# Regular bi-interpretability and finite axiomatizability of Chevalley groups

Elena Bunina and Pavel Gvozdevsky  Department of Mathematics, Bar-Ilan
University, 5290002 Ramat Gan, ISRAEL<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper we consider Chevalley groups over commutative rings with $1$,
constructed by irreducible root systems of rank $>1$. We always suppose that
for the systems $A_{2},B_{\ell},C_{\ell},F_{4},G_{2}$ our rings contain $1/2$
and for the system $G_{2}$ also $1/3$. Under these assumptions we prove that
the central quotients of Chevalley groups are regularly bi-interpretable with
the corresponding rings, the class of all central quotients of Chevalley
groups of a given type is elementarily definable and even finitely
axiomatizable. The same holds for adjoint Chevalley groups and for other
Chevalley groups with some special condition of bounded generation. We also
give an example of Chevalley group with infinite center, which is not bi-
interpretable with the corresponding ring and is elementarily equivalent to a
group that is not a Chevalley group itself.

###### Keywords:

Chevalley groups, commutative rings, first order logic, bi-interpretability,
finite axiomatizability, elementary definability

###### 2020 Mathematics Subject Classification:

20G35, 20A15, 03C60

The paper is written as part of the second author’s post-doctoral fellowship
at Bar-Ilan University, Department of Mathematics; and is supported by ISF
grant 1994/20.

##  1\. Historical overview

###  1.1. Elementary equivalence of groups, Maltsev-type theorems

Elementary (first-order) classification of algebraic structures goes back to
the works of Tarski and Maltsev. In general, the task is to characterize, in
somewhat algebraic terms, all algebraic structures elementarily equivalent to
a given one. Recall, that two algebraic structures $\mathfrak{A}$ and
$\mathfrak{B}$ in a language $\mathcal{L}$ are _elementarily equivalent_
($\mathfrak{A}\equiv\mathfrak{B}$) if they satisfy precisely the same first-
order sentences in $\mathcal{L}$.

Classification of objects up to an equivalence relation is one the principal
problems in mathematics. The equivalence relation on a structure is determined
by the type of property one would like to be preserved, namely in model theory
one classifies modulo elementary theory; in algebra, modulo isomorphism; in
geometric group theory, modulo quasi-isometries, etc. To determine which
groups (or other structures) are elementarily equivalent to a given one is
usually a very hard problem. The first powerful result for groups is the
classification of groups elementarily equivalent to an Abelian group was
obtained by W. Szmielew [61] (two Abelian groups are elementarily equivalent
if and only if they have the same special ‘‘characteristic numbers’’ like
$\,\mathrm{Exp}\,A$, $\dim p^{n}A[p]$, etc.). Nowadays the detailed
descriptions of the groups elementary equivalent to a given one are obtained
for wide list of groups, see for example the survey [21].

When we restrict ourselves to the class of finitely generated groups
elementarily equivalent to the given one, the problem becomes especially
challenging.

An outstanding result was the answer for the old problem raised by A. Tarski
around 1945: for non-Abelian finitely generated free groups the elementary
theory doesn’t distinguish these groups (see for details the series of works
of Kharlampovich–Myasnikov and Sela, e. g. [35], [55]). The similar situation
takes place for the torsion free hyperbolic groups (see Sela, [54]).

On the other hand, a theorem of F. Oger [49] states that two finitely
generated nilpotent groups $G$ and $H$ are elementarily equivalent if and only
if $G\times\mathbb{Z}\cong H\times\mathbb{Z}$. In paper [46], A. Myasnikov
gave a description of groups elementarily equivalent to a given finitely
generated nilpotent $K$-group, where $K$ is a field of characteristic zero.
Recently, A. Myasnikov and M. Sohrabi classified groups elementarily
equivalent to a free nilpotent $R$-group of finite rank, where $R$ is a domain
of characteristic zero, see [42]. M. Casals-Ruiz and I. Kazachkov in [21]
proved that all finitely generated groups elementarily equivalent to a
solvable Baumslag–Solitar group $BS(m,1)$ are isomorphic to it.

Certainly, any new such complete classification result is a rather rare
mathematical phenomenon.

Another way of studying elementary equivalence of algebraic models is to
establish a connection between derivative models (linear groups over rings and
fields, automorphism groups and endomorphism rigs of different structures,
etc.) and initial models (and ‘‘parameters’’) used for the construction. In
his classical paper [40] A.I. Maltsev studied elementary equivalence of matrix
groups $\mathcal{G}_{n}(F)$ where
$\mathcal{G}_{n}=\,\mathrm{GL}\,_{n},\,\mathrm{SL}\,_{n},\,\mathrm{PGL}\,_{n},\,\mathrm{PSL}\,_{n}$,
$n\geqslant 3$, and $F$ is a field. Namely, he showed that
$\mathcal{G}_{n}(F)\equiv\mathcal{G}_{m}(L)$ if and only if $n=m$ and $F\equiv
L$. His proof was based on two principal results. The first one states that
for any integer $k\geqslant 3$ and $\mathcal{G}_{n}$ as above there is a group
sentence $\Phi_{k,\mathcal{G}}$ such that for any $n$, and a field $F$,
$\Phi_{k,\mathcal{G}}$ holds in $\mathcal{G}_{n}(F)$ if and only if $k=n$. The
second one is that $F$ and $\mathcal{G}_{n}(F)$ are mutually interpretable in
each other. More precisely, $\mathcal{G}_{n}(F)$ is absolutely interpretable
in $F$ (i. e., no use of parameters), while $F$ is interpretable in
$\mathcal{G}_{n}(F)$ uniformly with respect to some definable subset of tuples
of parameters (so-called _regular_ interpretability). This implies that the
theories $Th(F)$ and $Th(\mathcal{G}_{n}(F))$ are reducible to each other in
polynomial time, hence $Th(\mathcal{G}_{n}(F))$ is decidable if and only if
$Th(F)$ is decidable. Later Beidar and Mikhalev introduced another general
approach to elementary equivalence of classical matrix groups [6]. Their proof
was based on Keisler–Shelah theorem (_two structures are elementarily
equivalent if and only if their ultrapowers over non-principal ultraflters are
isomorphic_ , see [33], [56]) and the description of the abstract isomorphisms
of the groups of the type $\mathcal{G}_{n}(F)$. E. Bunina extended their
results to unitary linear and Chevalley groups (see [17], [18], [10], [15]).
Note that in all the results above the first-order theories include only the
standard constants from the languages of groups and rings. The model theory of
the group $UT_{n}(R)$, where $n\geqslant 3$, and $R$ is an arbitrary unitary
associative ring, was studied in details by O. Belegradek [7]. He used heavily
that the ring $R$ is interpretable (with parameters) in $UT_{n}(R)$. A.
Myasnikov and M. Sohrabi studied model theory of groups
$\,\mathrm{SL}\,_{n}(\mathcal{O})$, $\,\mathrm{GL}\,_{n}(\mathcal{O})$, and
$T_{n}(\mathcal{O})$ over fields and rings of algebraic integers in [43] and
[44]. Their method exploits the mutual interpretability (and also bi-
interpretability) of the group and the ring. In a similar manner N. Avni, A.
Lubotsky, and C. Meiri in [4] studied the first order rigidity of non-uniform
higher rank arithmetic groups (see also [5]).

Recently, D. Segal and K. Tent (see [53]) showed that for Chevalley groups
$G_{\mathcal{P}}(\Phi,R)$ of rank $>1$ over an integral domain $R$ if
$G_{\mathcal{P}}(\Phi,R)$ has finite elementary width or is adjoint, then
$G_{\mathcal{P}}(\Phi,R)$ and $R$ are bi-interpretable. In [13] E. Bunina
proved that over local rings Chevalley groups $G_{\mathcal{P}}(\Phi,R)$ of
rank $>1$ are regularly bi-interpretable with the corresponding rings.

###  1.2. Chevalley groups, their automorphisms, isomorphisms and elementary
equivalence.

In 50-th years of the previous century Chevalley, Steinberg and others
introduced the concept of Chevalley groups over commutative rings, which
includes classical linear groups (special linear $\,\mathrm{SL}\,_{n}$,
special orthogonal $\,\mathrm{SO}\,_{n}$, symplectic $\,\mathrm{Sp}\,_{2n}$,
spinor $\,\mathrm{Spin}\,_{n}$, and also projective groups connected with
them).

Isomorphisms and automorphisms of Chevalley groups were also studied
intensively. The standard description of isomorphisms of Chevalley groups over
fields was obtained by R. Steinberg [58] for the finite case and by J.
Humphreys [31] for the infinite one. Many papers were devoted to description
of automorphisms of Chevalley groups over different commutative rings (see
Borel–Tits [8], Carter–Chen Yu [20], Chen Yu [22]–[23], E. Abe [1], A.
Klyachko [37] and others). E. Bunina described automorphisms of arbitrary
Chevalley groups over local rings (where these rings contain $1/2$ for the
root systems $A_{2},B_{\ell},C_{\ell}$ and $F_{4}$ and $1/3$ for the root
system $G_{2}$) and then in 2012 (see [11]) described all automorphisms of
adjoint Chevalley groups over arbitrary commutative rings. Recently (see [14])
the complete desription of automorphisms and isomorphisms of Chevalley groups
over almost all commutative rings was obtained:

######  Theorem 1.1.

Let $G=G_{\mathcal{P}}(\Phi,R)$ $(E_{\mathcal{P}}(\Phi,R))$ be an _(_
elementary _)_ Chevalley group with an irreducible root sustem $\Phi$ of rank
$>1$, $R$ be a commutative ring with $1$. Suppose that for
$\Phi=A_{2},B_{\ell},C_{\ell}$ or $F_{4}$ we have $1/2\in R$, for $\Phi=G_{2}$
we have $1/2,1/3\in R$. Then every automorphism of the group $G$ is standard
(the composition of ring, central, diagram and inner automorphisms). If $G$ is
an adjoint Chevalley groups, then the inner automorphism in the composition is
strictly inner and the central automorphism is trivial.

Description of automorphisms and isomorphisms of Chevalley groups over rings
is a magnificent instrument needed for solutions of diefferent model-theoretic
problems of Chevalley groups in general (including Maltsev-type theorems, bi-
interpretability, first order rigidity, elementary definability etc.). Talking
about Maltsev-type theorems, the following result (see [12] and [14]) almost
completes Maltsev-type theory for Chevalley groups:

######  Theorem 1.2.

Let $G=G_{\mathcal{P}}(\Phi,R)$ and
$G^{\prime}=G_{\mathcal{P}^{\prime}}(\Phi^{\prime},R^{\prime})$ _(_ or
$E_{\mathcal{P}}(\Phi,R)$ and
$E_{\mathcal{P}^{\prime}}(\Phi^{\prime},R^{\prime}))$ be two adjoint _(_
elementary _)_ Chevalley groups with indecomposable root systems
$\Phi,\Phi^{\prime}$ of ranks $>1$ over infinite commutative rings $R$ and
$R^{\prime}$ (for the cases of the roots systems $A_{2}$, $B_{\ell}$,
$C_{\ell}$, $F_{4}$ with $1/2$ and for the system $G_{2}$ with $1/2$ and
$1/3$).

Then the groups $G$ and $G^{\prime}$ are elementarily equivalent if and only
if the root systems $\Phi$ and $\Phi^{\prime}$ are isomorphic, the weight
lattices of $\mathcal{P}$ and $\mathcal{P}^{\prime}$ are isomorphic and the
rings $R$ and $R^{\prime}$ are elementarily equivalent.

###  1.3. First order rigidity and QFA property, bi-interpretability of linear
groups and their elementary definability.

Recent years have been characterized by a real breakthrough in model theory of
linear groups. Within quite a few years the papers of A. Nies ([48], [47],
etc.), N. Avni, A. Lubotzky, C. Meiri ([4], [5], also see [32]), A. Myasnikov,
O. Kharlampovich and M. Sohrabi ([34], [44], [45], etc.), D. Segal and K. Tent
([53]), B. Kunyavskii, E. Plotkin, N. Vavilov ([38]) and others appeared.

Since according to Löwenheim–Skolem theorem for any infinite structure
$\mathcal{A}$ there exists a structure $\mathcal{B}$ such that
$\mathcal{A}\equiv\mathcal{B}$ and they have different cardinalities, so
classification up to elementary equivalence is always strictly wider than
classification up to isomorphism. But for finitely generated case there exists
a chance that elementary equivalence could imply isomorphism. For example if
any finitely generated ring $A$ is elementarily equivalent to $\mathbb{Z}$,
then it is isomorphic to $\mathbb{Z}$. This phenomenon is not true for
infinitely generated rings.

A question dominating research in this area has been if and when elementary
equivalence between finitely generated groups (rings) implies isomorphism.
Recently, Avni–Lubotzky–Meiri [4] (see also [51]) found the term _first-order
rigidity_. Namely, a finitely generated group (ring or other structure)
$\mathcal{A}$ is _first-order rigid_ if any other finitely generated group
(ring or other structure) elementarily equivalent to $\mathcal{A}$ is
isomorphic to $\mathcal{A}$.

In many groups holds the more strong property when a single first-order axiom
is sufficient to distinguish a group among all the finitely generated groups.
The following concept was introduced in [47] for groups only, and then in [48]
for arbitrary structures. For a fixed finite signature an infinite finitely
generated structure $\mathcal{A}$ is _quasi finitely axiomatizable_ (_QFA_) if
there is a first order sentence $\varphi$ such that (1)
$\mathcal{A}\vDash\varphi$ and (2) if $H$ is a finitely generated structure in
the same signature such that $H\vDash\varphi$, then $H\cong\mathcal{A}$.

The most important example of QFA rings is the ring of integer
$(\mathbb{Z},+,\cdot)$, see [50].

The real breakthrough in this area was the application of bi-interpretability
towards finding new classes of QFA rings and groups.

######  Definition 1.3.

Suppose structures $\mathcal{A},\mathcal{B}$ in finite signatures are given,
as well as interpretations of $\mathcal{A}$ in $\mathcal{B}$, and vice versa.
Then an isomorphic copy $\widetilde{\mathcal{A}}$ of $\mathcal{A}$ can be
defined in $\mathcal{A}$, by ‘‘decoding’’ $\mathcal{A}$ from the copy of
$\mathcal{B}$ defined in $\mathcal{A}$. Similarly, an isomorphic copy
$\widetilde{\mathcal{B}}$ of $\mathcal{B}$ can be defined in $\mathcal{B}$. An
isomorphism $\Phi:{\mathcal{A}}\cong\widetilde{\mathcal{A}}$ can be viewed as
a relation on $\mathcal{A}$, and similarly for an isomorphism
$\mathcal{B}\cong\widetilde{\mathcal{B}}$. We say that $\mathcal{A}$ and
$\mathcal{B}$ are _bi-interpretable_ (with parameters) if there are such
isomorphisms that are first-order definable (see [28], Ch. 5).

The special interesting case is when one can prove that a given structure is
bi-interpretable with the ring $\mathbb{Z}$ of integers. A. Khelif ([36], see
also [34]) realized that one can use bi-interpretability of a finitely
generated structure $\mathcal{A}$ with $(\mathbb{Z},+\times)$ as a general
method to prove that $\mathcal{A}$ is QFA. Somewhat later, Scanlon
independently used this method to show that each finitely generated field is
QFA: _Suppose the structure $\mathcal{A}$ in a finite signature is bi-
interpretable with the ring $\mathbb{Z}$. Then $\mathcal{A}$ is QFA._

As it was mentioned above in [44] A. Myasnikov and M. Sohrabi proved that the
groups $\,\mathrm{SL}\,_{n}(\mathcal{O})$, $n\geqslant 3$, are bi-
interpretable (with parameters) with the ring $\mathcal{O}$ and therefore with
$\mathbb{Z}$; the groups $\,\mathrm{GL}\,_{n}(\mathcal{O})$ and
$T_{n}(\mathcal{O})$, $n\geqslant 3$, are bi-interpretable with $\mathcal{O}$
(and with $\mathbb{Z}$) only if $\mathcal{O}^{*}$ is finite. Consequently in
all good cases these linear groups are QFA. This result is crucial for proving
first order rigidity for such groups.

Along with Maltsev-type theorems, the problems of so-called _elementary
definability_ of classes of groups are quite important. These problems are
formulated as follows: _Suppose that we have a class $\mathcal{G}$ of groups
_(_ rings, etc._)_ and an arbitrary group _(_ rings, etc._)_ $H$ which is
elementarily equivalent to some $G\in\mathcal{G}$. Is it true that
$H\in\mathcal{G}$?_

For example the class of all linear groups over fields (commutative rings) is
elementary definable (A.I. Maltsev, [39]), but in the other hand for general
linear groups even over fields there is no elementary definability: there
exists a group $H\equiv\,\mathrm{GL}\,_{n}(F)$ which is not of type
$\,\mathrm{GL}\,$ itself.

In 1984 B.I. Zilber (see [66]) proved that for algebraic groups over
algebraically closed fields the problem of elementary definability has the
positive answer: _the class of all simple algebraic groups over algebraically
closed fields is elementary definable._

To extend this Zilber theorem for the cases of rings it is crucial to
introduce a notion of regular bi-interpretability. If $\mathcal{A}$ is
interpreted in $\mathcal{B}$ uniformly with respect to a $\emptyset$-definable
subset $D\subseteq\mathcal{B}^{k}$ then we say that $\mathcal{A}$ is
_regularly interpretable_ in $\mathcal{B}$ and write in this case
$\mathcal{A}\cong\Gamma(\mathcal{B},\varphi)$, provided $D$ is defined by
$\varphi$ in $\mathcal{B}$.

######  Definition 1.4.

Two algebraic structures $\mathcal{A}$ and $\mathcal{B}$ are called _regularly
bi-interpretable_ in each other if the following conditions hold:

(1) $\mathcal{A}$ and $\mathcal{B}$ are regularly interpretable in each other,
so $\mathcal{A}\cong\Gamma(\mathcal{B},\varphi)$ and
$\mathcal{B}\cong\Delta(\mathcal{A},\psi)$ for some interpretations (also
called _codes_) $\Gamma$ and $\Delta$ and the corresponding formulas
$\varphi,\psi$ (without parameters). By transitivity $\mathcal{A}$, as well as
$\mathcal{B}$, is regularly interpretable in itself, so we can write
$\mathcal{A}\cong(\Gamma\circ\Delta)(\mathcal{A},\varphi^{*})\text{ and
}\mathcal{B}\cong(\Delta\circ\Gamma)(\mathcal{B},\psi^{*}),$ where $\circ$
denotes composition of interpretations and $\varphi^{*},\psi^{*}$ are the
corresponding formulas.

(2) There is a formula $\theta(\overline{y},x,\overline{z})$ in the language
of $\mathcal{A}$ such that for every tuple $p^{*}$ satisfying
$\varphi^{*}(\overline{z})$ in $\mathcal{A}$ the formula
$\theta(\overline{y},x,p^{*})$ defines in $\mathcal{A}$ the isomorphism
$\overline{\mu}_{\Gamma\circ\Delta}:(\Gamma\circ\Delta)(\mathcal{A},p^{*})\to\mathcal{A}$
and there is a formula $\sigma(\overline{u},x,\overline{t})$ in the language
of $\mathcal{B}$ such that for every tuple $q^{*}$ satisfying
$\psi^{*}(\overline{v})$ in $\mathcal{B}$ the formula
$\sigma(\overline{v},x,q^{*})$ defines in $\mathcal{B}$ the isomorphism
$\overline{\mu}_{\Delta\circ\Gamma}:(\Delta\circ\Gamma)(\mathcal{B},q^{*})\to\mathcal{B}$.

So the existence of regular bi-interpretation plays the decisive role for
establishing elementary definability of given classes of groups. If
linear/Chevalley groups (or another derivative structures) of any concrete
types over some classes of fields/rings are regularly bi-interpretable with
the corresponding rings, then this class of groups (structures) is elementary
definable.

A. Myasnikov and M. Sohrabi proved that for all linear groups which they
considered in the paper [44] the bi-interpretation with the corresponding
rings is indeed a regular bi-interpretation, hence the elementary definability
problem has positive solution. This is also the case for the results of
Segal–Tent (see [53]) mentioned above.

##  2\. Main results

All mentioned recent results illustrate that together with bi-interpretability
with parameters it is of unprecedented importance to state regular bi-
interpretability of linear groups and corresponding rings.

In [13] E. Bunina proved the following theorem:

######  Theorem 2.1.

If $G(R)=G_{\mathcal{P}}(\Phi,R)$ $(E(R)=E_{\mathcal{P}}(\Phi,R))$ is an _(_
elementary _)_ Chevalley group of rank $>1$, $R$ is a local ring _(_ with
$\frac{1}{2}$ for the root systems ${B}_{\ell},{C}_{\ell},{F}_{4},{G}_{2}$ and
with $\frac{1}{3}$ for ${G}_{2})$, then the group $G(R)$ _(_ or $(E(R))$ is
regularly bi-interpretable with $R$. Therefore the class of Chevalley groups
over local rings _(_ with mentioned restrictions _)_ is elementary definable.

The given paper is devoted to the maximal possible generalization of Theorem
2.1. The best situation would be, if for all (almost all) commutative rings
with $1$ all (or almost all) corresponding Chevalley groups are regularly bi-
interpretable with their rings. Unfortunately it is not true: if a Chevalley
group has an infinite center, then in certain cases it is possible to prove
that this groups is not bi-interpretable with the corresponding rings and not
elementary definable in the class of all groups. To the contrary, if a center
is trivial (it holds for all adjoint Chevalley groups over all rings), then
regular bi-interpretability and elementary definability take place for
‘‘almost all’’ commutative rings: for all commutative rings with $\frac{1}{2}$
for the root systems ${B}_{\ell},{C}_{\ell},{F}_{4},{G}_{2}$ and with
$\frac{1}{3}$ for ${G}_{2}$.

The (positive) answer is not known yet for Chevalley groups over all rings, if
the center is not trivial and finite, but it is proved in the given paper in
the special case of bounded generation (Theorem 2.7) and for central quotients
of Chevalley groups (Theorem 2.6).

Moreover, in all positive cases we show that the classes of the corresponding
Chevalley groups (of a given type and with a given weight lattices) are
elementary definable by one formula, i. e., finitely axiomatizable.

To formulate short informal and detailed formal versions of two main theorems
and further conjectures we will introduce here just for simplicity several new
notations/definitions.

######  Definition 2.2.

For any group $G$ we will denote by $PG$ the quotient group by its center:

$PG:=G/Z(G).$

In particular, we will use the notation $PG_{\mathcal{P}}(\Phi,R)$ and will
call this group a _central quotient of a Chevalley group_.

######  Definition 2.3.

A commutative ring $R$ with $1$ is called _good_ , if

— we deal with the root system $\Phi=A_{2},B_{\ell},C_{\ell},F_{4}$ or $G_{2}$
and $1/2\in R$;

— we deal with the root system $\Phi=G_{2}$ and $1/3\in R$.

For the cases $\Phi=A_{\ell},D_{\ell},E_{\ell}$, $l\geqslant 3$, there are no
restrictions for $R$.

######  Definition 2.4.

Let $G$ be a group, $X\subseteq G$ be a subset, and $N$ be a non-negative
integer. We say that $G$ is $N$-boundedly generated by the set $X$, if every
element of $G$ is the product of at most $N$ elements from $X$.

######  Definition 2.5.

For a group $G$ and elements $g_{1}$,$\ldots$,$g_{l}\in G$ we set

$X_{g_{1},\ldots,g_{l}}=\\{[a,b]\colon a,b\in G\\}\cup\\{a^{m}\colon a\in
G\\}\cup\\{g_{1},\ldots,g_{l}\\}.$

Now we are ready to give short formulations of our results.

Let us always assume that

— a root system $\Phi$ is indecomposable and it’s rank is $>1$;

— a ring $R$ is a good commutative ring with $1$.

Also let us call results about bi-interpretability and elementary definability
_positive_ , if:

— the Chevalley group $G_{\mathcal{P}}(\Phi,R)$ (or some other group connected
with it) is regularly bi-interpretable with $R$;

— any group elementarily equivalent to $G_{\mathcal{P}}(\Phi,R)$ (or some
other group connected with it) is also $G_{\mathcal{P}}(\Phi,S)$ under some
ring $S$, which is elementarily equivalent to $R$;

— the class of all Chevalley groups $G_{\mathcal{P}}(\Phi,-)$ (or some other
group connected with it) over all good rings is finitely axiomatizable.

So, shortly speaking, in this paper we obtain positive results for the
following groups and rings:

1. All central quotients $PG_{\mathcal{P}}(\Phi,R)$ of Chevalley groups over good rings;

2. In particular, all adjoint Chevalley groups $G_{\mathcal{P}}(\Phi,R)$ over good rings;

3. All Chevalley groups $G_{\mathcal{P}}(\Phi,R)$ over good rings, $N$-boundedly generated by some set $X_{g_{1},\ldots,g_{l}}$ for some natural $N$.

Also we show the negative result (by negative we mean that all 1–3 do not
hold) for some $G_{\mathcal{P}}(\Phi,R)$ in the case of infinite center.

Our conjectures for today are the following:

1. If the center of $G_{\mathcal{P}}(\Phi,R)$ is infinite, usually the results are negative.

2. Even if the center of $G=G_{\mathcal{P}}(\Phi,R)$ is finite and $G$ is not $N$-boundedly generated by any $X_{g_{1},\ldots,g_{l}}$ and any $N\in\mathbb{N}$, then there exist negative examples.

So now we will give precise formulations of two main theorems of the given
paper. The first one is about central quotients of Chevalley groups:

######  Theorem 2.6.

Let $\Phi$ be an irreducible root system of rank $>1$, $\mathcal{P}$ be any
corresponding weight lattice. Then there exists a $|\Phi|$-ary logical formula
$\varphi(g_{\alpha}\colon\alpha\in\Phi)$ in the first order language of groups
such that the following holds:

  1. (1)

If $R$ is a good commutative ring, then for given elements $\\{g_{\alpha}\in
PG_{\mathcal{P}}(\Phi,R)\colon\alpha\in\Phi\\}$, we have

$PG_{\mathcal{P}}(\Phi,R)\models\varphi(g_{\alpha}\colon\alpha\in\Phi)$

if and only if there exists an automorphism
$\overline{f}\in\,\mathrm{Aut}\,(PG_{\mathcal{P}}(\Phi,R))$ such that
$\overline{f}(g_{\alpha})=\overline{x}_{\alpha}(1)$ for all $\alpha\in\Phi$.

  2. (2)

Under the assumptions of item  _1_ , there is a regular bi-interpretation of
$R$ and $PG_{\mathcal{P}}(\Phi,R)$ such that the interpretation of
$PG_{\mathcal{P}}(\Phi,R)$ in $R$ is absolute and the parameters for
interpretation of $R$ in $PG_{\mathcal{P}}(\Phi,R)$ are defined by $\varphi$.

  3. (3)

If $G$ is a group and $g_{\alpha}\in G$, $\alpha\in\Phi$ are such that
$G\models\varphi(g_{\alpha}\colon\alpha\in\Phi)$, then there exists a good
ring $R$ and an isomorphism $f\colon G\to PG_{\mathcal{P}}(\Phi,R)$ such that
$f(g_{\alpha})=\overline{x}_{\alpha}(1)$ for all $\alpha\in\Phi$.

  4. (4)

If $G$ is a group, $G\equiv G_{\mathcal{P}}(\Phi,R)$, where $R$ is good, then
their centers are elementarily equivalent and there exists a ring $S\equiv R$
such that

$G/Z(G)\cong PG_{\mathcal{P}}(\Phi,S).$

The second main theorem is about Chevalley groups themselves with the property
of $N$-bounded generation:

######  Theorem 2.7.

Let $\Phi$ be an irreducible root system of rank $>1$, $\mathcal{P}$ be any
corresponding weight lattice. Let $m$ be such number that the center of the
group scheme $G_{\mathcal{P}}(\Phi,-)$ is the group $\mu_{m}$ of the $m$-th
roots of unity. Let $N$, $l$ and $n$ be some non-negative integers. Let
$\alpha_{0}\in\Phi$ be some fixed root. Let $\varpi\colon
G_{\mathcal{P}}(\Phi,-)\to\,\mathrm{GL}\,(n,-)$ be some faithful
representation, i.e a closed embedding of algebraic group schemes. Then there
exists a $|\Phi|+l(n^{2}+1)$-ary logical formula
$\varphi^{N}(g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant k\leqslant
l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant i,j\leqslant n)$
in the first order language of groups such that the following holds:

  1. (1)

If $R$ is a good commutative ring, then for given elements
$\\{g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant k\leqslant
l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant i,j\leqslant
n\\}$, we have

$G_{\mathcal{P}}(\Phi,R)\models\varphi^{N}(g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon
1\leqslant k\leqslant l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant
l,\;1\leqslant i,j\leqslant n)$

if and only if the group $G_{\mathcal{P}}(\Phi,R)$ is $N$-boundedly generated
by the set $X_{g_{1},\ldots,g_{l}}$, and there exists an automorphism
$f\in\,\mathrm{Aut}\,(G_{\mathcal{P}}(\Phi,R))$ such that

$\bullet$ $f(g_{\alpha})=x_{\alpha}(1)$ for all $\alpha\in\Phi$;

$\bullet$ $f(g_{ij}^{(k)})=x_{\alpha_{0}}(\xi_{ij}^{(k)})$ for all $1\leqslant
k\leqslant l$ for some $\xi_{ij}^{(k)}\in R$;

$\bullet$ $\varpi(f(g_{k}))$ is the matrix $(\xi_{ij}^{(k)})$ for all
$1\leqslant k\leqslant l$.

  2. (2)

If $R$ is a good commutative ring, and there is a $l$-tuple
$g_{1},\ldots,g_{l}\in G_{\mathcal{P}}(\Phi,R)$ such that the group
$G_{\mathcal{P}}(\Phi,R))$ is $N$-boundednly generated by the set
$X_{g_{1},\ldots,g_{l}}$, there is a regular bi-interpretation of $R$ and
$G_{\mathcal{P}}(\Phi,R)$ such that the interpretation of
$G_{\mathcal{P}}(\Phi,R)$ in $R$ is absolute and the parameters for
interpretation of $R$ in $G_{\mathcal{P}}(\Phi,R)$ are defined by
$\varphi^{N}$.

  3. (3)

If $G$ is a group and $g_{\alpha},g_{k},g_{ij}^{(k)}\in G$,
$\alpha\in\Phi,\;1\leqslant k\leqslant l$, $1\leqslant i,j\leqslant n$ are
such that $G\models\varphi^{N}(g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon
1\leqslant k\leqslant l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant
l,\;1\leqslant i,j\leqslant n)$, then there exists a good ring $R$ and an
isomorphism $f\colon G\to G_{\mathcal{P}}(\Phi,R)$ such that

$\bullet$ $f(g_{\alpha})=x_{\alpha}(1)$ for all $\alpha\in\Phi$;

$\bullet$ $f(g_{ij}^{(k)})=x_{\alpha_{0}}(\xi_{ij}^{(k)})$ for all $1\leqslant
k\leqslant l$ for some $\xi_{ij}^{(k)}\in R$;

$\bullet$ $\varpi(f(g_{k}))$ is the matrix $(\xi_{ij}^{(k)})$ for all
$1\leqslant k\leqslant l$.

  4. (4)

Let $G$ be a group, and $G\equiv G_{\mathcal{P}}(\Phi,R)$, where $R$ is good.
Assume that for some $g_{1},\ldots,g_{l}\in G_{\mathcal{P}}(\Phi,R))$ the
group $G_{\mathcal{P}}(\Phi,R))$ is $N$-boundedly generated by the set
$X_{g_{1},\ldots,g_{l}}$, then there exists a ring $S\equiv R$ such that

$G\cong G_{\mathcal{P}}(\Phi,S).$

######  Remark 2.8.

The statement of Theorem 2.7 makes it natural to ask for what rings the group
$G_{\mathcal{P}}(\Phi,R)$ is boundedly generated by the set
$X_{g_{1},\ldots,g_{l}}$ as above. Let us list what we know.

$\bullet$ If $R$ is a Dedekind domain of arithmetic type then
$G_{\mathcal{P}}(\Phi,R)$ s boundedly generated by elementary unipotents, and
hence by commutators, see for example [19],[41],[38].

$\bullet$ Assume that $G_{\mathcal{P}}(\Phi,R)=G_{\mathrm{sc}}(\Phi,R)$ is
simply connected, and the ring $R$ is finitely generated, then by [27] the
group $K_{1}(\Phi,R)=G_{\mathrm{sc}}(\Phi,R)/E_{\mathrm{sc}}(\Phi,R)$ is
nilpotent. Further assume that $K_{1}(\Phi,R)$ is finitely generated (the
question whether this groups is always finitely generated for a finitely
generated ring $R$ is an open problem). Then $K_{1}(\Phi,R)$ is polycyclic,
see [52, Chapter 4A, Exercise 1.2]. It then easily follows that
$K_{1}(\Phi,R)$ is boundedly generated by $m$-th powers and some finite set.
Thus we reduce the question to an elementary group. Similarly, we can do this
for a non simply connected group $G_{\mathcal{P}}(\Phi,R)$, if we assume
additionally that its quotient by the image of $G_{\mathrm{sc}}(\Phi,R)$ is
finitely generated; this quotient is a subgroup in the cohomology group
$H^{1}_{\mathrm{fppf}}(R,\mu)$, where
$\mu=\mathrm{Ker}(G_{\mathrm{sc}}(\Phi,-)\to G_{\mathcal{P}}(\Phi,-))$.

$\bullet$ In papers [63],[25],[26] it is proved that an element from the
Chevalley group of type $\Phi$, where $\mathrm{rk}\,\Phi\geqslant 2$ and
$\Phi\neq F_{4},G_{2},E_{8}$, over a polynomial ring with coefficients in a
small-dimensional ring can be reduced to an element of certain proper
subsystem subgroup by a bounded number of elementary root elements (the bound
depends on the number of variables). By induction, we can reduce such an
element to a subsystem subgroup of type $A_{1}$. Since every finitely
generated ring is a quotient of $\mathbb{Z}[x_{1},\ldots,x_{n}]$, and the map
on corresponding elementary subgroups is surjective, we conclude that for a
finitely generated ring any element from the elementary subgroup can be
reduced to a subsystem subgroup of type $A_{1}$. Note that the systems
$\Phi\neq F_{4},G_{2},E_{8}$ are inessential for us here, because for them the
all Chevalley groups are adjoint; hence we can apply Theorem 2.6 to them, so
we don’t need bounded generation.

$\bullet$ The question whether the group
$G(\Phi,\mathbb{Z}[x_{1},\ldots,x_{n}])$ is boundedly generated by elementary
root elements is an open problem. The educated guesses of the specialists are
that the answer is negative. Now it is natural to formulate a weaker version
of that problem: is the group $G(\Phi,\mathbb{Z}[x_{1},\ldots,x_{n}])$
boundedly generated by commutators, $m$-th powers, and a finite set, where $m$
is as above?

##  3\. Basic notations and useful facts about Chevalley groups

We refer for the facts related to root systems and semisimple Lie algebras to
[9] and [30]. More detailed information about elementary Chevalley groups is
contained in the books [59], and about the Chevalley groups (also over rings)
in [64], [65], [24] (see also later references in these papers).

We always fix some arbitrary indecomposable root system $\Phi$ of the rank
$\ell>1$.

We consider an arbitrary Chevalley group $G_{\mathcal{P}}(\Phi,R)$,
constructed by the root system $\Phi$, a ring $R$ and a lattice $\mathcal{P}$
of a representation $\pi$ of the corresponding Lie algebra. It is known, that
Chevalley group is defined by these three parameters. If we consider an
elementary Chevalley group, we denote it by $E_{\mathcal{P}}(\Phi,R)$.

Relations between Chevalley groups and the corresponding elementary subgroups
is an important problem in the theory of Chevalley groups over rings. For
elementary Chevalley groups there exists a convenient system of generators
$x_{\alpha}(\xi)$, $\alpha\in\Phi$, $\xi\in R$, and all relations between
these generators are well-known. For general Chevalley groups it is not always
true.

If $R$ is an algebraically closed field, then

$G_{\mathcal{P}}(\Phi,R)=E_{\mathcal{P}}(\Phi,R)$

for any weight lattice $\mathcal{P}$. This equality is not true even for the
case of fields, which are not algebraically closed.

If $\Phi$ is an irreducible root system of rank $\ell>1$, then
$E_{\mathcal{P}}(\Phi,R)$ is always normal in $G_{\mathcal{P}}(\Phi,R)$ (see
[62], [27]) and even is characteristic in it.

We will use the following result about normal subgroups of elementary
Chevalley groups (see[3]).

Note that for every ideal $I$ of $R$ the natural mapping $R\to R/I$ induces a
homomorphism

$\lambda_{I}:G_{\mathcal{P}}(\Phi,R)\to G_{\mathcal{P}}(\Phi,R/I).$

If $I$ is a proper ideal of $R$, then the kernel of $\lambda_{I}$ is a non-
central normal subgroup of $G_{\mathcal{P}}(\Phi,R)$.

######  Definition 3.1.

We denote the inverse image of the center of $G_{\mathcal{P}}(R/I)$ under
$\lambda_{I}$ by $Z_{\mathcal{P}}(\Phi,R,I)$.

By $E_{\mathcal{P}}(\Phi,R,I)$ we denote the minimal normal subgroup of
$E_{\mathcal{P}}(\Phi,R)$ which contains all $x_{\alpha}(t)$, $\alpha\in\Phi$,
$t\in I$.

######  Theorem 3.2.

Let the rank of an indecomposable root system $\Phi$ is more than one. If a
subgroup $H$ of $E_{\mathcal{P}}(\Phi,R)$ is normal in
$E_{\mathcal{P}}(\Phi,R)$, then

$E_{\mathcal{P}}(\Phi,R,I)\leqslant H\leqslant Z_{\mathcal{P}}(\Phi,R,I)\cap
E_{\mathcal{P}}(\Phi,R)$

for some uniquely defined ideal $I$ of the ring $R$.

For adjoint Chevalley group (which always have trivial center) the group
$Z_{\mathcal{P}}(\Phi,R,I)$ coincides with the group
$G_{\mathcal{P}}(\Phi,R,I)$ consisting of all matrices $A$ from
$G_{\mathcal{P}}(\Phi,R)$ such that $A-E\in M_{N}(I)$.

To prove our main theorem, we will need to study isomorphisms between central
quotients of Chevalley groups. This result is valuable in itself.

######  Theorem 3.3.

If $\Phi$ is an irreducible root system of rank $>1$, $\mathcal{P}$ is any
corresponding lattice, $R$ and $S$ are good commutative rings with $1$, then
if the groups

$PG_{\mathcal{P}}(\Phi,R)=G_{\mathcal{P}}(\Phi,R)/Z(G_{\mathcal{P}}(\Phi,R))\text{
and
}PG_{\mathcal{P}}(\Phi,S)=G_{\mathcal{P}}(\Phi,S)/Z(G_{\mathcal{P}}(\Phi,S))$

are isomorphic, then the rings $R$ and $S$ are isomorphic.

###### Proof.

Since the (elementary Chevalley) subgroups $E_{\mathcal{P}}(\Phi,R)$ and
$E_{\mathcal{P}}(\Phi,S)$ are normal in the corresponding Chevalley groups
$G_{\mathcal{P}}(\Phi,R)$ and $G_{\mathcal{P}}(\Phi,S)$, then their quotients
by their centers are normal subgroups in $PG_{\mathcal{P}}(\Phi,R)$ and
$PG_{\mathcal{P}}(\Phi,S)$. Since for any elementary Chevalley group
$E_{\pi}(\Phi,R)$ the quotient group by its center is elementary adjoint
Chevalley group $E_{ad}(\Phi,R)$, then we can assume that

$E_{ad}(\Phi,R)\vartriangleleft PG_{\mathcal{P}}(\Phi,R)\text{ and
}E_{ad}(\Phi,S)\vartriangleleft PG_{\mathcal{P}}(\Phi,S).$

Since the groups $PG_{\mathcal{P}}(\Phi,R)$ and $PG_{\mathcal{P}}(\Phi,S)$ are
isomorphic, then we can suppose that the group $PG_{\mathcal{P}}(\Phi,S)$
contains two normal subgroups: $G_{1}=E_{ad}(\Phi,S)$ and
$G_{2}=f(E_{ad}(\Phi,R))$.

The subgroups $G_{1}$ and $G_{2}$ satisfy the property $(*)$:
$C_{PG_{\mathcal{P}}(\Phi,S)}(G_{i})=e,$ $i=1,2$.

Since both $G_{1}$ and $G_{2}$ are normal, if their intersection is trivial,
then any $g_{1}\in G_{2}$ commutes with any $g_{2}\in G_{2}$, which is
impossible by this property $(*)$. Therefore $H:=G_{1}\cap G_{2}$ is a
nontrivial normal subgroup of $G_{1}$, $G_{2}$ and $PG_{\mathcal{P}}(\Phi,S)$.
According to Theorem 3.2

$E_{ad}(\Phi,S,I)\leqslant H\leqslant G_{\mathcal{P}}(\Phi,S,I)\cap
E_{P}(\Phi,S)$

and

$f(E_{ad}(\Phi,R,J))\leqslant H\leqslant f(G_{\mathcal{P}}(\Phi,R,J)\cap
E_{P}(\Phi,R)),$

where $I$ is an ideal of $S$ and $J$ is an ideal of $R$.

Let us factorize the group $G_{\mathcal{P}}(\Phi,S)$ by the ideal $I$: every
matrix from $G_{\mathcal{P}}(\Phi,S)$ will be taken modulo $I$. Clearly the
groups $PG_{\mathcal{P}}(\Phi,S)$ and $G_{1}=E_{ad}(\Phi,S)$ also will be
factorized and will be mapped to $PG_{\mathcal{P}}(\Phi,S/I)$ and
$G_{1}^{I}=E_{ad}(\Phi,S/I)$, respectively. The normal subgroup $G_{2}$ also
will be mapped to the normal subgroup $G_{2}^{I}$ of
$PG_{\mathcal{P}}(\Phi,S/I)$ under this factorization, and the subgroup $H$
will be mapped to $\\{e\\}$.

Therefore we have the following situation: the group
$PG_{\mathcal{P}}(\Phi,S/I)$ contains a normal subgroup $G_{2}^{I}$
intersecting with $E_{ad}(\Phi,S/I)$ by $\\{e\\}$. By property $(*)$
$G_{2}^{I}$ is trivial, which means $G_{2}=H$ and $f(E_{ad}(\Phi,R))\subseteq
E_{ad}(\Phi,S)$ in the initial group. From symmetry
$f(E_{ad}(\Phi,R))=E_{ad}(\Phi,S)$ and therefore $R\cong S$.

∎

##  4\. Proof of Theorem 2.6

######  Lemma 4.1.

Let $\beta\in\Phi$, then there is a $(|\Phi|+1)$-ary formula
$\operatorname{IsElUnip}_{\beta}(g,g_{\alpha}\colon\alpha\in\Phi)$ such that
for any commutative ring $R$ and any $g\in PG_{\mathcal{P}}(\Phi,R)$ we have

$\left[PG_{\mathcal{P}}(\Phi,R)\models\operatorname{IsElUnip}_{\beta}(g,Zx_{\alpha}(1)\colon\alpha\in\Phi)\right]\Leftrightarrow\left[\exists\xi\in
R\colon g=Zx_{\beta}(\xi)\right].$

###### Proof.

Follows from the proofs of [16, Propositions 3 and 4] in case of adjoint
group. Note that the difference between $G_{\,\mathrm{ad}\,}(\Phi,R)$ and
$PG_{\mathcal{P}}(\Phi,R)$ does not affect these proofs. ∎

######  Lemma 4.2.

Let $\beta\in\Phi$, then there are $(|\Phi|+3)$-ary formulas
$\operatorname{Add}_{\beta}(g_{1},g_{2},g_{3},g_{\alpha}\colon\alpha\in\Phi)$
and
$\operatorname{Mult}_{\beta}(g_{1},g_{2},g_{3},g_{\alpha}\colon\alpha\in\Phi)$
such that for any commutative ring $R$ and any $\xi_{1}$,$\xi_{2}$,$\xi_{3}\in
R$ we have

$\left[PG_{\mathcal{P}}(\Phi,R)\models\operatorname{Add}_{\beta}(Zx_{\beta}(\xi_{1}),Zx_{\beta}(\xi_{2}),Zx_{\beta}(\xi_{3}),Zx_{\alpha}(1)\colon\alpha\in\Phi)\right]\Leftrightarrow\left[\xi_{1}+\xi_{2}=\xi_{3}\right]\text{,}$

and

$\left[PG_{\mathcal{P}}(\Phi,R)\models\operatorname{Mult}_{\beta}(Zx_{\beta}(\xi_{1}),Zx_{\beta}(\xi_{2}),Zx_{\beta}(\xi_{3}),Zx_{\alpha}(1)\colon\alpha\in\Phi)\right]\Leftrightarrow\left[\xi_{1}\xi_{2}=\xi_{3}\right]\text{.}$

###### Proof.

Follows from the proof of [16, Theorem 4]. ∎

Let us fix some faithful representation of $G_{\mathcal{P}}(\Phi,-)$, i.e a
closed embedding of algebraic group schemes
$G_{\mathcal{P}}(\Phi,-)\to\,\mathrm{GL}\,(n,-)$ for some $n$; and let us fix
some root $\alpha_{0}\in\Phi$,

######  Lemma 4.3.

There is an $(|\Phi|+n^{2}+1)$-ary formula
$\operatorname{MatUpToCent}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi)$ such that for any commutative ring $R$ and
any $g$, $g_{i,j}\in PG_{\mathcal{P}}(\Phi,R)$, we have
$PG_{\mathcal{P}}(\Phi,R)\models\operatorname{MatUpToCent}(g,g_{i,j}\colon
1\leqslant i,j\leqslant n,Zx_{\alpha}(1)\colon\alpha\in\Phi)$ if and only if
there are $\xi_{i,j}\in R$ such that the following holds:

  1. (1)

$g_{i,j}=Zx_{\alpha_{0}}(\xi_{i,j})$ for all $1\leqslant i,j\leqslant n$;

  2. (2)

The matrix $(\xi_{i,j})$ is in the image of $G_{\mathcal{P}}(\Phi,R)$;

  3. (3)

The element $g$ in the group $PG_{\mathcal{P}}(\Phi,R)$ can be represented in
the group $G_{\mathcal{P}}(\Phi,R)$ by the matrix $(\xi_{i,j})$.

###### Proof.

We build this formula as the conjunction of the following.

Firstly, we use Lemma 4.1 and take the conjunction of
$\operatorname{IsElUnip}_{\alpha_{0}}(g_{i,j},g_{\alpha}\colon\alpha\in\Phi)$
for all $1\leqslant i,j\leqslant n$. So we ensure that
$g_{i,j}=Zx_{\alpha_{0}}(\xi_{i,j})$ for some $\xi_{i,j}\in R$.

Secondly, we use Lemma 4.2 to state that the matrix $(\xi_{i,j})$ is in the
image of $G_{\mathcal{P}}(\Phi,R)$. We can do it because this image is defined
by polynomial equations.

Now by [60, Corollary 5.2] there exists $N\in\mathbb{N}$ such that for any
commutative ring $R$, any $g\in G_{\mathcal{P}}(\Phi,R)$, and any
$\beta\in\Phi$ the element $x_{\beta}(1)^{g}$ is a product of at most $N$
elementary root elements.

The last part of our formula will be as follows:

$\bigwedge_{\beta\in\Phi}\bigvee_{\beta_{1},\ldots,\beta_{N}\in\Phi}\left[\exists
h_{1},\ldots,h_{N}\colon\left(\bigwedge_{i=1}^{N}\operatorname{IsElUnip}_{\beta_{i}}(h_{i},g_{\alpha}\colon\alpha\in\Phi)\right)\wedge(g_{\beta}^{g}=h_{1}\ldots
h_{N})\wedge\\\
\wedge\psi_{\beta,\beta_{1},\ldots,\beta_{N}}(h_{1},\ldots,h_{N},g_{i,j}\colon
1\leqslant i,j\leqslant n,g_{\alpha}\colon\alpha\in\Phi)\right]\text{,}$

where the formula
$\psi_{\beta,\beta_{1},\ldots,\beta_{N}}(h_{1},\ldots,h_{N},g_{i,j}\colon
1\leqslant i,j\leqslant n,g_{\alpha}\colon\alpha\in\Phi)$ is such that for any
commutative ring $R$, any $\zeta_{1},\ldots,\zeta_{N}\in R$ and any $h\in
G_{\mathcal{P}}(\Phi,R)$ with the matrix of $h$ in our representation being
$(\xi_{i,j})$, we have

$G_{\mathcal{P}}(\Phi,R)\models\psi_{\beta,\beta_{1},\ldots,\beta_{N}}(x_{\beta_{1}}(\zeta_{1}),\ldots,x_{\beta_{N}}(\zeta_{N}),x_{\alpha_{0}}(\xi_{i,j})\colon
1\leqslant i,j\leqslant n,x_{\alpha}(1)\colon\alpha\in\Phi)\Leftrightarrow\\\
\Leftrightarrow x_{\beta}(1)^{h}=x_{\beta_{1}}(\zeta_{1})\ldots
x_{\beta_{N}}(\zeta_{N})\text{.}$

In order to construct such formula $\psi_{\beta,\beta_{1},\ldots,\beta_{N}}$,
note that the equality $x_{\beta}(1)^{h}=x_{\beta_{1}}(\zeta_{1})\ldots
x_{\beta_{N}}(\zeta_{N})$ is equivalent to certain polynomial equality in $R$
on $\zeta_{1},\ldots,\zeta_{N}$, and $\xi_{i,j}$. So we can turn elements
$x_{\beta_{i}}(\zeta_{i})$ into $x_{\alpha_{0}}(\pm\zeta_{i})$ via conjugation
with an element of the extended Weyl group, which is expressed through
elements $x_{\alpha}(1)$; and then we use Lemma 4.2 in order to write the
required polynomial equality in the language of groups.

Therefore, we achieved that the element $g$ and the element represented by the
matrix $(\xi_{i,j})$ differs by multiplying by such an element $c$ that it
commutes with all the $Zx_{\alpha}(1)$. Since
$PG_{\mathcal{P}}(\Phi,R)\leqslant G_{\,\mathrm{ad}\,}(\Phi,R)$, it follows
then from [16, Theorem 3] that $c$ is trivial. ∎

Now we prove Theorem 2.6.

###### Proof.

Note that item (4) directly follows from item (3), so we will prove only items
(1)–(3).

We construct the formula $\varphi$ as a conjunction of formulas that
essentially says the following:

  1. (1)

The center of $G$ is trivial.

  2. (2)

The formulas $\operatorname{Add}_{\alpha_{0}}$ and
$\operatorname{Mult}_{\alpha_{0}}$ define binary operation on the set of such
elements $g$ that
$\operatorname{IsElUnip}_{\alpha_{0}}(g,g_{\alpha}\colon\alpha\in\Phi)$ holds;
and this set is a ring with respect to these operations.

  3. (3)

The ring $R$ from the item 2 is good.

  4. (4)

For any tuple $g_{i,j}$, $1\leqslant i,j\leqslant n$ such that
$\operatorname{IsElUnip}_{\alpha_{0}}(g_{i,j},g_{\alpha}\colon\alpha\in\Phi)$
holds and that satisfy the equations that define $G_{\mathcal{P}}(\Phi,-)$
(i.e. satisfy the formula that is obtained from these equations by rewriting
it in the language of groups using Lemma 4.2) there exists a unique $g$ such
that $\operatorname{MatUpToCent}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi)$ holds.

  5. (5)

For any $g$ there exists a tuple $g_{i,j}$, $1\leqslant i,j\leqslant n$ such
that $\operatorname{MatUpToCent}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi)$ holds.

  6. (6)

For any
$g$,$g^{\prime}$,$g^{\prime\prime}$,$g_{i,j}$,$g_{i,j}^{\prime}$,$g_{i,j}^{\prime\prime}$,
if $\operatorname{MatUpToCent}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi)$,
$\operatorname{MatUpToCent}(g^{\prime},g^{\prime}_{i,j}\colon 1\leqslant
i,j\leqslant n,g_{\alpha}\colon\alpha\in\Phi)$ and
$\operatorname{MatUpToCent}(g^{\prime\prime},g^{\prime\prime}_{i,j}\colon
1\leqslant i,j\leqslant n,g_{\alpha}\colon\alpha\in\Phi)$ hold and the matrix
$(g_{i,j})$ is the product of matrices $(g^{\prime}_{i,j})$ and
$(g^{\prime\prime}_{i,j})$ in the sense of operations defined by
$\operatorname{Add}_{\alpha_{0}}$ and $\operatorname{Mult}_{\alpha_{0}}$, then
$g=g^{\prime}g^{\prime\prime}$.

  7. (7)

For any tuple $g_{i,j}$, $1\leqslant i,j\leqslant n$ we have
$\operatorname{MatUpToCent}(e,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi)$ holds if and only if it represents the
matrix from the center of $G_{\mathcal{P}}(\Phi,R)$ (note that the center is
defined by equations).

  8. (8)

In the correspondence between elements and matrices defined by items 3 and 4,
matrices of elements $x_{\alpha}(1)$ correspond to the elements $g_{\alpha}$
(these are matrices with integer entries, and any particular integer is
definable in the language of rings).

Now let us prove that the formula constructed above satisfies items (1)–(3) of
the Theorem 2.6.

Firstly, by construction, for any good commutative ring $R$ we have

$PG_{\mathcal{P}}(\Phi,R)\models\varphi(Zx_{\alpha}(1)\colon\alpha\in\Phi)\text{.}$

Therefore, the implication from the right to the left in item (1) holds true.

Secondly, item (3) also holds by construction.

Now let us prove the implication from the left to the right in item (1).
Assume that for some $g_{\alpha}\in PG_{\mathcal{P}}(\Phi,R)$ we have

$PG_{\mathcal{P}}(\Phi,R)\models\varphi(g_{\alpha}\colon\alpha\in\Phi)\text{.}$

By item (3), there exists an isomorphism $f:PG_{\mathcal{P}}(\Phi,S)\to
PG_{\mathcal{P}}(\Phi,R)$ for some ring $S$ such that
$f(Zx_{\alpha}(1))=g_{\alpha}$ for all $\alpha\in\Phi$. By Theorem 3.3 the
(good) rings $R$ and $S$ are isomorphic. Now composing $f$ with the ring
isomorphism from $PG_{\mathcal{P}}(\Phi,S)$ to $PG_{\mathcal{P}}(\Phi,R)$, we
obtained the required automorphism.

It remains to prove the item (2). By item (1) it is enough to prove that there
exists a bi-interpretation of $R$ and $PG_{\mathcal{P}}(\Phi,R)$ such that the
interpretation of $PG_{\mathcal{P}}(\Phi,R)$ in $R$ is absolute and the
parameters for interpretation of $R$ in $PG_{\mathcal{P}}(\Phi,R)$ are
$Zx_{\alpha}(1)$.

We interpret $PG_{\mathcal{P}}(\Phi,R)$ in $R$ by an obvious way: elements of
$PG_{\mathcal{P}}(\Phi,R)$ are matrices over $R$ that satisfy certain
equations; equiality of elements is an equivalence up to the center; and the
multiplication is the multiplication of matrices.

Further we interpret $R$ in $PG_{\mathcal{P}}(\Phi,R)$ using Lemmas 4.1 and
4.2.

It is easy to see that the set of pairs $(\xi,g)\in R\times
G_{\mathcal{P}}(\Phi,R)$ such that $g$ differs from $x_{\alpha_{0}}(\xi)$ by a
central element is a set of points of a closed subscheme in
$\mathbb{A}_{1}\times G_{\mathcal{P}}(\Phi,-)$; hence the isomorphism between
$R$ and its interpretation inside $PG_{\mathcal{P}}(\Phi,R)$ is defined by
polynomial formulas.

The isomorphism between $PG_{\mathcal{P}}(\Phi,R)$ and its interpretation
inside $R$ can be defined using Lemma 4.3. ∎

######  Corollary 4.4.

Let $\Phi$ be an irreducible root system of rank $>1$, $R$ be a good
commutative ring. Then the adjoint Chevalley group $G_{ad}(\Phi,R)$ satisfies
all conclusions of Theorem 2.6.

In particular, is any group $G$ is elementarily equivalent to
$G_{ad}(\Phi,R)$, then there exists a good ring $S$ such that $G\cong
G_{ad}(\Phi,S)$.

###### Proof.

This corollary follows from Theorem 2.6 and the fact that for any adjoint
Chevalley group its center is trivial (see [2]). ∎

######  Corollary 4.5.

Let $\Phi$ be an irreducible root system of rank $>1$, $\mathcal{P}$ be any
corresponding weight lattice, $\mathbf{G}_{\Phi,\mathcal{P}}$ and
$\mathbf{PG}_{\Phi,\mathcal{P}}$ be the classes of

  1. (1)

all Chevalley groups $G_{\mathcal{P}}(\Phi,R)$ with trivial centers and groups
$PG_{\mathcal{P}}(\Phi,R)$ over arbitrary commutative rings for
$\Phi=A_{\ell},D_{\ell},E_{\ell}$, $\ell\geqslant 3$;

  2. (2)

all Chevalley groups $G_{\mathcal{P}}(\Phi,R)$ with trivial centers and groups
$PG_{\mathcal{P}}(\Phi,R)$ over arbitrary commutative rings with $1/2$ for
$\Phi=A_{2},B_{\ell},C_{\ell},F_{4}$, $\ell\geqslant 2$;

  3. (3)

all Chevalley groups $G_{\mathcal{P}}(\Phi,R)$ over arbitrary commutative
rings with $1/2$ and $1/3$ for $\Phi=G_{2}$.

Then the classes $\mathbf{G}_{\Phi,\mathcal{P}}$ and
$\mathbf{PG}_{\Phi,\mathcal{P}}$ are finitely axiomatizable.

###### Proof.

Directly follows from Theorem 2.6. ∎

######  Corollary 4.6.

Let $\Phi$ be an irreducible root system of rank $>1$, $\mathcal{P}$ be any
corresponding weight lattice. Let $T$ be a collection (possibly infinite) of
first order sentences in the language of rings such that $T$ includes the
sentence "$1/2\in R$" in case $\Phi=A_{2},B_{\ell},C_{\ell},F_{4}$, $G_{2}$
$\ell\geqslant 2$, and $T$ includes the sentence "$1/3\in R$" in case
$\Phi=G_{2}$. Then the class of groups that are isomorphic to
$PG_{\mathcal{P}}(\Phi,R)$ for some ring $R$ that satisfy all sentences from
$T$ is axiomatizable, i.e. can be described by (possibly infinite) collection
of first order sentences.

###### Proof.

In case where $T$ consist of one sentence, the statement follows directly from
Theorem 2.6; moreover in this case the class in question can be describe by
one sentence. In general case we take each sentence in $T$, construct the
corresponding sentence for groups and take the collection of all such
sentences. Clearly, the groups in question must satisfy all these sentences.
Conversely, if a group $G$ satisfies all these sentences, then for every
sentence from $T$ there exists a ring $R$ such that $G\simeq
PG_{\mathcal{P}}(\Phi,R)$ and $R$ satisfy this sentence; but by Theorem 3.3
all these rings must be isomorphic; hence, they satisfy all the sentences from
$T$. ∎

######  Example 4.7.

The class of groups that are isomorphic to $PG_{\mathcal{P}}(\Phi,K)$, where
$K$ is a pseudo-finite field (of admissible characteristic) is axiomatizable
(for details about logical properties of groups over pseudo-finite fields see
[29]).

##  5\. Proof of Theorem 2.7

######  Lemma 5.1.

Let $\beta\in\Phi$, then there is a $(|\Phi|+1)$-ary formula
$\operatorname{IsElUnip}_{\beta}(g,g_{\alpha}\colon\alpha\in\Phi)$ such that
for any commutative ring $R$ such that $1/2\in R$ in case $\Phi=C_{2}$ and any
$g\in G_{\mathcal{P}}(\Phi,R)$ we have

$\left[G_{\mathcal{P}}(\Phi,R)\models\operatorname{IsElUnip}_{\beta}(g,x_{\alpha}(1)\colon\alpha\in\Phi)\right]\Leftrightarrow\left[\exists\xi\in
R\colon g=x_{\beta}(\xi)\right].$

###### Proof.

Follows from the proofs of [16, Propositions 3 and 4]. ∎

######  Lemma 5.2.

Let $\beta\in\Phi$, then there is a $(|\Phi|+3)$-ary formulas
$\operatorname{Add}_{\beta}(g_{1},g_{2},g_{3},g_{\alpha}\colon\alpha\in\Phi)$
and
$\operatorname{Mult}_{\beta}(g_{1},g_{2},g_{3},g_{\alpha}\colon\alpha\in\Phi)$
such that for any commutative ring $R$ such that $1/2\in R$ in case
$\Phi=C_{2}$ and any $\xi_{1}$,$\xi_{2}$,$\xi_{3}\in R$ we have

$\left[G_{\mathcal{P}}(\Phi,R)\models\operatorname{Add}_{\beta}(x_{\beta}(\xi_{1}),x_{\beta}(\xi_{2}),x_{\beta}(\xi_{3}),x_{\alpha}(1)\colon\alpha\in\Phi)\right]\Leftrightarrow\left[\xi_{1}+\xi_{2}=\xi_{3}\right]\text{,}$

and

$\left[G_{\mathcal{P}}(\Phi,R)\models\operatorname{Mult}_{\beta}(x_{\beta}(\xi_{1}),x_{\beta}(\xi_{2}),x_{\beta}(\xi_{3}),x_{\alpha}(1)\colon\alpha\in\Phi)\right]\Leftrightarrow\left[\xi_{1}\xi_{2}=\xi_{3}\right]\text{.}$

###### Proof.

Follows from the proof of [16, Theorem 4]. ∎

######  Lemma 5.3.

There is an $(|\Phi|+n^{2}+1)$-ary formula
$\operatorname{MatUpToCent}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi)$ such that for any commutative ring $R$ such
that $1/2\in R$ in case $\Phi=C_{2}$ and any $g$, $g_{i,j}\in
G_{\mathcal{P}}(\Phi,R)$, we have
$G_{\mathcal{P}}(\Phi,R)\models\operatorname{MatUpToCent}(g,g_{i,j}\colon
1\leqslant i,j\leqslant n,x_{\alpha}(1)\colon\alpha\in\Phi)$ if and only if
there are $\xi_{i,j}\in R$ such that the following holds:

  1. (1)

$g_{i,j}=x_{\alpha_{0}}(\xi_{i,j})$ for all $1\leqslant i,j\leqslant n$;

  2. (2)

The matrix $(\xi_{i,j})$ is in the image of $G_{\mathcal{P}}(\Phi,R)$ under
the representation $\varpi$;

  3. (3)

The element $g$ differs from the element represented by the matrix
$(\xi_{i,j})$ by multiplication by an element from the center of
$G_{P}(\Phi,R)$.

###### Proof.

Similar to the proof of Lemma 4.3. ∎

######  Lemma 5.4.

There is an $|\Phi|+(l+1)(n^{2}+1)$-ary formula
$\operatorname{Matrix}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant k\leqslant
l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant i,j\leqslant n)$
such that for any commutative ring $R$ such that $1/2\in R$ in case
$\Phi=C_{2}$ and any $g$, $g_{i,j}$,$g_{k}\in G_{\mathcal{P}}(\Phi,R)$ such
that the group $G_{\mathcal{P}}(\Phi,R))$ is $N$-boundedly generated by the
set $X_{g_{1},\ldots,g_{l}}$, we have
$G_{\mathcal{P}}(\Phi,R)\models\operatorname{Matrix}(g,g_{i,j}\colon
1\leqslant i,j\leqslant n,x_{\alpha}(1)\colon\alpha\in\Phi;\;g_{k}\colon
1\leqslant k\leqslant l;\;x_{\alpha_{0}}(\varpi(g_{k})_{ij})\colon 1\leqslant
k\leqslant l,\;1\leqslant i,j\leqslant n)$ if and only if there are
$\xi_{i,j}\in R$ such that the following holds:

  1. (1)

$g_{i,j}=x_{\alpha_{0}}(\xi_{i,j})$ for all $1\leqslant i,j\leqslant n$;

  2. (2)

The matrix $(\xi_{i,j})$ equals $\varpi(g)$.

###### Proof.

We build this formula as the conjunction of two parts.

In the first part, we use Lemma 4.1 and take the conjunction of
$\operatorname{IsElUnip}_{\alpha_{0}}(g_{i,j},g_{\alpha}\colon\alpha\in\Phi)$
for all $1\leqslant i,j\leqslant n$. So we ensure that
$g_{i,j}=x_{\alpha_{0}}(\xi_{i,j})$ for some $\xi_{i,j}\in R$.

Now let $\Omega$ be the set of all words of length $N$ in letters
$\\{c,d,x_{1},\ldots,x_{l}\\}$. To each $\omega\in\Omega$ we put in
correspondence a term in the language of groups build as follows: we take two
variables for each letter $c$ and one variable for each letter $d$, all
variables are distinct from each other and from the letters
$x_{1}$,$\ldots$,$x_{l}$. Now we replace each letter $c$ with the commutator
of corresponding variables, and each letter $d$ with the $m$-th power of the
corresponding variable.

The second part is constructed as a disjunction by the set $\Omega$ of the
formulas that says the following:

  1. (1)

the element $g$ can be represented by the term constructed from the
corresponding element $\omega\in\Omega$ as described above with elements
$g_{1}$,$\ldots$,$g_{l}$ being substituted as $x_{1}$,$\ldots$,$x_{l}$;

  2. (2)

the elements $g_{ij}$ satisfy the formula constructed using Lemmas 5.1 and 5.2
that essentially says that the matrix $(\xi_{ij})$ can be represented by the
same term as above with matrices
$(\xi_{ij}^{(1)})$,$\ldots$,$(\xi_{ij}^{(l)})$ being substituted as
$x_{1}$,$\ldots$,$x_{l}$, assuming $\varpi(g_{ij}^{(k)})=(\xi_{ij}^{(k)})$ for
$1\leqslant k\leqslant l$;

  3. (3)

for any variable that comes from the letter $c$ or $d$ in the corresponding
element $\omega\in\Omega$ the corresponding element used in item 1 differs
from the element represented by the corresponding matrix used in item 2 by an
element from the center of $G_{P}(\Phi,R)$ (this item uses Lemma 5.3).

Note that if $a_{1}$ differs from $a_{2}$ by a central element and $b_{1}$
differs from $b_{2}$ by a central element, then $[a_{1},b_{1}]=[a_{2},b_{2}]$
and $a_{1}^{m}=a_{2}^{m}$. Therefore, our formula does what required. ∎

Now we prove Theorem 2.7.

###### Proof.

Note that item (4) directly follows from items (1)–(3), so we will prove only
items (1)–(3).

We construct the formula $\varphi^{N}$ as a conjunction of formulas that
essentially says the following:

  1. (1)

$G$ is $N$-boundedly generated by the set $X_{g_{1},\ldots,g_{l}}$.

  2. (2)

The formulas $\operatorname{Add}_{\alpha_{0}}$ and
$\operatorname{Mult}_{\alpha_{0}}$ define binary operation on the set of such
elements $g$ that
$\operatorname{IsElUnip}_{\alpha_{0}}(g,g_{\alpha}\colon\alpha\in\Phi)$ holds;
and this set is a ring with respect to these operations.

  3. (3)

The ring $R$ from the item 2 is good.

  4. (4)

For any tuple $g_{i,j}$, $1\leqslant i,j\leqslant n$ such that
$\operatorname{IsElUnip}_{\alpha_{0}}(g_{i,j},g_{\alpha}\colon\alpha\in\Phi)$
holds and that satisfy the equations that define $G_{\mathcal{P}}(\Phi,-)$
(i.e. satisfy the formula that is obtained from these equations by rewriting
it in the language of groups using Lemma 4.2) there exists a unique $g$ such
that $\operatorname{Matrix}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant k\leqslant
l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant i,j\leqslant n)$
holds.

  5. (5)

For any $g$ there exists a unique tuple $g_{i,j}$, $1\leqslant i,j\leqslant n$
such that it satisfy the equations as above and
$\operatorname{Matrix}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant k\leqslant
l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant i,j\leqslant n)$
holds.

  6. (6)

For any
$g$,$g^{\prime}$,$g^{\prime\prime}$,$g_{i,j}$,$g_{i,j}^{\prime}$,$g_{i,j}^{\prime\prime}$,
if $\operatorname{Matrix}(g,g_{i,j}\colon 1\leqslant i,j\leqslant
n,g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant k\leqslant
l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant i,j\leqslant n)$,
$\operatorname{Matrix}(g^{\prime},g_{i,j}^{\prime}\colon 1\leqslant
i,j\leqslant n,g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon 1\leqslant
k\leqslant l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant l,\;1\leqslant
i,j\leqslant n)$ and
$\operatorname{Matrix}(g^{\prime\prime},g_{i,j}^{\prime\prime}\colon
1\leqslant i,j\leqslant n,g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon
1\leqslant k\leqslant l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant
l,\;1\leqslant i,j\leqslant n)$ hold and the matrix $(g_{i,j})$ is the product
of matrices $(g^{\prime}_{i,j})$ and $(g^{\prime\prime}_{i,j})$ in the sense
of operations defined by $\operatorname{Add}_{\alpha_{0}}$ and
$\operatorname{Mult}_{\alpha_{0}}$, then $g=g^{\prime}g^{\prime\prime}$.

  7. (7)

In the correspondence between elements and matrices defined by items 3 and 4,
matrices of elements $x_{\alpha}(1)$ correspond to the elements $g_{\alpha}$
(these are matrices with integer entries, and any particular integer is
definable in the language of rings); and for any $1\leqslant k\leqslant l$ the
matrix represented by the tuple $(g_{i,j}^{(k)})$ correspond to the element
$g_{k}$.

Now let us prove that the formula constructed above satisfies items (1)–(3) of
the Theorem 2.7.

Firstly, the implication from the right to the left in item (1) holds true by
construction of $\varphi^{N}$.

Secondly, item (3) also holds by construction.

Now let us prove the implication from the left to the right in item (1).
Assume that for some $g_{\alpha},g_{k},g_{ij}^{(k)}\in
G_{\mathcal{P}}(\Phi,R)$ we have

$G_{\mathcal{P}}(\Phi,R)\models\varphi^{N}(g_{\alpha}\colon\alpha\in\Phi;\;g_{k}\colon
1\leqslant k\leqslant l;\;g_{ij}^{(k)}\colon 1\leqslant k\leqslant
l,\;1\leqslant i,j\leqslant n)\text{.}$

By item (3), there exists an isomorphism $f\colon G_{\mathcal{P}}(\Phi,S)\to
G_{\mathcal{P}}(\Phi,R)$ for some ring $S$ such that

$\bullet$ $f(g_{\alpha})=x_{\alpha}(1)$ for all $\alpha\in\Phi$;

$\bullet$ $f(g_{ij}^{(k)})=x_{\alpha_{0}}(\xi_{ij}^{(k)})$ for all $1\leqslant
k\leqslant l$ for some $\xi_{ij}^{(k)}\in R$;

$\bullet$ $\varpi(f(g_{k}))$ is the matrix $(\xi_{ij}^{(k)})$ for all
$1\leqslant k\leqslant l$.

By Theorem 3.3 the rings $R$ and $S$ are isomorphic (here we use that they are
good). Now composing $f$ with the ring isomorphism from
$G_{\mathcal{P}}(\Phi,S)$ to $G_{\mathcal{P}}(\Phi,R)$, we obtained the
required automorphism.

It remains to prove the item (2). By item (1) it is enough to prove that there
exists a bi-interpretation of $R$ and $G_{\mathcal{P}}(\Phi,R)$ such that the
interpretation of $G_{\mathcal{P}}(\Phi,R)$ in $R$ is absolute and the
parameters for interpretation of $R$ in $G_{\mathcal{P}}(\Phi,R)$ are
$x_{\alpha}(1)$,$g_{k}$, and $x_{\alpha_{0}}(\xi_{ij}^{(k)})$, where
$\varpi(g_{k})=(\xi_{ij}^{(k)})$.

We interpret $G_{\mathcal{P}}(\Phi,R)$ in $R$ by an obvious way: elements of
$G_{\mathcal{P}}(\Phi,R)$ are matrices over $R$ that satisfy certain equations
and the multiplication is the multiplication of matrices.

Further we interpret $R$ in $G_{\mathcal{P}}(\Phi,R)$ using Lemmas 5.1 and
5.2.

It is easy to see that the set of pairs $(\xi,g)\in R\times
G_{\mathcal{P}}(\Phi,R)$ such that $\varpi(g)=x_{\alpha_{0}}(\xi)$ is a set of
points of a closed subscheme in $\mathbb{A}_{1}\times
G_{\mathcal{P}}(\Phi,-)$; hence the isomorphism between $R$ and its
interpretation inside $G_{\mathcal{P}}(\Phi,R)$ is defined by polynomial
formulas.

The isomorphism between $G_{\mathcal{P}}(\Phi,R)$ and its interpretation
inside $R$ can be defined using Lemma 5.4. ∎

######  Corollary 5.5.

Let $\Phi$ be an irreducible root system of rank $>1$, $\mathcal{P}$ be any
corresponding weight lattice, and $l$,$N$ be positive integers. Let $T$ be a
collection (possibly infinite) of first order sentences in the language of
rings such that $T$ includes the sentence "$1/2\in R$" in case
$\Phi=A_{2},B_{\ell},C_{\ell},F_{4}$, $G_{2}$ $\ell\geqslant 2$, and $T$
includes the sentence "$1/3\in R$" in case $\Phi=G_{2}$, and such that $T$
includes the sentence "there is a $l$-tuple $g_{1},\ldots,g_{l}\in G$ such
that the group $G$ is $N$-boundednly generated by the set
$X_{g_{1},\ldots,g_{l}}$" . Then the class of groups that are isomorphic to
$G_{\mathcal{P}}(\Phi,R)$ for some ring $R$ that satisfy all sentences from
$T$ is axiomatizable, i.e. can be described by (possibly infinite) collection
of first order sentences.

###### Proof.

Similar to Corollary 4.6. ∎

######  Example 5.6.

The class of groups that are isomorphic to $G_{\mathcal{P}}(\Phi,K)$, where
$K$ is a pseudo-finite field (of admissible characteristic) is axiomatizable.

##  6\. One bad example

In this section we will show the example of a Chevalley group with infinite
center, which is not bi-interpretable with the corresponding ring and which is
not elementary definable in the class of all groups.

Consider the Chevalley group $G_{\mathcal{P}}(A_{3},R)$, where $\mathcal{P}$
is the only suitable lattice that is neither root lattice nor weight lattice.
That is the special orthogonal group $\,\mathrm{SO}\,(6,R)$, i.e. the group of
$6\times 6$ matrices that preserves the quadratic form
$x_{1}x_{6}+x_{2}x_{5}+x_{3}x_{4}$, and have trivial Dickson invariant. The
corresponding elementary subgroup will be denoted by $EO(6,R)$.

The toric subgroup $TO(6,R)\leqslant\,\mathrm{SO}\,(6,R)$ consists of diagonal
matrices of shape ${\rm
diag}(\xi_{1},\xi_{2},\xi_{3},\xi_{3}^{-1},\xi_{2}^{-1},\xi_{1}^{-1})$. where
$\xi_{i}\in R^{*}$.

######  Lemma 6.1.

The element ${\rm
diag}(\xi_{1},\xi_{2},\xi_{3},\xi_{3}^{-1},\xi_{2}^{-1},\xi_{1}^{-1})$ belong
to $EO(6,R)$ if and only if $\xi_{1}\xi_{2}\xi_{3}\in(R^{*})^{2}$.

###### Proof.

The simply connected Chevalley group $G_{\mathrm{sc}(A_{3},R)}$ is the group
$\,\mathrm{SL}\,(4,R)$. The element from $TO(6,R)$ belong to $EO(6,R)$ if and
only if it belong to the image of the toric subgroup of $\,\mathrm{SL}\,(4,R)$
under the natural map ${\,\mathrm{SL}\,(4,R)}\to{\,\mathrm{SO}\,(6,R)}$.
Indeed, the toric elements of $\,\mathrm{SL}\,(4,R)$ are elementary; hence so
are their images. Conversely, every elementary element belongs to the image of
$\,\mathrm{SL}\,(4,R)$ and the preimage of $TO(6,R)$ is the torus.

Now the map on tori is given by

${\rm diag}(\zeta_{1},\zeta_{2},\zeta_{3},\zeta_{4})\mapsto{\rm
diag}(\zeta_{1}\zeta_{2},\zeta_{1}\zeta_{3},\zeta_{1}\zeta_{4},\zeta_{3}\zeta_{4},\zeta_{2}\zeta_{4},\zeta_{2}\zeta_{3})\text{.}$

Since $\zeta_{1}\zeta_{2}\zeta_{3}\zeta_{4}=1$, it follows that the product of
the first three diagonal entries of the element from the image is equal to
$\zeta_{1}^{2}\in(R^{*})^{2}$. Conversely, suppose we have
$\xi_{1}\xi_{2}\xi_{3}=\zeta_{1}^{2}$ for some $\zeta_{1}\in R^{*}$, Then we
set $\zeta_{2}=\xi_{1}\zeta_{1}^{-1}$, $\zeta_{3}=\xi_{2}\zeta_{1}^{-1}$,
$\zeta_{4}=\xi_{3}\zeta_{1}^{-1}$; hence we have
$\zeta_{1}\zeta_{2}\zeta_{3}\zeta_{4}=1$. and

${\rm
diag}(\xi_{1},\xi_{2},\xi_{3},\xi_{3}^{-1},\xi_{2}^{-1},\xi_{1}^{-1})={\rm
diag}(\zeta_{1}\zeta_{2},\zeta_{1}\zeta_{3},\zeta_{1}\zeta_{4},\zeta_{3}\zeta_{4},\zeta_{2}\zeta_{4},\zeta_{2}\zeta_{3})\text{.}$

∎

######  Lemma 6.2.

We have

$\,\mathrm{SO}\,(6,\mathbb{F}_{2}[t])=EO(6,\mathbb{F}_{2}[t])$

###### Proof.

Follows from [57]. ∎

######  Proposition 6.3.

Set $R=\mathbb{F}_{2}[t,\varepsilon]/(\varepsilon^{2})$. Then

$\,\mathrm{SO}\,(6,R)=EO(6,R)\times(\mathbb{Z}/2\mathbb{Z})^{\infty}\text{.}$

###### Proof.

The center of $\,\mathrm{SO}\,(6,R)$ consists of scalar matrices with the
scalar being the square root of one. Thus the center is isomorphic to
$\mu_{2}(R)=(\mathbb{Z}/2\mathbb{Z})^{\infty}$. Clearly, the center commutes
with $EO(6,R)$. Therefore, it remains to show that $EO(6,R)\cap
Z(\,\mathrm{SO}\,(6,R))=e$, and that $\,\mathrm{SO}\,(6,R)$ is generated by
$EO(6,R)$ and $Z(\,\mathrm{SO}\,(6,R))$.

Let us prove the first claim. Let $g\in EO(6,R)\cap Z(\,\mathrm{SO}\,(6,R))$.
Since $g\in Z(\,\mathrm{SO}\,(6,R))$, we have $g={\rm diag}(\xi,\ldots,\xi)$,
with $\xi^{2}=1$. By Lemma 6.1 we have $\xi^{3}\in(R^{*})^{2}=\\{1\\}$; hence
$\xi=\xi^{3}=1$.

Let us prove the second claim. Clearly, $\,\mathrm{SO}\,(6,R)$ is generated by
$\,\mathrm{SO}\,(6,\mathbb{F}_{2}[t])$ and the congruence subgroup
$\,\mathrm{SO}\,(6,\varepsilon R,R)$. By Lemma 6.2 the group
$\,\mathrm{SO}\,(6,\mathbb{F}_{2}[t])$ is contained in the elementary
subgroup. Since $\varepsilon$ is nilpotent, it follows that the congruence
subgroup $\,\mathrm{SO}\,(6,\varepsilon R,R)$ is contained in the big Gauss
cell; hence it is contained in $TO(6,R)EO(6,R)$. Therefore, it remains to show
that any element of $TO(6,R)$ is a product of an elementary element and a
central element. Note that all the elements of $R^{*}$ are square roots of
one. So given an element ${\rm
diag}(\xi_{1},\xi_{2},\xi_{3},\xi_{3},\xi_{2},\xi_{1})\in TO(6.R)$ we can
multiply it by the central element ${\rm
diag}(\xi_{1}\xi_{2}\xi_{3},\ldots\xi_{1}\xi_{2}\xi_{3})$, and by Lemma6.1 the
result will be in the elementary subgroup. ∎

######  Corollary 6.4.

For $R=\mathbb{F}_{2}[t,\varepsilon]/(\varepsilon^{2})$ the group
$\,\mathrm{SO}\,(6,R)$ is not bi-interpretable with $R$.

###### Proof.

Assume the converse. Then it is easy to see that the subgroup of
$\,\mathrm{Aut}\,(\,\mathrm{SO}\,(6,R))$ that stabilises the corresponding
parameters must be isomorphic to a certain subgroup of $\,\mathrm{Aut}\,(R)$.
However, the group $\,\mathrm{Aut}\,(R)$ is countable (because $R$ is
countable and the automorphism is uniquely defined by the images of $t$ and
$\varepsilon$); and it follows from Proposition 6.3 that for any finite subset
of $\,\mathrm{SO}\,(6,R)$ the subgroup of
$\,\mathrm{Aut}\,(\,\mathrm{SO}\,(6,R))$ that stabilises the elements of this
set is uncountable. This is a contradiction. ∎

######  Corollary 6.5.

For $R=\mathbb{F}_{2}[t,\varepsilon]/(\varepsilon^{2})$ there is the group $G$
that is elementary equivalent to $\,\mathrm{SO}\,(6,R)$, but not isomorphic to
any Chevalley group over any ring.

###### Proof.

Set $G=EO(6,R)\times(\mathbb{Z}/2\mathbb{Z})^{I}$, where $I=[0,1]$. It follows
from Proposition 6.3 that $G$ is elementary equivalent to
$\,\mathrm{SO}\,(6,R)$.

It is easy to see that an infinite Chevalley group over a ring must have the
same cardinality as its derived subgroup. Therefore, $G$ is not a Chevalley
group. ∎

Acknowledgements. Our sincere thanks go to Eugene Plotkin for very useful
discussions regarding various aspects of this work and permanent attention to
it.

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