ is called a _root subgroup_ if
it is normalized by the torus $T$. In this situation, the character $e$ of the
torus $T$ by which $T$ acts by conjugation on the ${\mathbb{G}}_{a}$-subgroup
is called the _root_ of the corresponding subgroup:

$ts(a)t^{-1}=s(\chi^{e}(t)a),\quad t\in T,\quad a\in{\mathbb{K}}.$

Many research projects are devoted to the study of root subgroups of
automorphism groups or, equivalently, homogeneous locally nilpotent
derivations of graded algebras. This topic deserves a special survey, and we
will not consider it in detail here.

Finally, we note that in the recent paper [18], flexibility and rigidity
criteria for non-normal affine toric varieties are obtained.

###  3.3. Homogeneous and almost homogeneous varieties

Following [76], we consider a class of connected linear algebraic groups that
are generated by their ${\mathbb{G}}_{a}$-subgroups. It is not difficult to
see that this class coincides with the class of connected linear algebraic
groups that do not admit nontrivial characters. In turn, these are exactly the
groups that are semidirect products of a connected semisimple group and a
unipotent radical. If such a group $G$ acts on a variety $X$, then the image
of $G$ in ${\rm Aut}(X)$ is contained automatically in the group ${\rm
SAut}(X)$. In particular, if $G$ acts on the smooth locus of a variety $X$
transitively, then $X$ is flexible.

As an example of such a situation, we consider a simple finite-dimensional
module $V$ of a semisimple group $G$ and let $X$ be equal to the closure of
the orbit of a highest weight vector. It is known from [78] that the variety
$X$ consists of two orbits, an open orbit and a fixed point coinciding with
the origin. If the variety $X$ does not coincide with the space $V$, the
origin is a singular point of the variety $X$. This shows that $X$ is
flexible. The variety $X$ is a normal affine cone over the generalized flag
variety $G/P$, where $P$ is a parabolic subgroup of the group $G$. The
flexibility of such cones was first proved in [13, Theorem 1.1].

Also, the above proposition can be applied to homogeneous spaces $G/H$ of
groups $G$ without nontrivial characters. It is well known that for a
semisimple group $G$ such a homogeneous space is affine if and only if the
subgroup $H$ is reductive [79, Theorem 4.17]. Along the way we obtain many
important examples of smooth flexible varieties [6, Proposition 5.4]. For
instance, such examples are homogeneous spaces of the groups ${\rm SL}_{n}$,
${\rm SO}_{n}$ and ${\rm Sp}_{2n}$.

Now we suppose that a connected linear algebraic group $G$ acts on an
irreducible variety $X$ with an open orbit. In this situation, the variety $X$
is called _almost homogeneous_. It turns out that under certain conditions it
is possible to guarantee the flexibility of such varieties.

####  3.3.1. The smooth case.

Suppose that an almost homogeneous affine variety $X$ is smooth. In the case
of a semisimple group $G$, using Luna’s Slice Theorem, we show in [6, Theorem
5.6] that $X$ is a homogeneous variety with respect to a larger group which is
a semidirect product $G\rightthreetimes V$, where $V$ is a finite-dimensional
$G$-module. Since the group $G\rightthreetimes V$ has no characters, the
variety $X$ is flexible.

In the case of an arbitrary reductive group $G$ under the assumption that
there is no non-constant invertible regular function on the variety, a similar
result is proved in [40, Theorem 2].

####  3.3.2. ${\rm SL}_{2}$-embeddings.

Suppose that the group ${\rm SL}_{2}$ acts with an open orbit on a normal
affine three-dimensional variety $X$. The classification of such varietiess
was obtained by Popov [75] in the early 70s. In this situation, the stabilizer
of a general position for the action of the group ${\rm SL}_{2}$ is finite. If
such a stabilizer is not commutative, the variety turns out to be ${\rm
SL}_{2}$-homogeneous, and we come to the situation considered above. In the
remaining cases, the stabilizer of a general position is cyclic, and two cases
are possible here. In the first case, the variety consists of no more than two
${\rm SL}_{2}$-orbits, it is smooth and therefore flexible. In the second
case, the variety $X$ consists of three orbits, one is a fixed point, and it
is the only singular point on $X$. In [14] it is shown that $X$ is realized as
a categorical quotient of the affine hypersurface
$x_{0}^{b}=x_{1}x_{4}-x_{2}x_{3}$ by the action of a one-dimensional
diagonalizable group. This hypersurface is a suspension over
${\mathbb{A}}^{3}$. Using this circumstance, in [6, Theorem 5.7] we construct
a ${\mathbb{G}}_{a}$-subgroup in ${\rm Aut}(X)$ that connects three-
dimensional and two-dimensional ${\rm SL}_{2}$-orbits on $X$, and thereby
prove the flexibility of the variety $X$.

As a result, we obtain that any normal affine variety with an action of the
group ${\rm SL}_{2}$ with an open orbit is flexible.

####  3.3.3. Horospherical and spherical varieties

We recall that a normal algebraic variety $X$ on which a connected reductive
group $G$ acts with an open orbit is _horospherical_ if the stabilizer in $G$
of a point in the open orbit contains a maximal unipotent subgroup of $G$.
Affine horospherical varieties were studied in detail in [78].

The flexibility of affine horospherical varieties of a semisimple group $G$ is
proved in [84]. The idea of the proof is to construct
${\mathbb{G}}_{a}$-subgroups in ${\rm Aut}(X)$ that connect different
$G$-orbits on $X$ consisting of smooth points. These subgroups are realized as
exponents of locally nilpotent derivations whose kernels, as in the toric
case, correspond to the faces of a cone associated with a horospherical
variety.

In the case of an arbitrary reductive group $G$, the flexibility of affine
horospherical varieties without non-constant invertible regular functions is
proved in [40, Theorem 3]. In particular, a new proof of the flexibility of
non-degenerate affine toric varieties is obtained here.

A natural generalization of horospherical varieties are spherical varieties.
Recall that a normal algebraic variety $X$ with an action of a reductive group
$G$ is called _spherical_ if the restriction of the action to a Borel subgroup
$B$ has an open orbit in $X$. Spherical varieties are considered as the most
adequate analogue of toric varieties in the case of actions of noncommutative
reductive groups. There is also a combinatorial, albeit much more complex
description of spherical varieties in terms of so-called colored cones.

The following conjecture in formulates in [7, Section 6].

######  Conjecture 2.

Any affine spherical variety without non-constant invertible regular functions
is flexible.

The above results confirm this conjecture, but in general it remains open.

###  3.4. Vector bundles

Let $\pi\colon E\to X$ be a reduced irreducible linear space over a flexible
variety $X$, which is a vector bundle over $X^{{\rm reg}}$. Suppose there is
an action of the group ${\rm SAut}(X)$ on $E$ such that the action of each
${\mathbb{G}}_{a}$-subgroup in ${\rm SAut}(X)$ is algebraic and the morphism
$\pi$ is equivariant. In [6, Corollary 4.5] it is shown that the total space
$E$ is a flexible variety. In particular, the total space of the tangent
bundle $TX$ and, more generally, of any tensor bundle $E=(TX)^{\times
a}\times(T^{*}X)^{\otimes b}$ is flexible.

In [32] one can find an example of a smooth rational affine surface $S$ with a
finite automorphism group such that the automorphism group of the cylinder
$S\times{\mathbb{A}}^{2}$ acts infinitely transitively on the complement in
this cylinder to a closed subset of codimension $\geq 2$. This example shows
that there are rigid varieties such that the cylinders over them are
generically flexible.

###  3.5. Affine cones over projective varieties

Let $X$ be an affine cone over a projective variety $Y$ polarized by an ample
divisor $H$. By an affine cone we mean the spectrum of the homogeneous
coordinate ring

$\bigoplus_{n\geq 0}H^{0}(Y,nH).$

If the divisor $H$ is very ample and we consider the image of the variety $Y$
under embedding into a projective space by means of a complete linear system
$|H|$, the variety $X$ coincides with the affine cone over the projective
subvariety $Y$.

A natural question is the characterization of flexibility of the variety $X$
in terms of the pair $(Y,H)$. The study of this question began in the work of
Perepechko [73]. It is based on the results of a series of papers by
Kishimoto-Prokhorov-Zaidenberg. We briefly recall the relevant concepts and
results. A detailed explanation can be found, for example, in a recent survey
[21].

An open subset $U$ of a variety $Y$ is called a _cylinder_ if $U\cong
Z\times{\mathbb{A}}^{1}$ for some smooth affine variety $Z$. A cylinder $U$ is
called _$H$ -polar_ if $U=Y\setminus\text{Supp}\,D$ for some effective
${\mathbb{Q}}$-divisor $D$ linearly equivalent to $H$. In the works of
Kishimoto-Prokhorov-Zaidenberg it is shown that an $H$-polar cylinder $U$ on
$Y$ determines a ${\mathbb{G}}_{a}$-action on the affine cone $X$ over $Y$.

We call a subset $W\subseteq Y$ _invariant_ with respect to a cylinder $U\cong
Z\times{\mathbb{A}}^{1}$ if $W\cap U=\pi^{-1}(\pi(W))$, where $\pi\colon U\to
Z$ is the projection along ${\mathbb{A}}^{1}$. In other words, a subset $W$ is
invariant if every ${\mathbb{A}}^{1}$-fiber of the cylinder is either
contained in $W$ or does not intersect $W$. A variety $Y$_i_ s _transversally
covered_ by cylinders $U_{i}$, $i=1,\ldots,s$ if $Y=\bigcup_{i}U_{i}$ and no
proper subset $W\subseteq Y$ is invariant with respect to all elements of the
covering $\\{U_{i}\\}$.

Theorem 2.5 in [73] states that if for some very ample divisor $H$ on a normal
projective variety $Y$ there is a transversal cover by $H$-polar cylinders,
then the corresponding affine cone $X$ over $Y$ is flexible. This result
allows to prove that any affine cone over a del Pezzo surface of degree $\geq
5$ is flexible. In [73] flexibility is also proved for some affine cones over
del Pezzo surfaces of degree $4$, including the pluri-anticanonical cone. In
[72], the result on flexibility is obtained for all affine cones over del
Pezzo surfaces of degree $4$.

On the other hand, it is known that pluri-canonical cones over del Pezzo
surfaces of degrees $1$ and $2$ are rigid, that is, they do not admit
nontrivial ${\mathbb{G}}_{a}$-actions, see [23, Corollary 1.8] and [55,
Theorem 1.1]. The question of the rigidity of a pluri-canonical cone over del
Pezzo surfaces of degree $3$ has remained open for more than 15 years. The
proof of this fact is obtained in [23]. It is all the more surprising that any
very ample divisor on a del Pezzo surface of degree $3$ that is not
proportional to the anticanonical divisor defines an affine cone $X$ on which
the group ${\rm SAut}(X)$ acts with an open orbit [74]. The presence of
nontrivial ${\mathbb{G}}_{a}$-actions on affine cones corresponding to the
anticanonical divisor on del Pezzo surfaces with singularities is investigated
in [22].

In [70], the following result is obtained.

######  Proposition 8 ([70, Theorem 1.4]).

Suppose that a normal projective variety $Y$ is covered by flexible affine
charts $U_{i}$, ${i\in I}$ and we fix a very ample divisor $H$ on $Y$. If each
subset $Y\setminus U_{i}$ is the support of an effective
${\mathbb{Q}}$-divisor $D_{i}$ that is linearly equivalent to the divisor $H,$
then the affine cone $\text{AffCone}_{H}Y$ over $Y$ is a flexible variety.

By the _Segre-Veronese variety_
$v_{s_{1}}({\mathbb{P}}^{d_{1}})\times\ldots\times
v_{s_{k}}({\mathbb{P}}^{d_{k}})$ we mean the image of embedding of the direct
product ${\mathbb{P}}^{d_{1}}\times\ldots\times{\mathbb{P}}^{d_{k}}$ of
projective spaces, where each of the factors is embedded into a larger
projective space by the Veronese map with the parameters $s_{1},\ldots,s_{k}$,
respectively, and the product is embedded by the Segre map. Further, for each
projective subvariety $X\subseteq{\mathbb{P}}^{n}$ we call the _variety of
secants_ (respectively, the _variety of tangents_) of $X$ the Zariski closure
of the union of secant (respectively, tangent) lines of the subvariety $X$ in
${\mathbb{P}}^{n}$. In [70, Theorem 2.20] it is shown that the affine cone
over the variety of secants to a Segre-Veronese variety is flexible. Moreover,
if $s_{1}=\ldots=s_{k}=1$, then the affine cone over the variety of tangents
of a Segre-Veronese variety turns out to be flexible.

Theorem 4.5 in [70] guarantees the flexibility of affine cones over some
three-dimensional Fano varieties. In [80], the flexibility of affine cones
over four-dimensional Fano-Mukai varieties of genus $10$ is proved.

###  3.6. Universal torsors

The concept of a universal torsor originated in the works of Colliot-Thélène
and Sansuc [25, 26] on arithmetic algebraic geometry and was used to study
rational points on algebraic varieties. In particular, universal torsors
demonstrated their effectiveness for obtaining positive results on the Manin
Conjecture that provides an asymptotic formula for the number of rational
points of bounded height.

Let us recall briefly the definition. Let $X$ be a normal algebraic variety
with a free finitely generated divisor class group ${\rm Cl}(X)$. Denote by
${\rm WDiv}(X)$ the group of Weil divisors on $X$ and fix a subgroup
$K\subseteq{\rm Div}(X)$ that projects onto ${\rm Cl}(X)$ isomorphically.
Consider the Cox sheaf $\mathcal{R}$ of the variety $X$ associated with the
subgroup $K$: to each open subset $U\subseteq X$ this sheaf put in
correspondence the algebra of sections

$\bigoplus_{D\in K}H^{0}(U,D).$

The algebra of global sections of this sheaf is called the _Cox ring_ $R(X)$
of the variety $X$. It is easy to see that the objects defined here up to
isomorphism do not depend on the choice of the subgroup $K$.

The relative spectrum of the sheaf $\mathcal{R}$ determines the morphism

$q\colon\widehat{X}:={\rm Spec\,}_{X}\mathcal{R}\to X.$

In the case when the variety $X$ is smooth, the map $q$ is a principal locally
trivial bundle whose fiber is a torus $H$ with the character group ${\rm
Cl}(X)$. This mapping has certain universal properties and is called the
_universal torsor_ over the variety $X$. It is known that $\widehat{X}$ is a
smooth quasi-affine variety.

If the Cox ring $R(X)$ is finitely generated, we can consider its spectrum
${\overline{X}:={\rm Spec\,}R(X)}$. It is called the _total coordinate space_
of the variety $X$. This is an affine factorial variety, and the variety
$\widehat{X}$ is embedded into $\overline{X}$ as an open subset whose
complement does not contain divisors. For more information on this
construction and its generalizations, see [27] and [5, Chapter I].

Following the work [11], we say that a variety $X$ is _$A$ -covered_ if there
is a cover of $X$ by open subsets $U_{i}$ isomorphic to the affine space
${\mathbb{A}}^{n}$. It is clear that an $A$-covered variety is a smooth
rational variety with a free finitely generated divisor class group. In [11]
examples of wide classes of $A$-covered varieties are given. In particular,
these are all smooth complete toric varieties, all flag varieties, and, more
generally, all smooth complete spherical varieties. Using the fact that the
property to be $A$-covered is preserved under blowing up a point, it can be
shown that all smooth complete rational surfaces are $A$-covered. In [11,
Appendix] it is proved that all smooth complete rational varieties with an
action of a torus of complexity one are $A$-covered.

######  Theorem 16 ([11, Theorem 3]).

Let $X$ be an $A$-covered variety of dimension $\geq 2$ and
$q\colon\widehat{X}\to X$ be a universal torsor over $X$. Then the group ${\rm
SAut}(\widehat{X})$ acts on the variety $\widehat{X}$ infinitely transitively.

The proof of this result is based on the fact that every open chart on $X$
isomorphic to ${\mathbb{A}}^{n}$ carries $n$ transversal structures of
cylinders, each of which defines a ${\mathbb{G}}_{a}$-action on the universal
torsor.

Theorem 16 allows to obtain many interesting examples of smooth flexible
quasi-affine varieties. For instance, in this way one can construct a flexible
quasi-affine variety with a non finitely generated algebra of regular
functions [11, Section 5].

Note that Theorem 16 implies only that the group ${\rm SAut}(\overline{X})$
has a big open orbit on the total coordinate space $\overline{X}$ [11, Theorem
4]. The question of the flexibility of the variety $\overline{X}$ remains
open. Partial results are known in this direction: in [70, Theorem 5.4] it is
proved that the total coordinate space over a smooth del Pezzo surface is
flexible, and [70, Theorem 5.9] guarantees the flexibility property for total
coordinate spaces over complete smooth rational $T$-varieties of complexity
one.

###  3.7. Gizatullin surfaces

This is what normal affine surfaces are called, which admit a completion by a
chain of smooth rational curves. Gizatullin’s Theorem [43, Theorems 2 and 3]
(see also [31]) states that a normal affine surface $X$ other than

${\mathbb{A}}^{1}\times\big{(}{\mathbb{A}}^{1}\setminus\\{0\\}\big{)}$

admits such a completion if and only if the group ${\rm SAut}(X)$ acts on $X$
with an open orbit; in this case the complement to the open orbit is finite.

Gizatullin [43] conjectured that if the ground field ${\mathbb{K}}$ has
characteristic zero then the open orbit coincides with the set of smooth
points, that is, $X$ is flexible. At that time, it was known that this is not
the case in positive characteristic, since here the full automorphism group of
a surface $X$ can have smooth fixed points [44]. As we have seen above,
Gizatullin’s conjecture is true for Gizatullin surfaces defined in
${\mathbb{A}}^{3}$ by equations of the form $xy-f(z)=0$; in this case we
obtain a suspension over the line ${\mathbb{A}}^{1}$. A wide class of surfaces
for which Gizatullin’s conjecture holds form the so-called Danilov-Gizatullin
surfaces [42].

However, in [58] Kovalenko constructed a counterexample to Gizatullin’s
conjecture over the field of complex numbers: in [58, Theorem 3.11] classes of
smooth Gizatullin surfaces $X$ are described for which the action of the group
${\rm Aut}(X)$ on $X$ is not transitive.

###  3.8. Calogero-Moser spaces

Recall that the _Calogero-Moser space_ is the space of the categorical
quotient

$C_{n}=\big{\\{}(X,Y)\in{\rm Mat}_{n}({\mathbb{C}})\times{\rm
Mat}_{n}({\mathbb{C}})\,:\,{\rm rk}([X,Y]+{\rm Id})=1\big{\\}}/\\!/{\rm
GL}_{n}({\mathbb{C}}),$

where the group ${\rm GL}_{n}({\mathbb{C}})$ acts on pairs of matrices by
conjugation:

$g(X,Y)=(gXg^{-1},gYg^{-1}).$

Such spaces play an important role in representation theory and other branches
of mathematics. It is known that $C_{n}$ is a smooth irreducible affine
variety of dimension $2n$. It carries a hyperkähler structure and is an
example of a Nakajima quiver variety. This variety arises as a partial
completion of the integrable Calogero-Moser system. These and other facts with
links to the original works can be found in [15, Section 1].

Let us consider on the space of pairs of matrices the mappings of the form

$(X,Y)\mapsto(X+p(Y),Y)\quad\text{and}\quad(X,Y)\mapsto(X,Y+q(X)),$

where $p(x)$ and $q(x)$ are arbitrary polynomials in one variable. It is easy
to see that these mappings induce automorphisms of the space $C_{n}$. Let us
denote by $G$ the group of automorphisms of the space $C_{n}$ generated by
these automorphisms. In [15, Theorem 1] it is proved that the action of $G$ on
$C_{n}$ is $2$-transitive. In the same paper, the authors consider the
diagonal action of the group $G$ on the direct product
$C_{n_{1}}\times\ldots\times C_{n_{k}}$ for any pairwise distinct positive
integers $n_{1},\ldots,n_{k}$. We call such an action _collectively infinitely
transitive_ if for any positive integers $m_{1},\ldots,m_{k}$ and any sets of
pairwise distinct $m_{1}$ points on $C_{n_{1}}$, pairwise distinct $m_{2}$
points on $C_{n_{2}}$ and so on, there is an element of the group $G$ that
translates these sets to any other predefined sets of pairwise distinct points
of the same cardinalities.

In [15], the authors conjectured that the action of the group $G$ on the space
$C_{n}$ is infinitely transitive, and the diagonal action described above has
the property of collective infinite transitivity. These conjectures are proved
in [62, Theorem 3]. Independently, in [2] the author checks the flexibility of
the variety $C_{n}$, which implies infinite transitivity of the group ${\rm
SAut}(C_{n})$ containing $G$ as a proper subgroup.

##  4\. Infinite transitivity and finite generation

In this section we discuss a recently discovered nontrivial effect. It would
seem that the infinite transitivity of the action of the group of special
automorphisms on the set of smooth points of a flexible affine variety is
caused by the fact that we include saturated sets of
${\mathbb{G}}_{a}$-subgroups, in particular, all replicas of
${\mathbb{G}}_{a}$-subgroups, in the acting group. In this context, the
following conjecture, formulated in [9], may look unexpected.

######  Conjecture 3.

Let $X$ be an affine variety of dimension $\geq 2,$ where the set of flexible
points forms a nonempty open subset $O$. Then there is such a finite
collection $\\{H_{1},\ldots,H_{k}\\}$ of ${\mathbb{G}}_{a}$-subgroups in the
automorphism group ${\rm Aut}(X)$ that the subgroup $G=\langle
H_{1},\ldots,H_{k}\rangle$ generated by these subgroups as an abstract group
acts on $O$ infinitely transitively.

###  4.1. The case of an affine toric variety

In [9] and subsequent publications, a number of confirmations of Conjecture 3
were obtained. The most general result in this direction is the following
theorem.

######  Theorem 17 ([9, Theorem 5.20]).

Let $X$ be a non-degenerate affine toric variety of dimension $\geq 2$. Assume
that $X$ is smooth in codimension two. Then there is a finite collection
$H_{1},\ldots,H_{k}$ of root subgroups on $X$ such that the group $G=\langle
H_{1},\ldots,H_{k}\rangle$ acts on the smooth locus $X^{{\rm reg}}$ infinitely
transitively.

Below we describe a scheme of the proof of this and similar results. The proof
is divided into four stages. At the first stage, a description of certain
countable sets of ${\mathbb{G}}_{a}$-subgroups of the group ${\rm Aut}(X)$ is
obtained for which the group $G$ generated by them acts on $X$ with an open
orbit and the action of $G$ on the open orbit is infinitely transitive, see
[9, Theorem 2.2, Corollary 2.8].

At the second stage we take the closure $\overline{G}$ of the subgroup $G$ in
the automorphism group ${\rm Aut}(X)$ considered as an ind-group. Let us
discuss this stage in more detail. Following [59, Proposition 2.1], we fix an
embedding $X\hookrightarrow{\mathbb{A}}^{n}$ and consider the degree function
on the algebra ${\mathbb{K}}[X]$ induced by the embedding. For each
automorphism $\alpha\in{\rm Aut}(X)$, we define ${\rm deg}(\alpha)$ as a
maximum of degrees of homogeneous components of the images of elements of a
fixed generating system under the action of the automorphism $\alpha$.

Then we have ${\rm Aut}(X)=\varinjlim\Sigma_{s}$, where

  * •

for $s\geq 1$ the subset $\Sigma_{s}:=\\{\alpha\in{\rm Aut}(X)\,|\,{\rm
deg}(\alpha),{\rm deg}(\alpha^{-1})\leq s\\}$ is a closed subvariety of the
variety $\Sigma_{s+1}$;

  * •

for $r,s\geq 1$ the composition defines a morphism
$\Sigma_{r}\times\Sigma_{s}\to\Sigma_{rs}$;

  * •

the passage to the inverse element defines an automorphism of $\Sigma_{s}$.

The Zariski closure of a subset $F\subseteq{\rm Aut}(X)$ is defined as

$\overline{F}=\varinjlim\overline{(F\cap\Sigma_{s})},$

where the dash means the Zariski closure in the variety $\Sigma_{s}$. The
Zariski closure of a subset $F$ is a closed ind-subvariety in the ind-variety
${\rm Aut}(X)$. In these terms, an algebraic subgroup in ${\rm Aut}(X)$ is a
subgroup that is a closed subvariety in some $\Sigma_{s}$.

It is not difficult to verify that the closure $\overline{G}$ of a subgroup
$G\subseteq{\rm Aut}(X)$ is a closed ind-subgroup in ${\rm Aut}(X)$. Further,
if $\rho\colon\mathbb{A}^{1}\to{\rm Aut}(X)$ is such a morphism that
$\rho(t)\in G$ for $t\neq 0$, then we have $\rho(0)\in\overline{G}$.

######  Lemma 3 ([9, Lemma 3.2]).

Any $G$-invariant closed subset $Y\subseteq X$ is $\overline{G}$-invariant. If
the group $G$ acts on $X$ with an open orbit $O_{G},$ then $O_{G}$ coincides
with the open orbit $O_{\overline{G}}$ of the group $\overline{G}$.

These observations in the case of algebraically generated subgroups allow us
to prove the following.

######  Proposition 9 ([9, Proposition 3.4]).

Let $G$ be an algebraically generated subgroup of the group ${\rm Aut}(X)$.
Then

  * (a)

the orbits of the groups $G$ and $\overline{G}$ on $X$ coincide. In
particular, if $\overline{G}$ acts with an open orbit $O_{\overline{G}},$ then
the same is true for $G$ and $O_{G}=O_{\overline{G}};$

  * (b)

if $\overline{G}$ acts $m$-transitively on $O_{\overline{G}},$ then the same
is true for $G;$

  * (c)

if $\overline{G}$ acts infinitely transitively on $O_{\overline{G}},$ then the
same is true for $G$.

Thus, the results of the second stage allow us to prove the infinite
transitivity not for the action of the group $G$, but for the action of its
closure.

At the third stage we guarantee that $\overline{G}$ contains extra root
subgroups with respect to $G$. This requires a certain degeneration technique.
Namely, consider the locally nilpotent derivation $\partial$ of the algebra
${\mathbb{K}}[X]$ corresponding to a ${\mathbb{G}}_{a}$-subgroup of the group
$G$, and define its Newton polyhedron $N(\partial)$ as the convex hull of
weights of homogeneous components of the derivation $\partial$. Vertices of
this polyhedron correspond to root subgroups, and these subgroups are
contained in the group $\overline{G}$, see [9, Corollary 4.17].

In order to find a non-root subgroup with which to start this process, we
conjugate one root subgroup in $G$ by another root subgroup that does not
centralize the first one. The Newton polyhedron $N(\partial)$ for locally
nilpotent derivation corresponding to the result of such conjugation turns out
to be a segment whose ends correspond to new root subgroups in $\overline{G}$.
This segment can be calculated explicitly using the Baker-Campbell-Hausdorff
formula, see [9, Corollary 4.14]. On this path, we come to the next technical
result.

Let $X$ be an affine toric variety of dimension $\geq 2$. Below we use the
terminology and notation introduced in Section 3.2. Consider the Demazure
roots $e_{1},e_{2}$ of the cone of the variety $X$ corresponding to primitive
vectors $p_{1},p_{2}$ on the edges of the cone, and the corresponding root
subgroups $H_{e_{1}},H_{e_{2}}$ in the group ${\rm Aut}(X)$. Let $d=\langle
p_{1},e_{2}\rangle+1$ and assume that $e_{2}+de_{1}$ is a Demazure root
associated with the vector $p_{1}$, or, equivalently, that $\langle
p_{2},e_{1}\rangle>0$.

######  Lemma 4 ([9, Lemma 4.18]).

The root subgroup $H_{e_{1}+de_{2}}$ lies in the subgroup $\overline{\langle
H_{e_{1}},H_{e_{2}}\rangle}$.

Finally, at the fourth stage [9, Section 5.3], we find some finite collection
of root subgroups $H_{1},\ldots,H_{k}$ in the group ${\rm Aut}(X)$ and
generate a subgroup $G$ by them. Repeatedly applying Lemma 4, we find a
countable set of root subgroups in the group $\overline{G}$, which satisfies
the conditions of the first stage. This completes the proof of Theorem 17.

######  Remark 3.

The condition of smoothness in codimension two is essentially used at the
fourth stage of the proof of Theorem 17 when constructing a collection of
subgroups $H_{1},\ldots,H_{k}$. Unfortunately, we cannot find a way to
construct the required collection that would not use this condition.

Let us formulate a generalization of Lemma 4 as a conjecture.

######  Conjecture 4 ([9, Conjecture 5.23]).

Let $X$ be an irreducible affine variety. Consider the group $G=\langle
H_{1},\ldots,H_{k}\rangle$ generated by a finite collection of
${\mathbb{G}}_{a}$-subgroups $H_{i}={\rm exp}({\mathbb{K}}\partial_{i})$ in
${\rm Aut}(X),$ where $\partial_{i}$ is a locally nilpotent derivation of the
algebra ${\mathbb{K}}[X]$. Then a ${\mathbb{G}}_{a}$-subgroup $H={\rm
exp}({\mathbb{K}}\partial)$ lies in $\overline{G}$ if and only if the locally
nilpotent derivation $\partial$ belongs to the Lie algebra generated by
$\partial_{1},\ldots,\partial_{k}$.

Currently, this conjecture has not been proved.

###  4.2. The case of an affine space

The scheme of the proof of Theorem 17 described above allows us to obtain more
explicit results in the case $X={\mathbb{A}}^{n}$.

Let us start with the case $n\geq 3$. We fix a positive integer $l$ and
consider a ${\mathbb{G}}_{a}$-subgroup $F_{l}$ of transformations of the space
${\mathbb{A}}^{n}$, which transforms coordinates according to the formula
$(x_{1}+tx_{2}^{l},x_{2},\ldots,x_{n})$, $t\in{\mathbb{K}}$. Let $S_{n}$ be
the symmetric group acting on ${\mathbb{A}}^{n}$ by permutations of
coordinates. Denote by $G_{l}$ the subgroup in ${\rm Aut}({\mathbb{A}}^{n})$
generated by the subgroups $F_{l}$ and $S_{n}$.

######  Theorem 18 ([9, Theorem 5.2]).

The group $G_{2}$ acts on the set ${\mathbb{A}}^{n}\setminus\\{0\\}$
infinitely transitively.

Note that for $l\neq 2$ the action of the group $G_{l}$ on
${\mathbb{A}}^{n}\setminus\\{0\\}$ is not even $2$-transitive. Indeed, for
$l=1$, this action is linear and, therefore, preserves the property of a pair
of vectors to be collinear. For $l>2$, we fix a primitive root of unity
$\omega$ of degree $l-1$ and consider the set of pairs of points on
${\mathbb{A}}^{n}\setminus\\{0\\}$ of the form $(P,\omega P)$. It is easy to
check that transformations from the group $G_{l}$ send pairs of this form to
pairs of the same form.

Now we come to the case $n=2$. Here we define two one-parameter subgroups of
the automorphism group: the subgroup $F_{l}$ acts according to the formula
$(x_{1}+t_{1}x_{2}^{l},x_{2})$, and the subgroup $R_{s}$ acts as
$(x_{1},x_{2}+t_{2}x_{1}^{s})$, $t_{1},t_{2}\in{\mathbb{K}}$. Let $G_{l,s}$ be
the subgroup in ${\rm Aut}({\mathbb{A}}^{2})$ generated by the subgroups
$F_{l}$ and $R_{s}$.

With $ls\\!\neq\\!2$ the action of the group $G_{l,s}$ on ${\mathbb{A}}^{2}$
is not even $2$-transitive. Indeed, for $l=0$ or $s=0$, the action preserves
the differences of the corresponding coordinates. If $l=s=1$, then the action
is linear. Finally, if $ls>2$, we fix the primitive root of unity $\omega$ of
degree $ls-1$ and consider the set of pairs of points on
${\mathbb{A}}^{2}\setminus\\{0\\}$ of the form $((x_{1},x_{2}),(\omega
x_{1},\omega^{s}x_{2}))$. A direct check shows that transformations from the
group $G_{l,s}$ send such pairs to pairs of the same type.

Thus, it remains to consider the case $ls=2$, or, up to permutation, $l=2$ and
$s=1$. So far we have not been able to prove the infinite transitivity of the
action of the group $G_{2,1}$ on the set ${\mathbb{A}}^{2}\setminus\\{0\\}$
within the framework of the plan described above. After we formulated this
property in a conjectural form, its proof was obtained using a different
technique.

######  Theorem 19 ([65, Corollary 21]).

The action of the group $G_{2,1}$ on the set
${\mathbb{A}}^{2}\setminus\\{0\\}$ is infinitely transitive.

The main idea of [65] is to study relationships between two structures on the
group ${\rm Aut}({\mathbb{A}}^{2})$, the structure of the ind-group and the
structure of the amalgamated free product. Using a technique related to the
so-called Polydegree Conjecture, the authors show that the closure of the
subgroup $G_{2,1}$ contains all subgroups $F_{l}$ for $l\geq 2$, see [65,
Theorem 20]. This result implies Theorem 19. It would be interesting to find a
more elementary proof of this theorem. 111A new proof and generalizations of
this result are given in a recent preprint Chistopolskaya A. and Taroyan G.,
Infinite transitivity for automorphism groups of the affine plane,
arXiv:2202.02214

Let us return to the case of an affine space ${\mathbb{A}}^{n}$ of arbitrary
dimension. We denote by ${\rm Aff}_{n}$ the group of affine transformations of
the space ${\mathbb{A}}^{n}$. We also consider the group of tame automorphisms
${\rm Tame}_{n}$, see Section 1.3.

######  Definition 5.

An automorphism $h\in{\rm Aut}({\mathbb{A}}^{n})$ is called _co-tame_ if
$\langle{\rm Aff}_{n},h\rangle={\rm Tame}_{n}$, and _topologically co-tame_ if
$\overline{\langle{\rm Aff}_{n},h\rangle}={\rm Tame}_{n}$.

The Edo Theorem, generalizing the results of Bodnarchuk and Furter, claims
that for $n\geq 2$ any automorphism $h\in{\rm
Aut}({\mathbb{A}}^{n})\setminus{\rm Aff}_{n}$ is topologically co-tame. This
result leads to the following theorem.

######  Theorem 20 ([9, Theorem 5.12]).

For $n\geq 2$ and any automorphism $h\in{\rm
Aut}({\mathbb{A}}^{n})\setminus{\rm Aff}_{n}$ the group $G=\langle{\rm
Aff}_{n},h\rangle$ acts on ${\mathbb{A}}^{n}$ infinitely transitively.

Bodnarchuk, Levis and Edo showed that for $n\geq 3$ any triangular non-affine
automorphism of the space ${\mathbb{A}}^{n}$ is co-tame, whereas for $n=2$ a
triangular automorphism cannot be co-tame. A detailed presentation of these
results and links to original works can be found in [9, Section 5.2]. It
should be noted that back in 1995 in the work [16] Bodnarchuk related these
results to the infinite transitivity of the action of subgroups of the
automorphism group of an affine space.

It is natural to ask for what minimum value $k$ there is a group $G=\langle
H_{1},\ldots,H_{k}\rangle$ generated by ${\mathbb{G}}_{a}$-subgroups that acts
on the space ${\mathbb{A}}^{n}$ infinitely transitively. It turns out that the
value $k=3$ is suitable. For example, in the case $n=2$ one may consider the
subgroup generated by $G_{2,1}$ and an arbitrary one-dimensional subgroup of
parallel translations. Also for any $n\geq 2$ we construct the required triple
of subgroups $H_{1},H_{2},H_{3}$ in [9, Theorem 5.17]. This construction is
based on a result of Chistopolskaya [24], which states that for an arbitrary
nonzero nilpotent matrix $x$, there is such a nilpotent matrix $y$ that $x$
and $y$ generate the Lie algebra $\mathfrak{sl}_{n}$. Another triple of
subgroups having the same property is constructed independently in [1, Theorem
10].

The question of whether the value $k=2$ is achievable for any $n\geq 2$
remains open.

######  Remark 4.

In [33, Proposition 1.14] it is shown that for any flexible variety $X$ in the
group ${\rm SAut}(X)$ there are two ${\mathbb{G}}_{a}$-subgroups $H_{1},H_{2}$
such that the group $G$ generated by the subgroups $H_{1},H_{2}$ and all their
replicas acts on $X$ with an open orbit $O$, and the action of $G$ on $O$ is
infinitely transitive.

###  4.3. Groups generated by root subgroups and the Tits alternative

At the end of this section, we outline briefly the results of the work [12].
The starting point was Demailly’s question about whether it is possible to
characterize the infinite transitivity property of the action of the group
$G=\langle H_{1},\ldots,H_{k}\rangle$ on its open orbit in terms of the growth
of finitely generated subgroups of the group $G$.

In this context, it is natural to mention the classical result of Tits [85]:
any finitely generated subgroup in a linear algebraic group over an arbitrary
field contains either a solvable subgroup of finite index (the virtual
solvability property) or a non-commutative free subgroup. If the ground field
has characteristic zero, this alternative holds for any, not necessarily
finitely generated subgroup.

We say that a group $G$ satisfies the _Tits alternative_ if any its subgroup
is either virtually solvable or contains a non-commutative free subgroup. It
is known that the Tits alternative holds for the group ${\rm
Aut}({\mathbb{A}}^{2})$ and, more generally, for the group of birational
automorphisms of a compact complex Kähler surface, in particular, for the two-
dimensional Cremona group. The alternative also holds for the automorphism
group of a three-dimensional smooth affine quadric and for a number of other
automorphism groups; references to the original works can be found, for
example, in the Introduction to [12].

For groups generated by root subgroups, there is a weaker result.

######  Theorem 21 ([12, Theorem 1.1]).

Let $X$ be an affine toric variety and $G=\langle H_{1},\ldots,H_{k}\rangle$
be a subgroup of ${\rm Aut}(X)$ generated by a finite set of root subgroups.
Then either $G$ is a unipotent linear algebraic group, or $G$ contains a non-
commutative free subgroup.

The question whether the group $G$ from Theorem 21 satisfies the Tits
alternative is open. At the same time, Theorem 21 shows that the property of
infinite transitivity of an action of a group on its open orbit cannot be
characterized only in terms of maximal growth of subgroups: any non-unipotent
group $G=\langle H_{1},\ldots,H_{k}\rangle$ contains a non-commutative free
subgroup, that is, a subgroup of the (maximal possible) exponential growth.
Nevertheless, the following conjecture remains open.

######  Conjecture 5 ([12, Conjecture 1]).

Let $X$ be an affine variety of dimension $\geq 2$. Consider a subgroup
$G=\langle H_{1},\ldots,H_{k}\rangle$ in ${\rm Aut}(X)$ generated by a finite
collection of ${\mathbb{G}}_{a}$-subgroups. Suppose that $G$ acts
$2$-transitively on some of its orbits. Then $G$ contains a non-commutative
free subgroup.

In the proof of Theorem 21, we start with the case $k=2$. If the subgroups
$H_{1}$ and $H_{2}$ commute elementwise, the group $G$ is commutative and
unipotent. Therefore, we can assume that the Demazure roots $e_{1},e_{2}$
corresponding to the subgroups $H_{1},H_{2}$ are associated with primitive
vectors $p_{1},p_{2}$ on different rays of the cone of the affine toric
variety $X$. In this case, everything is determined by the values of the
pairings $\langle p_{1},e_{2}\rangle$ and $\langle p_{2},e_{1}\rangle$: if at
least one of these pairings is zero, the group $G$ turns out to be unipotent,
and in the case when both pairings are positive, we prove the existence of a
non-commutative free subgroup in the group $G$. For example, if both pairings
are $\geq 2$, the group $G$ turns out to be a free product of the subgroups
$H_{1},H_{2}$ and any two nonunit elements $h_{1}\in H_{1}$ and $h_{2}\in
H_{2}$ generate a free subgroup; see details in [12, Section 3].

In the case $k\geq 3$, we can assume without loss of generality that for any
pair $e_{i},e_{j}$ of Demazure roots corresponding to root subgroups of the
group $G$, at least one of the pairings, $\langle p_{i},e_{j}\rangle$ or
$\langle p_{j},e_{i}\rangle$, is zero. In this situation, we prove that the
group $G$ is unipotent [12, Proposition 4.8]. This result is based on the
study of the Lie algebra generated by a finite collection of homogeneous
locally nilpotent derivations.

The methods used here are close to the methods of [10], where a criterion of
the finite dimensionality of a Lie algebra generated by a finite collection of
homogeneous locally nilpotent derivations is obtained, and the structure of
such finite-dimensional Lie algebras is described. In particular, the results
of [10] generalize the well-known Demazure Theorem [28]: the Lie algebras
arising here have type A, that is, the semisimple parts of such Lie algebras
are direct sums of Lie algebras $\mathfrak{sl}$.

##  5\. Appendix A. Infinitely transitive actions in complex analysis

In this section we consider complex-analytic versions of flexibility and
infinite transitivity of the action of the automorphism group, and also
discuss connections of these concepts and the constructions studied above with
the Andersen-Lempert theory and the concept of Gromov sprays.

######  Definition 6 ([46, Section 1.1.B]).

Let us consider a complex manifold $X$.

  * (i)

A _dominating spray_ on $X$ is a holomorphic vector bundle $\rho\colon E\to X$
with a holomorphic map $s\colon E\to X$ such that its restriction $s$ to the
zero section $Z$ is the identical map, and for each point ${x\in Z\cong X}$
the tangent mapping $d_{x}s$ sends the fiber $E_{x}\\!:=\\!\rho^{-1}(x)$
(viewed as a subspace of the tangent space $T_{x}E$) to the tangent space
$T_{x}X$ surjectively.

  * (ii)

Let $h\colon X\to B$ be a surjective submersion of complex manifolds. We say
that $h$ admits a _fiber dominating spray_ if there exists a holomorphic
vector bundle $E$ over $X$ and a holomorphic map $s\colon E\to X$ such that
the restriction of $s$ on each fiber $h^{-1}(b),b\in B$ yields a dominating
spray on this fiber.

In these terms, the well-known Oka-Grauert-Gromov principle can be stated as
follows.

######  Theorem 22 ([46, Section 4.5]).

Consider a surjective submersion

${h\colon X\to B}$

of Stein manifolds. If $h$ admits a fiber dominating spray, then

  * (a)

any continuous section of the mapping $h$ is homotopic to a holomorphic
section;

  * (b)

if two holomorphic sections are homotopic in the class of continuous maps,
then they are also homotopic holomorphically.

The following result shows that this principle is applicable to smooth affine
algebraic $G$-bundles with flexible fibers.

######  Proposition 10 ([6, Proposition A.3]).

  * (a)

Every smooth flexible affine complex algebraic variety $X$ admits a dominating
spray.

  * (b)

Let $h\colon X\to B$ be a surjective submersion of smooth affine complex
algebraic varieties. Assume that there exists an algebraically generated
subgroup $G\subseteq{\rm Aut}(X),$ whose orbits coincide with the fibers of
$h$. Then the mapping ${X\to B}$ admits a fiber dominating spray.

###### Proof.

It is clear that assertion (a) follows from assertion (b). To prove (b), we
consider a sequence of algebraic subgroups

${\mathcal{H}}=(H_{1},\ldots,H_{s})$

in the group $G$ such that the tangent space to the orbit $Gx$ at each point
$x\in X$ is generated by tangent vectors to the orbits $H_{i}x$,
${i=1,\ldots,s}$. Let $\text{exp}\colon T_{1}(H_{i})\to H_{i}$ be the
exponential map. Consider the trivial vector bundle

$E=\prod_{i=1}^{s}T_{1}(H_{i})\times X$

over $X$ and the morphism

$s\colon E\to X,\quad\big{(}(h_{1},\ldots,h_{s}),x\big{)}\mapsto{\rm
exp}(h_{1})\ldots{\rm exp}(h_{s})x.$

It is not difficult to see that this is a fiber dominating spray. ∎

Now we come to the definition of holomorphic flexibility. Recall that a
holomorphic vector field on a complex manifold $X$ is called _completely
integrable_ if its phase flow determines a holomorphic action of the group
$({\mathbb{C}},+)$ on $H$.

######  Definition 7 ([6, Definition A.4]).

  * (i)

A Stein manifold $\\!X\\!$ is called _holomorphically flexible_ if completely
integrable holomorphic vector fields on $X$ generate the tangent space
$T_{x}X$ at every smooth point on $X$.

  * (ii)

Consider a holomorphic submersion $h\colon X\to B$ of Stein manifolds. We say
that $X$ is _holomorphically flexible over $B$_ if completely integrable
relative holomorphic vector fields on $X$ generate the relative tangent bundle
for $X\to B$ at each point $x\in X$. In this case, each fiber $h^{-1}(b),b\in
B$ is a holomorphically flexible Stein manifold.

For example, the vector field $\delta=z\frac{d}{dz}$ on
$X={\mathbb{C}}\setminus\\{0\\}$ is completely integrable. However, the
derivation $\delta$ is not locally nilpotent. The manifold
${\mathbb{C}}\setminus\\{0\\}$ is not flexible in the sense of the main part
of this work, but it is holomorphically flexible.

In the context of holomorphic actions, it also can be shown that if $X$ is a
connected Stein manifold that is holomorphically flexible over a Stein
manifold $B$, then the relative tangent bundle to $X$ over $B$ is generated by
a finite number of completely integrable relative holomorphic vector fields on
$X$ [6, Lemma A.6]. Repeating the reasoning given above, we show that if a
Stein manifold $X$ is holomorphically flexible over a Stein manifold $B$, then
the map $h\colon X\to B$ admits a fiber dominating spray [6, Corollary A.7].
In particular, the Oka-Grauert-Gromov principle applies to the map $h\colon
X\to B$.

It should be noted that the question of whether the group of holomorphic
automorphisms acts on a flexible connected Stein manifold $X$ of dimension
$\geq 2$ infinitely transitively remains open [6, Problem A.8]. In [6] it is
shown that such an action is transitive: it follows from the implicit function
theorem that all orbits of the group of holomorphic automorphisms are open in
$X$ in the standard Hausdorff topology, and so there is only one orbit.

At the same time, the infinite transitivity of an action can be proved under
stronger restriction. To formulate the corresponding result, we need to
introduce several concepts from the Andersen-Lempert theory; see, for example,
[51, 86].

######  Definition 8.

  * (i)

A complex manifold $X$ has the _density property_ if the Lie algebra generated
by all completely integrable holomorphic vector fields is dense in the Lie
algebra of all holomorphic vector fields on $X$ in the compact-open topology.

  * (ii)

An affine algebraic variety has the _algebraic density property_ if the Lie
algebra generated by all completely integrable algebraic vector fields on $X$
coincides with the Lie algebra of all algebraic vector fields on $X$.

######  Theorem 23 ([51, 86]).

If a connected Stein manifold $X$ of dimension ${\geq 2}$ has the density
property, then the group of holomorphic automorphisms of the manifold $X$ acts
on $X$ infinitely transitively. Moreover, for any discrete subset $Z\subseteq
X$ and any Stein space $Y$ of positive dimension that admits a proper
embedding into $X,$ there is another proper embedding $\varphi\colon
Y\hookrightarrow X$ such that $Z\subseteq\varphi(Y)$.

A similar result can be proved for holomorphic automorphisms and holomorphic
vector fields preserving a volume form on a Stein manifold, see [51, Corollary
2.1 and Remark 2.2] and [6, Theorem A.12].

The study of relationships between the flexibility of varieties and different
versions of the concept of Gromov sprays has been conducted actively in recent
years. Recall that a complex manifold $X$ is called _elliptic_ (respectively,
_subelliptic_) if it admits a dominating holomorphic spray (respectively, a
dominating family of holomorphic sprays; see [52, Definition 1.1] for a
precise definition).

We say that a smooth algebraic variety $X$ is _locally stably flexible_ if $X$
is a union $\bigcup_{i}X_{i}$ of a finite number of Zariski-open quasi-affine
subsets $X_{i}$, and for some positive integer $m$ the varieties
$X_{i}\times{\mathbb{C}}^{m}$ are flexible for all $i$. In [52] it is shown
that the blow-up of a locally stably flexible variety in a smooth algebraic
subvariety is a subelliptic manifold. This result is obtained in [52] as a
consequence of a more general result concerning so-called $k$-flexible
varieties.

Let us fix a positive integer $k$. A flexible quasi-affine variety $X$ is
called _$k$ -flexible_ if there exists such a morphism $\varphi\colon X\to Q$
to a normal affine variety $Q$ that for some Zariski-open subset $Q_{0}$ in
$Q$ the preimage $\varphi^{-1}(Q_{0})$ is isomorphic to the direct product
$Q_{0}\times{\mathbb{C}}^{k}$ over $Q_{0}$. Theorem 6.1 from [52] states that
if $X$ is a locally $k$-flexible variety for some $k\geq 2$ and $Z$ is a
closed subvariety in $X$ of codimension at least $k$, then the result of the
blow up of $X$ along $Z$ is a subelliptic manifold.

In particular, varieties studied in [52] are Oka manifolds. Recall that a
complex manifold $X$ is called an _Oka manifold_ if, for any positive integer
$n$, any holomorphic map from a neighborhood of a compact convex subset $K$ in
the space ${\mathbb{C}}^{n}$ to $X$ is a uniform limit on $K$ of holomorphic
maps from ${\mathbb{C}}^{n}$ to $X$. For more information on Oka manifolds and
the connection of this theory with the concept of flexibility one can see, for
example, the work [61] and the recent survey [36]. The known facts on
flexibility of toric varieties allow to apply the Oka theory and to obtain
interesting results on extension, approximation and interpolation of maps and
other geometric consequences both for toric varieties and for classes close to
toric varieties, see, for example, [64].

An exceptionally interesting object for research is the action of the group of
holomorphic automorphisms of a Stein manifold on countable sets of points of
such a manifold, see [83, 3, 88, 89] and references therein. In this
situation, the orbits of the group of holomorphic automorphisms on the set of
infinite discrete sequences form continuum families. Among such sets, there
are those on which any permutation is obtained by restricting a holomorphic
automorphism of the ambient space. Such sets are called _tame_. In the case
when the ambient space is the space ${\mathbb{C}}^{n}$, the property of a set
to be tame is equivalent to the fact that by a suitable holomorphic
automorphism of ${\mathbb{C}}^{n}$ this set can be sent to a standard
position, for example, to the set of points with positive integer coordinates
on the first coordinate axis. Every tame set in ${\mathbb{C}}^{n}$ is
_disposable_ in the sense that for such a set $E$ there exists a biholomorphic
map ${\mathbb{C}}^{n}\to{\mathbb{C}}^{n}\setminus E$. On the other hand, there
are examples of _rigid_ sets, that is, sets that are preserved only by the
identical holomorphic automorphism.

##  6\. Appendix B. Infinitely transitive actions of finitely generated groups

Let us start with elementary properties of multiply transitive actions of
abstract groups on arbitrary sets.

######  Lemma 5.

A commutative group $G$ cannot act $2$-transitively on a set $X$ with at least
$3$ elements.

###### Proof.

Let $G$ act $2$-transitively on $X$. Let us choose three distinct elements
$x_{1},x_{2},x_{3}\in X$. An element of $G$ sending the pair $(x_{1},x_{2})$
to the pair $(x_{1},x_{3})$ is contained in the stabilizer of the point
$x_{1}$. From commutativity of the group it follows that the stabilizer of the
point $x_{1}$ acts identically on the orbit of the point $x_{1}$, and hence on
the entire set $X$, a contradiction. ∎

In particular, no cyclic group can act $2$-transitively. Now, following [69],
we construct an example of an infinitely transitive action of a $2$-generated
group on the set of integers ${\mathbb{Z}}$.

We say that a permutation $g$ of an infinite set $X$ is an _infinite cycle_ if
the cyclic group $\langle g\rangle$ acts transitively on $X$. Consider two
transformations on the set ${\mathbb{Z}}$: let
$h\colon{\mathbb{Z}}\to{\mathbb{Z}}$ send $i$ to $i+1$, and $f$ be an infinite
cycle on the set of positive integers ${\mathbb{N}}$ that leaves all other
integers fixed.

######  Proposition 11 ([69, Lemma 1]).

The group $G:=\langle f,h\rangle$ acts on the set ${\mathbb{Z}}$ infinitely
transitively.

Let us start with an auxiliary statement. For each non-negative integer $n$,
we consider the subgroup $G_{n}:=\langle
f,h^{-1}fh,\ldots,h^{-n}fh^{n}\rangle$ in $G$. It is easy to see that this
subgroup leaves all integers not exceeding $-n$ fixed, and, therefore, it acts
on the set $X_{n}$ consisting of all integers greater than $-n$.

######  Lemma 6.

The group $G_{n}$ acts on the set $X_{n}$ $(n+1)$-transitively.

###### Proof.

We use the induction on $n$. For $n=0$, the subgroup $G_{0}$ coincides with
$\langle f\rangle$. This group acts transitively on $X_{0}={\mathbb{N}}$ by
construction.

Assume that the assertion is proved for all $i$ less than some $n$. Note that

$(h^{-n}fh^{n})(-n+1)=f(1)-n\neq-n+1.$

Taking into account the inductive hypothesis, we conclude that the action of
$G_{n}$ on $X_{n}$ is transitive. The stabilizer of the point $-n+1$ contains
the subgroup $G_{n-1}$, which acts $n$-transitively on
${X_{n-1}=X_{n}\setminus\\{-n+1\\}}$. This proves the lemma. ∎

###### Proof of Proposition 11.

Let us fix a positive integer $m$ and consider two sets of pairwise distinct
integers of $m$ elements each. We choose $n\geq m-1$ so that any element from
these sets is greater than $-n$. By Lemma 6, the first set can be sent to the
second one by an element of the subgroup $G_{n}$. ∎

It is shown in [69, Theorem 1] that by a suitable choice of the infinite cycle
$h$ one can ensure that $G$ is isomorphic to the free group with two
generators $f$ and $h$. It is well known that a free group with two generators
contains a normal subgroup, which is a free group with $n$ generators for any
positive integer $n$, as well as a normal subgroup, which is a free group with
a countable number of generators, see, for example, [69, Section 1]. Thus,
Theorem 24 below allows us to construct an effective infinitely transitive
action of a free group with any finite or countable number of generators on
the set ${\mathbb{Z}}$.

Let us prove some more general statements.

######  Lemma 7.

Suppose that a group $G$ acts on a set $X$ with at least $3$ elements
effectively and $2$-transitively. Let $N$ be a nontrivial normal subgroup in
$G$. Then the action of $N$ on $X$ is transitive. In particular, the center
$Z(G)$ of the group $G$ is trivial.

###### Proof.

Since the group $G$ permutes $N$-orbits and at least one such orbit contains
at least two points, choosing a pair of points from $X$ lying in the same
$N$-orbit and a pair of points lying in different $N$-orbits, we obtain a
contradiction with $2$-transitivity.

If the center of $G$ is nontrivial, then it acts on $X$ transitive. This shows
that the stabilizers in $G$ of points in $X$ are conjugated by elements of the
center and so they coincide. This again contradicts to $2$-transitivity. ∎

######  Remark 5.

Consider the group $G$ of all affine transformations of a vector space $V$ of
dimension $\geq 2$ over the field ${\mathbb{Z}}/2{\mathbb{Z}}$. Then $G$ acts
on $V$ $3$-transitively, and its normal subgroup $N$ of parallel translations
acts on $V$ transitively, but not $2$-transitively. This shows that when
passing to a normal subgroup, the transitivity degree can decrease by more
than $1$.

The following result can be found, for example, in [30, Corollary 7.2A]. For
convenience of the reader, we give a proof of this result taken from [12,
Lemma 5.6].

######  Theorem 24.

Let a group $G$ act on a set $X$ effectively and infinitely transitively, and
$N$ be a nontrivial normal subgroup in $G$. Then the action of $N$ on $X$ is
also infinitely transitive.

###### Proof.

Consider a set $\alpha=\\{x_{1},\ldots,x_{m}\\}$ of pairwise distinct elements
in $X$. Let $G_{\alpha}$ be the intersection in $G$ of the stabilizers of
points from $\alpha$ and set $N_{\alpha}=N\cap G_{\alpha}$. Then $N_{\alpha}$
is a normal subgroup in $G_{\alpha}$. It is sufficient to prove that for any
$m$ and any $\alpha$, the action of $N_{\alpha}$ on
$X\setminus\\{x_{1},\ldots,x_{m}\\}$ is transitive. For $m=0$ we obtain
$G_{\alpha}=G$, $N_{\alpha}=N$, and the transitivity follows from Lemma 7.

Suppose that the required property does not hold for some $m$ and some
$\alpha$, and the value $m\geq 1$ is minimal possible. We know that the group
$G_{\alpha}$ acts on ${X\setminus\\{x_{1},\ldots,x_{m}\\}}$ infinitely
transitively. It follows from Lemma 7 that if the action of $N_{\alpha}$ on
${X\setminus\\{x_{1},\ldots,x_{m}\\}}$ is not transitive, then
$N_{\alpha}=\\{e\\}$.

Let $\beta=\\{x_{1},\ldots,x_{m-1}\\}$. From the minimality of $m$ it follows
that the group $N_{\beta}$ acts transitively on
$X\setminus\\{x_{1},\ldots,x_{m-1}\\}$. The condition $N_{\alpha}=\\{e\\}$
implies that this action is free, and we can identify the sets
$X\setminus\\{x_{1},\ldots,x_{m}\\}$ and $N_{\beta}\setminus\\{e\\}$: an
element $y$ is identified with an element $h\in N_{\beta}\setminus\\{e\\}$
such that $y=hx_{m}$. Under this identification, the action of $G_{\alpha}$ on
$N_{\beta}\setminus\\{e\\}$ is the action by conjugation:

$ghx_{m}=ghg^{-1}gx_{m}=ghg^{-1}x_{m}.$

But a conjugation sends a pair of the form $(h,h^{-1})$ to a pair of the same
form. This contradicts to $2$-transitivity of the action of $G_{\alpha}$ on
$N_{\beta}\setminus\\{e\\}$, except in the case when all elements of the group
$N_{\beta}$ have order not higher than $2$. In the latter case, the group
$N_{\beta}$ can be identified with the additive group $(V,+)$ of some vector
space $V$ over the field ${\mathbb{Z}}/2{\mathbb{Z}}$. Then the group
$G_{\alpha}$ acts on $V$ by linear transformations, such an action preserves
linear dependencies and, therefore, is not $3$-transitive. The resulting
contradiction completes the proof of the theorem. ∎

In conclusion, we list several results on infinitely transitive actions of
abstract groups.

It is clear that the group of all permutations with a finite support of an
infinite set $X$ acts on $X$ infinitely transitively. This group consists of
elements of finite order and it is not finitely generated.

It is easy to see that a group $G$ admits an effective infinitely transitive
action on a countable set if and only if $G$ can be embedded as a dense
subgroup into the infinite symmetric group $S({\mathbb{N}})$ equipped with the
topology of pointwise convergence. In [29] it is shown that most (in the
topological sense) infinitely transitive finitely generated subgroups in the
group $S({\mathbb{N}})$ are free.

In [56] it is proved that the fundamental group of a closed oriented surface
of genus $g>1$ admits an effective infinitely transitive action. A
characterization of fundamental groups of three-dimensional compact manifolds
admitting such an action is obtained in [49].

The existence of effective infinitely transitive actions of free products
$G_{1}*G_{2}$ was studied in [45, 47, 48, 71]. It turned out that if one of
the groups $G_{i}$ has order at least $3$, then the free product admits an
effective infinitely transitive action, whereas the group
$({\mathbb{Z}}/2{\mathbb{Z}})*({\mathbb{Z}}/2{\mathbb{Z}})$ does not allow
such an action. The existence of effective infinitely transitive actions for
groups acting on trees was studied in [34, 35].

It is shown in [49] that any group given by at least three generators and one
defining relation admits an effective infinitely transitive action.

In [12, Question 2], it is asked whether a finitely generated group $G$ acting
infinitely transitively on some infinite set $X$ can have intermediate growth.
Currently, this issue remains open.

An effective infinitely transitive action of the group of outer automorphisms
$\text{Out}{F_{n}}$ of the free group $F_{n}$ with $n\geq 4$ is constructed in
[41]. For $n=3$, the same result is obtained in [49]. It is also shown there
that for $n=2$ this group does not allow the required action.

In [20, Section IV.4], an effective infinitely transitive action of a non-
elementary hyperbolic group without nontrivial normal finite subgroups is
constructed. In [49] it is shown that every countable acylindrically
hyperbolic group admits an infinitely transitive action with a finite kernel.
This result has many important consequences. Also the work [49] contains a
detailed survey of recent results on multiply transitive actions of countable
groups.

## References

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