# Graphyne as the anode material of magnesium-ion batteries: ab initio study

E. Shomalia I. Abdolhosseini Sarsari*a,b F. Tabatabaeia MR. Mosaferia N.
Serianic a) Department of Physics, Isfahan University of Technology, Isfahan,
84156-83111, Iran
b) Computational Physical Sciences Research Laboratory, School of Nano-
Science, Institute for Research in Fundamental Sciences (IPM), P.O. Box
19395-5531, Tehran, Iran
c) The Abdus Salam ICTP, Strada Costiera 11, 34151 Trieste, Italy

###### Abstract

Graphyne, a single atomic layer structure of the carbon six-member rings
connected by one acetilenic linkage, is a promising anode of rechargeable
batteries. In this paper, a first-principle study has been carried out on
graphyne as a new candidate for the anode material of magnesium-ion batteries,
using density functional theory calculations. The main focus is on the
magnesium adsorption on graphyne surface. The structural properties such as
adsorption height and energy, the most stable adsorption sites, the Band
structure and DOS of the pristine graphyne the diverse Mg-decorated graphyne
structures, and energy barrier against Mg diffusion are also calculated. As a
consequence of the band structure and DOS of graphyne structures, it is found
that the pristine graphyne and the Mg-decorated graphyne structures show a
semiconducting nature and metallic behavior, respectively. Moreover, the
migration behavior of Mg on graphyne for the main diffusion paths is
determined by Nudged Elastic Band (NEB) method.

Rechargeable batteries, Graphyne, Mg-decorated graphyne, DOS spectrum, Band
structure, NEB method

##  I INTRODUCTION

Nowadays, with the acceleration of global energy demands, the depletion of
fossil fuels, and their environmental impacts; interest in renewable energies
has increased considerably over the last decades. In the meantime, efforts to
develop energy storage devices as a renewable energy technology become more
prevalent. One of the most popular energy storage system, lithium-ion
batteries, have received remarkable attention. A battery is an electrochemical
device which converts chemical energy directly into electrical energy and
consists of three components, i.e., the anode (negative electrode), the
cathode (positive electrode), and electrolyte. During discharge, lithium ions
depart from the anode to the cathode through the non-aqueus liquid electrolyte
and carry the current which leads to passing electrons around the external
circuit to power various systems. When the battery is charging, an external
electrical power source make the current to transfer in the reverse direction
and lithium ions migrate to the anode across the electrolyte Xianxia Yuan
(2011); Scrosati and Garche (2010).
Recently, rechargeable batteries have entered our daily life and have been
widely applied in portable and telecommunication electronic devices,
transportation, and household-type items. Basically, rechargeable batteries
have a significant role in the development of electronic devices such as;
hybrid electric vehicles, notebook, mobile, flash lights, radios, watches,
etc. They became very common because of their high energy density, enhanced
rate capabilities, low self-discharging, long cycle life, excellent safety
features, and etc. A great deal of attention has been paid to Li-ion battery
technologies which are conquering the market, but due to having a better
chance of reaching maturity, we decide to apply a different material for this
technology Meng and Arroyo-de Dompablo (2012)Etacheri  _et al._ (2011).
Since the batteries coming into existence, graphite-based materials have been
proposed for anodes. Graphite, the most stable form of carbon, has a layered,
planar structure. The carbon atoms in each layer, have been placed in a
honeycomb lattice and the individual layers are called graphene. Both graphite
and graphene are allotropic forms of carbon. It is well known that graphite
has a low capacity and that is not good enough for providing endless needs of
the current society. Graphene, a two dimensional (2D) sheet of sp2-hybridized
carbon, has a higher capacity than graphite, mainly because lithium can be
adsorbed on both sides of graphene, thus lithium has a larger space to diffuse
and it can be led to higher rate of diffusion Wu  _et al._ (2011)Liu  _et al._
(2016). Graphene has many extraordinary properties which have attracted the
attention of all researchers for a while. It is very strong and conducts heat
and electricity efficiently Neto  _et al._ (2009). However, nowadays
scientists are investigating that “Could graphyne be better than
graphene?”Graphyne (Gy), a novel carbon allotrope, is the single atomic layer
structure of the carbon six-member rings connected by one acetylenic linkage
($-C\equiv C-$), which introduces a diverse range of optical and electronic
properties. Moreover, the thing that makes graphyne special is the presence of
both sp and sp2 hybridized carbon atoms. Also, it can be said that graphyne
has two different Dirac cones lying slightly above and below the Fermi level
which means it is self-doped and naturally includes conducting charge carriers
(electrons and holes) without any external doping. Therefore, graphyne is a
promising semiconductor which can be used in electronic devices Sarkar  _et
al._ (2015)Peng  _et al._ (2014).
In this paper, we have focused on graphyne and Mg-decorated graphyne-based
systems. Magnesium, an inexpensive and environmentally friendly metal with
great raw material abundance and good operational safety, is a relatively new
option for use as an anode material in rechargeable batteries Yang  _et al._
(2011). (To begin, we examined stability of graphyne by calculating the phonon
spectrum using FHI-aims package). Then, we have conducted our study finding
possible sites of Mg adsorption on graphyne by calculating the adsorption
energy and its corresponding adsorption height. Moreover, we examined the
electronic structure of graphyne and Mg-decorated graphyne systems at
different sites by calculating the band structure and DOS spectrum. Finally,
the main target of this work; study of in-plane and out-plane diffusion paths
of Mg on a graphyne layer and out-plane diffusion pathway of Mg in bulk
graphyne, corresponding energy curves of adsorption sites of Mg on graphyne
monolayers, and corresponding transition states (TS); was determined using
Nudged Elastic Band (NEB) method.

##  II Computational Details

Our electronic structure calculations and geometry optimizations have been
performed in the framework of density functional theory (DFT) by using plane
wave pseudo-potential as well as numerical orbital - full potential
techniques, implemented in the Quantum Espresso Giannozzi  _et al._ (2009) and
FHI-aims Blum  _et al._ (2009) computational packages, respectively. We used
GGA-PBE exchange-correlation functional Perdew  _et al._ (1996). The
7$\times$7$\times$1 Monkhorest-pack scheme was used for sampling first
Brillouin zone. A vacuum thickness of about 22 Bohr was utilized to avoid
interaction of adjacent molecules. Energy cutoffs of 35 Ry and 350 Ry, were
used for plane wave expansion of wave functions and electron density. The
Nudged Elastic Band (NEB) method Caspersen and Carter (2005), implemented in
Quantum Espresso, was also used to determine the reaction path. In this
method, a path with the lowest energy is determined for the reaction then a
string of images of the system are created. Finally, the images are connected
together through the hypothetical springs with the same spring constant. It
results in an energy for each image. In this research, minimum Energy Path was
calculated as a function of adsorption sites of Mg on Gy monolayer by
considering 59 and 19 images for in-plane and out-plane diffusion and 23
images for diffusion pathway of Mg in bulk graphyne, respectively. The vdW-DF,
non-local correlation functional, proposed by Dion et al. Dion  _et al._
(2004) utilized to calculate distances between layers and distance between ion
and layer.

One of the most important issues in condensed matter physics and in
theoretical chemistry is the determination of the reaction path. Many
different methods have been presented for finding Minimum Energy Paths (MEPs)
and saddle points like the Drag method, the NEB method, the CPR method, the
Ridge method, the DHS method, and the Dimer method; which the Nudged Elastic
Band (NEB) method was used in this study. In the NEB method, which initial and
final states of the reaction are known, a string of images of the system are
created and connected together through the hypothetical springs with the same
spring constant from the reaction configuration, R, to the product
configuration, P. In fact, images of the system are the intermediate
configurations of system. Afterward, an optimized algorithm is applied to
relax the images and put them in the MEP. By using NEB method, it is possible
to determine Minimum Energy Path (MEP), saddle point energy (maximum potential
energy along the MEP), activation energy barrier, estimated transition rate
and also the configuration of atoms during the transition Henkelman  _et al._
(2002) Jónsson  _et al._ (1998).

##  III RESULTS AND DISCUSSIONS

To examine graphyne as a reliable anode material of magnesium-ion battery, we
not only look at the pristine graphyne and its properties, but also focus on
the adsorption behavior of the magnesium atoms onto the graphyne surface. We
first relaxed a single-layer graphyne, which is displayed in Fig.2. The bond
length between the sp- and sp2-hybridized C atoms are 1.22 (Å) and 1.42 (Å),
respectively and the C–C bond length that connect sp- and sp2-hybridized C
atoms is 1.39 (Å) which is in agreement with other researchers Zhang  _et al._
(2011).

Figure 1:  Optimized structure of single-layer graphyne. The blue spheres
represent C atoms.

###  III.1 Mg adsorption on graphyne surface

To survey the adsorption behavior, we considered the most stable locations for
Mg decoration in graphyne. The Mg atom can adsorb above the centers of large
and small hexagons which consisting of 12 and 6 carbon atoms, respectively and
they are denoted as H- and h-sites hereafter. Another adsorption site for Mg
named H-beside-site, H-bond-site, and h-bond-site in which Mg atom stands
right above one of the corners of large hexagons, right above one of the
carbon bonds of large hexagons, right above one of the carbon bonds of small
hexagons, respectively. The different Mg-decorated graphyne structures are
shown in Fig. 2.

We turn to investigate the electronic properties of graphyne with Mg atom in
different positions. Table LABEL:Table1 summarizes the calculated binding
energy, $(E_{b})$, the distance between Mg and graphyne, $(d_{Mg-graphyne})$,
and the charge transfer of the Mg atom, where $E_{b}$ and $(d_{Mg-graphyne})$
are defined by equation (1) and van der Waals correctionDion  _et al._ (2004),
respectively. The $d_{Mg-graphyne}$ in h-bond-site, H-bond-site and h-site
structures are 4.15(Å), 4.02(Å), and 4.07(Å), respectively, while in the case
of the H-site structure, the distance between Mg and graphyne decreases to
1.34(Å). Among the four structure, the H-site is the most favorable due to
more negative binding energy in comparison to others. The charge transfer from
Mg atom to graphyne, is calculated by Bader analysisHenkelman  _et al._
(2006). The results in LABEL:Table1 show that as the Mg distance decreases the
charge transfer increase. Mg atom in H-site structure transfers 1.35 electrons
to the substrate and More than others. Which would be responsible for the
greater negative binding energy and consequently more stable structure.

Table 1: Calculated binding energies $(E_{b})$, distances between Mg and graphyne $d_{Mg-graphyne}$ and the charge transfer of Mg atom Structures | $E_{b}(eV)$ | distance(Å) | $\Delta Q|e|$
---|---|---|---
h-bond-site | -0.47 | 4.15 | 0.07
H-bond-site | -0.48 | 4.02 | 0.08
h-site | -0.47 | 4.07 | 0.07
H-site | -0.87 | 1.34 | 1.35

$\displaystyle E_{b}=(E_{Gy}+nE_{Mg}-E_{Mg-Gy})/n$ (1)

To approach reality, a supercell containing two graphyne layers with an AB
stacking sequence was also considered. The interlayer spacing in graphyne was
set to be 3.45 (Å), which has been recognized in the range of the most stable
interlayer distances Narita  _et al._ (2000). The top view of bulk graphyne
with an AB stacking sequence is shown in Fig. 5-a . The Mg atom can be
intercalated into the bulk graphyne in two sites, which are named HH- and hH-
sites. In HH-site the Mg atom is take placed between centers of large hexagons
of two adjacent layers and in hH-site the Mg atom is placed between centers of
a small hexagon and a large hexagon of two adjacent layers. The binding energy
of HH- and hH-sites are … and …, respectively. The corresponding adsorption
height for HH-site is about 1.8 (Å), in which the Mg atom is nearly in the
middle of the spacing of the two adjacent layers. In the hH-site, Mg atom is
closer to the large hexagon with a distance of 1.24 (Å). Comparison reveals
that the intercalated Mg prefers to occupy an H-site rather than other sites,
due to its larger binding energy.
It is worth noting that due to the existence of a van der Waals force between
the Mg atom and the graphyne surface, the van der Waals effects have been
considered in the above-mentioned calculations.

Figure 2: The band structure of a) Gy, b) h-bond-site, c)H-bond-site, d)
h-site and e) H-site. The Optimized structure of single-layer graphyne labled
with prime. The blue and red spheres represent C atoms and Mg atom,
respectively.

###  III.2 Electronic properties

The band structure and density of state (DOS) have been calculated to study
the electronic properties of pristine graphyne and Mg adsorbed graphyne layer
systems. The band structure, and Dos of the pristine graphyne and the
different Mg-decorated graphyne structures are shown in Fig. 2. For pristine
graphyne, the band structure and Dos imply a semiconducting behavior with the
direct band gap of 0.358 (eV) due to the same location of the valence band
maximum (VBM) and conduction band minimum (CBM) of graphyne. This gap is in
good agreement with the results of 0.38 (eV) for Na-decorated graphyne
batteries from other DFT calculations. Moreover, it can be seen from the band
structure and Dos of the Mg-decorated graphyne structures that there is a
comparable downward shift of the conduction band for each structure (and,
alternatively, an upward shift of the Fermi level), indicating that the Mg
adsorption makes graphyne metallic. These results are consistent with the
density of state of all structures considered in this paper. It is noteworthy
that the downward shift of the conduction band for H-site is the highest among
all structures considered in this research.

The calculated band structure and density of state of graphyne by replacing Mg
atom on the top of the large and small hexagons (H and h sites) are presented
in Fig. 2 which exhibit an overlapping between conduction band and Fermi
energy. All the other Mg-decorated graphyne structures show the same metallic
nature but the amount of overlapping between the conduction band and Fermi
energy is different Sarkar  _et al._ (2015).
In order to determine the constituents of the electronic band, the projected
density of states was calculated for graphyne with different Mg-decoration.
The state near the Fermi level are mainly dominated by Mg s orbital in h-bond-
site, H-bond-site, and h-site. Decreasing distance between Mg and substrate in
H-site structure results in the Mg s orbital contribution reduction in the
Fermi state. It is found that the Mg p and s orbitals have influenced in Fermi
level, But the graphyne states play a very important role in the Fermi state.
The charge transfer on the substrate in this decoration is extremely more than
others.

There is a large difference in electronegativity between Mg and C atoms which
cause a charge transfer from Mg to C atoms and thus the metallic behavior can
be seen, that is an important characteristic for some applications such as
electrode materials in rechargeable batteries Seyed-Talebi  _et al._ (2015)
Liu  _et al._ (2015).

###  III.3 Mg diffusion in Graphyne

Considering the significance of Mg mobility on graphyne monolayer, the
diffusion path is calculated on the way to the most stable adsorption sites
using the Nudged Elastic Band (NEB) method. We investigated Mg migration
behaviors for two main diffusion paths; in-plane diffusion paths (h1-site to
H1-site, H1-site to H2-site, H2-site to h2-site) and out-plane diffusion paths
(across the hollow of H-site and h-site) as depicted in Fig. 3, respectively.
Since the calculation of energy curves is meaningful and very determinative on
battery applications, the corresponding Minimum Energy Paths (MEPs) are also
shown as a function of adsorption sites of Mg on graphyne monolayer and two
layers in Fig. 4. Moreover, the Mg migration behavior in a supercell
containing two graphyne layers with an AB stacking sequence, for the main
diffusion path (Hh-hH-HH1-HH2) was investigated which is shown in Fig. 5.

Figure 3:  The schematic drawings of a,b) in-plane diffusion paths
(h1-H1-H2-h2) Side image and image from above and two out-plane diffusion
paths c) across the hollow of H-site d) across the hollow of h-site

Figure 4:  Mg migration behaviors a) in-plane diffusion paths (h1-H1-H2-h2)
and two out-plane diffusion paths b) across the hollow of H-site c) across the
hollow of h-site

Figure 5:  The schematic drawings of a,b) out-plane diffusion pathway of Mg in
bulk graphyne (Hh-hH-HH-HH) Side image and image from above and b) the
corresponding energy profile as a function of adsorption sites.

The results from Fig. 4 indicate that the activation energy barrier of Mg
diffusion along the h-site to H-site direction is identified to be 0.072 (eV)
and the transition state (TS) is located on the image. 9; where is near the
top of the midpoint of the C(sp2)-C(sp2) bond between two adjoining hollows of
large and small hexagons but more leaning to the large hexagon. Moreover, Mg
migration between two neighboring H-site is as high as 0.835 (eV), more than
that in h-site to H-site diffusion path, and it can be seen that the
transition state (TS) is specified near the top of midpoint of the
C(sp)$\equiv$C(sp) bond between two adjacent hollows of large hexagons; where
is known by image.30. As the reaction pathway is followed, there is a high
energy barrier of about 0.783 (eV) for Mg diffusion along the H-site to h-site
path; which is reside on image.51 and represent the energetic favorability of
the H-site over the h-site. In addition, the h1-site to H-site diffusion path
with the lowest energy barrier of about 0.072 (eV) is mainly dominated in the
in-plane Mg migration on Gy.
In the following, the energy profiles of two typical out-plane diffusion paths
are also studied, which can be seen in Fig. 4 and 4, respectively. In out-
plane diffusion paths Mg migrates from one side of a graphyne layer to another
side along the direction perpendicular to the graphyne plane. The energy
barrier for Mg diffusion through a large and small hexagon is about 0.32 (eV)
and 15.68 (eV), respectively. Therefore, Mg can easily migrate through the
large hexagon by overcoming a relatively small energy barrier of 0.32 (eV).
Moreover, it can be seen that the out-plane diffusion of Mg across the small
hexagon is energetically and kinetically prohibited because of such a high
energy barrier. Eventually, the large hexagon (H-site), formed by the sp2\-
and sp-hybridized C atoms in graphyne, is recommended for both Mg adsorption
and out-plane Mg migration. In addition, according to Fig. 5 the energy
barrier of Mg diffusion along the Hh-site to hH-site and hH-site to HH1-site
are calculated to be 0.927 (eV) and 1.001 (eV), respectively, whereas that
from HH1-site to hH-site are more than 1.001 (eV). Furthermore, there is a
large barrier energy of 1.829 (eV) from HH1-site to HH2-site, which is
energetically unfavorable. But the energy barrier along HH2-site to HH1-site
is about 0.284 (eV), that is related to the reduction of the density of
graphyne layers in this area Zhang  _et al._ (2011) Xu  _et al._ (2016).

##  IV CONCLUSION

In this work, density functional calculations were employed to systematically
investigate the pristine graphyne and Mg-decorated graphyne structures as a
new anode material of magnesium-ion batteries, using Quantum Espresso package.
The obtained results from calculated adsorption energy and adsorption height
determined the most stable adsorption sites of magnesium on graphyne.
Moreover, exploring on the DOS and band structure of the pristine graphyne and
Mg-decorated graphyne structures, it was observed that the Mg adsorption makes
graphyne metallic. we calculated a direct band gap of 0.358 (eV) for pristine
graphyne, which is in good agreement with the results of 0.38 (eV) for Na-
decorated graphyne batteries from other DFT calculations Sarkar  _et al._
(2015). Also, we determined the Mg-migration behaviors on graphyne monolayer
on the way to the most stable adsorption sites for two main in-plane and out-
plane diffusion paths, in addition, an out-plane diffusion pathway of Mg in
bulk graphyne along the Hh-hH-HH1-HH2 direction, using Nudged Elastic Band
method. The activation barrier of this process for in-plane diffusion path
along h1-H1, H1-H2, and H2-h2 directions are 0.072 (eV), 0.835 (eV), and 0.783
(eV), respectively. Besides, the energy barrier for Mg out-plane diffusion
through H-site and h-site is about 0.32 (eV) and 15.68 (eV), respectively.
Furthermore, for diffusion of Mg in bulk graphyne along the Hh-hH, hH-HH1, and
HH1-HH2 paths, energy barriers are calculated to be 0.927 (eV), 1.001 (eV),
and 1.829 (eV), respectively. In summary, investigation of details about Mg
intercalation and diffusion into graphyne monolayer is highly desirable to
shed light on this topic and to achieve the better understanding of battery
technology.

##  V ACKNOWLEDGMENTS

The authors gratefully acknowledge the Sheikh Bahaei National High Performance
Computing Center (SBNHPCC) for providing computing facilities and time.
SBNHPCC is supported by scientific and technological department of
presidential office and Isfahan University of Technology (IUT).

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On verbally closed subgroups of free solvable groups

V. A. ROMAN’KOV111This research is supported by program of fundamental
investigations of SB RAS: I. 1.1.4, project 0314-2019-0004, E. I.
TIMOSHENKO222This research is supported by RFBR, project18-01-00100a

Abstract: We study verbally closed subgroups of free solvable groups. A number
of results is proved that give sufficient conditions under whose a verbally
closed subgroup is turned to be a retract and so algebraically closed of the
full group.

INTRODUCTION

Algebraically closed objects play an important role in the modern algebra. In
this paper we study verbally and $l$-verbally closed subgroups of free
solvable groups. We establish a connection of them with retracts and
algebraically closed subgroups.

Let ${\cal K}$ be a class of algebraic structures of language $\mathbf{L}$. A
structure $A\in{\cal K}$ is said to be algebraically closed in ${\cal K}$ if
and only if for any positive $\exists$-sentence $\varphi(x_{1},...,x_{n})$ in
$\mathbf{L}$ with constants in $A,$ if $\varphi(x_{1},...,x_{n})$ is true in a
structure $B\in{\cal K},$ that has $A$ as a substructure, then
$\varphi(x_{1},...,x_{n})$ is true in $A.$ See for instances, [1], [2].

An interesting specific case arises when we take as ${\cal K}$ the class
sub$(B)$ of all substructures of a structure $B.$ Moreover, we can take as
${\cal K}$ a pair $\\{A,B\\},$ where $A$ is a substructure of $B.$ Then we can
say about algebraic closeness of $A$ in $B.$

Now let $\mathbf{L}$ be the group-theoretical language and ${\cal K}$ be a
class of groups. In this case, the notion above can be presented in pure
group-theoretical terminology. For groups $H$ and $G$ we write $H\leq G,$ if
$H$ is a subgroup of $G,$ and we call $G$ extension of $H.$ By an equation
$w(x_{1},...,x_{n};H)=h,h\in H,$ in variables (unknowns) $x_{1},...,x_{n}$
with constants in $H$ we mean an expression in which $w(x_{1},...,x_{n};H)$ is
a group word in $x_{1},...,x_{n}$ and constants in $H.$ We say that an
equation $w(x_{1},...,x_{n};H)=h$ has a solution (is solvable) in $H$ (or more
generally, in an extension $G\geq H$) if there is a tuple of elements
$g_{1},...,g_{n}$ of $H$ (or more generally of $G$) if after a substitution
them instead of $x_{1},...,x_{n},$ respectively, we have the true equality
$w(g_{1},...,g_{n};H)=h.$ An equation of the form $w(x_{1},...,x_{n})=h$, in
which the left-hand side does not include constants, but only variables, is
called, split equation.

In a natural way we define a system of equation and a solution  of it.

A notion of an algebraically closed subgroup of a group is rewritten as
follows.

######  Definition 1.

Let $G$ be a group. A subgroup $H\leq G$ is called algebraically closed in
$G,$ if any finite system of equations with constants in $H$ that has a
solution in $G$ has a solution in $H.$

In [3], the following definition was given.

######  Definition 2.

Let $G$ be a group. A subgroup $H\leq G$ is called verbally closed in $G,$ if
for any group word $w(x_{1},...,x_{n})$ in variables $x_{1},...,x_{n}$ without
constants and each $h\in H$ a split equation

$w(x_{1},...,x_{n})=h,$ (1)

that has a solution in $G,$ is solvable in $H.$

This notion, more weak than algebraic closeness, meanwhile corresponds to much
more strong notion of retract.

######  Definition 3.

Retract of a group $G$ is a subgroup $H\leq G$ such that there is an
endomorphism $\rho:G\rightarrow H$ that is identical on $H.$ Such an
endomorphism $\rho$ is called retraction.

Let $F_{r}$ be a free group of rank $r.$ Obviously, each free factor of
$F_{r}$ is retract. For $r\geq 2$, $F_{r}$ admits retracts that are not free
factors. For example, any element of the form $g=f_{1}u$ of $F_{r}$ with basis
$\\{f_{1},...,f_{r}\\}$ where $u$ belongs to the normal closure of
$f_{2},...,f_{r},$ generates a cyclic retract $R$. In this case, a retraction
$\rho:F_{r}\rightarrow R,$ is defined by map $f_{1}\mapsto g,f_{j}\mapsto 1$
for $j=2,...,r,$ but a cyclic group generated by $g,$ is a free factor of
$F_{r}$ if and only if $g$ is primitive, i.e., it can be included in some
basis of $F_{r}.$ If $r=2$ and $u$ lies in the commutant of $F_{2}$, then
$g=f_{1}u$ is primitive if and only if it conjugates to $f_{1}$. This
assertion is followed from a classical Nielsen’s result, that every
automorphism of $F_{2}$ identical modulo commutant is inner. Element
$g=f_{1}f_{2}^{-1}f_{1}^{-1}f_{2}f_{1}$ of $F_{2}$ generates retract but not a
free factor.

######  Remark 1.

If $H$ is a retract of $G$ then $H$ is algebraically closed in $G$. Indeed, to
obtain a solution in $H$ of an equation $w(x_{1},...,x_{n};H)=h$ from a
solution $x_{i}=g_{i}\,(i=1,...,n)$ in $G$, it is sufficient to apply a
retraction $\rho:G\rightarrow H.$ Then $w(g_{1},...,g_{n},H)=h$ implies
$\rho(w(g_{1},...,g_{n});\rho(H))=w(\rho(g_{1}),...,\rho(g_{n});H)=\rho(h)=h.$
Obviously that any algebraically closed subgroup $H$ of $G$ is verbally closed
in $G$. At the same time, among these three concepts there are no two
equivalent one to other. In thesis [11], there are examples of algebraically
closed subgroups that are not retracts, as well as examples of verbally closed
subgroups that are not algebraically closed.

In [4], the author proves that intersection of a finite set of retracts of a
free group $F_{r}$ is retract. In [5], there is a question 17.19 (the same as
the question F11 in [6]): If $H$ is a finitely generated subgroup of $F_{r}$
and $R$ is a retract of $F_{r},$ is it followed that the intersection $R\cap
H$ is retract of $H$? Recently this question was answered negatively in [7].

Thus the concept of verbal closeness is weaker than algebraic closeness.
However in [3] the authors proved that any verbally closed subgroup $H$ of a
free group $F_{r}$ ($r\in\mathbb{N}$) is a retract of $F_{r}$. It follows that
the sets of verbally closed subgroups, algebraically closed subgroups and
retracts of $F_{r}$ coincide.

In [8], a similar statement was shown for free nilpotent groups. Let $N_{r,c}$
denote a free nilpotent group of a finite rank $r$ and nilpotency class $c.$
Then the following properties of a subgroup $H\leq N_{r,c}$ are equivalent one
to other: verbal closeness, algebraic closeness, the property ”to be a
retract” and the property ”to be a free factor” with respect to the variety
$\mathfrak{N}_{c}$ of all nilpotent groups of nilpotency class $\leq c.$

In the series of papers [9], [10], [11], [12] and [13] verbal closeness was
studied. The following property was introduced. A group $H$ is a strongly
verbally closed if $H$ is an algebraically closed subgroup in any group
containing $H$ as a verbally closed subgroup. The number of results was
presented showing that many groups are strongly verbally closed.

Now we introduce the following new notion.

######  Definition 4.

Let $G$ be a group. For any $l\in\mathbb{N},$ a subgroup $H\leq G$ is said to
be $l$-verbally closed in $G$ if every system of $l$ split equations on any
$n$ variables

$w_{i}(x_{1},...,x_{n})=h_{i},h_{i}\in H,i=1,...,l,$

that has a solution in $G,$ is solvable in $H$.

In the case $l=1$ this notion means verbal closeness.

######  Remark 2.

Note that the verbal closeness does not imply $l$-verbal closeness of $H$ in
$G$. Indeed, every equation $w(x_{1},...,x_{n};H)=h,h\in H,$ is equivalent to
a system of split equations with right sides in $H$. This system can be
obtained from the initial equation if we change each coefficient $h$ to new
variable $x_{h}$ and add the corresponding split equation $x_{h}=h$ to the
system. Hence any non-algebraically closed subgroup $H\leq G$ is non-l-
verbally closed for some $l.$ We noted in Remark 1, that there are verbally
closed but non-algebraically closed subgroups of some groups.

Further in the paper we denote a conjugate by $g^{f}=fgf^{-1}$ , and
commutator by $[f,g]=fgf^{-1}g^{-1}$. Here $f,g$ are elements of arbitrary
group $G.$ By $G^{\prime}$ we denote the derived subgroup (commutant) of $G$.
By $\gamma_{i}G$ we mean the $i$-th member of the low central series in $G$
(in particular, $\gamma_{1}G=G,\gamma_{2}G=G^{\prime}$). By $G^{(i)}$ we
denote the $i$-th member of the derived series in $G$ (in particular,
$G^{(0)}=G,G^{(1)}=G^{\prime}$). By $\langle g_{1},...,g_{k}\rangle$ we denote
the subgroup generated by elements $g_{1},...,g_{k}$ in $G.$ The variety of
all abelian grous is denoted as $\mathfrak{A}$, the variety of all nilpotent
groups of nilpotency class $c$ ($c\in\mathbb{N}$) is denoted as
$\mathfrak{N}_{c}$. By $\mathfrak{A}^{d}$ ($d\in\mathbb{N}$) we mean the
variety of all solvable groups of the solvability class $\leq d.$

For any tuple of varieties
$\mathfrak{B}_{i},i=1,...,k,\,\prod_{i=1}^{k}\mathfrak{B}_{i}$ denotes their
product. We write $F_{r}(\mathfrak{B})$ to denote a free group of rank $r$ in
$\mathfrak{B}.$ Further in the paper we assume that any rank $r$ is finite.
For some relatively free groups we use specific denotions:
$A_{r}=F_{r}(\mathfrak{A}),\,N_{r,c}=F_{r}(\mathfrak{N}_{r,c}),\,S_{r,d}=F_{r}(\mathfrak{A}^{d}).$

Let $H$ be a finitely generated subgroup of a group $G.$ Then $r_{ab}(H)$
denotes a rank, i.e., the minimal number of generators of
$HG^{\prime}/G^{\prime}.$

If $H$ is verbally closed then $G$ $\cap G^{\prime}=H^{\prime}.$ Indeed, if
$h\in G^{\prime}\cap H$, then some equation of the form
$h=\prod_{j=1}^{t}[x_{j},x_{j}^{\prime}]$ is solvable in $G$. Then this
equation has a solution in $H$, i.e., $h\in H^{\prime}.$ Then
$H/H^{\prime}\cong H/H\cap G^{\prime}\cong HG^{\prime}/G^{\prime}.$

The following assertion is followed from the statements above.

######  Proposition 1.

Let $G=\langle g_{1},...,g_{k},...\rangle$ be a solvable group and $H$ be a
subgroup of $G$ such that $r_{ab}(H)=0,$ i.e., $H\leq G^{\prime}.$ If $H$ is
verbally closed in $G,$ then $H=1$.

###### Proof.

Suppose that $h\in H,h\not=1.$ Let $h\in G^{(n)}\setminus G^{(n+1)},n\geq 1.$
Then $h$ can be expressed in the form $h=\prod_{i=1}^{m}h_{i},$ where every
factor $h_{i}$ is written as $[v_{i},w_{i}],\,v_{i},w_{i}\in G^{(n-1)}.$ Let
$v_{i}=v_{i}(g_{1},...,g_{r})$ and $w_{i}=w_{i}(g_{1},...,g_{r})$ are the
corresponding expressions of the elements $v_{i},w_{i}$ in generators of $G$.
The words $v_{i}(x_{1},...,x_{n})$ and $w_{i}(x_{1},...,x_{n})$ are chosen in
$F_{n}^{(n-1)}$, where $F_{n}$ is the free group with basis
$\\{x_{1},...,x_{n}\\}.$ Hence, the equation
$\prod_{i=1}^{m}[v_{i}(x_{1},\ldots,x_{r}),w_{i}(x_{1},\ldots,x_{r})]=h$ has a
solution in $G.$ Obviously this equation has no solutions in $G^{\prime}$
because any substitution of elements from $G^{\prime}$ instead of variables
gives an element in $G^{(n+1)}.$ Especially, this equation is not solvable in
$H.$

∎

The following statements are main results of this paper.

######  Theorem 1.

Let $H$ be a finitely generated verbally closed subgroup of a free solvable
group $S_{r,d}$. If $r_{ab}(H)=1,$ then $H$ is a retract.

######  Theorem 2.

Let $H$ be a finitely generated verbally closed subgroup of a free solvable
group $S_{r,d}$. If $r_{ab}(H)=r,$ then $H=S_{r,d}.$

######  Theorem 3.

Every $2$-generated verbally closed subgroup $H$ of a free solvable group
$S_{r,d}$ is a retract.

######  Theorem 4.

For any $l\in\mathbb{N},$ every finitely generated $l$-verbally closed
subgroup $H$ of a free metabelian group $M_{r}$ such that $r_{ab}(H)=l+1$ is a
retract.

§1. PRELIMINARY RESULTS

Recall that a set of elements $\\{g_{1},\ldots,g_{n}\\}$ of a group $G$ is
test set if any endomorphism $\varphi:G\rightarrow G,$ that fixes every
element $g_{i}\ (i=1,...,n),$ is an automorphism of $G.$ An element $g\in G$
is said to be test element if $\\{g\\}$ is test set. Test rank $tr(G)$ is
defined as the minimal number of elements in a test set of $G.$ See [14],
[15], [16], [17], [18] for details and results about test sets and test ranks
of groups.

Let $F_{r}$ be a free group of rank $r$ and let $V$ be a normal subgroup of
$F_{r}.$ A test set $\\{g_{1},\ldots,g_{n}\\}\subseteq G=F_{r}/V^{\prime}$ is
said to be strongly test set if any endomorphism $\varphi:G\rightarrow G$
fixing every element $g_{i}$ is inner automorphism that acts as conjugate by
element from $V/V^{\prime}.$

Recall, that a variety of the form
$\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{m}},m\geq 1$ is called
polynilpotent of class  $\bar{c}=(c_{1},...,c_{m}).$ We consider a
polynilpotent variety of the form
P$\mathfrak{N}_{\bar{c}}=\mathfrak{A}\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{m}},\,m\geq
1,$ of class $\bar{c}=(2,c_{1},...,c_{m})$. Note that class
$\bar{c}=(2,2,...,2)$ corresponds to $\mathfrak{A}^{m+1}.$ The following lemma
follows from results and their proofs of the papers ([16], [17]) by C.K. Gupta
and the second author.

######  Lemma 1.

A relatively free group $F_{r}($P$\mathfrak{N}_{\bar{c}})$ of rank $r\geq 2$
of the variety P$\mathfrak{N}_{\bar{c}},\,m\geq 1,$ admits a strongly test set
consisting of $(r-1)$ elements.

###### Proof.

Let $F_{r}/V$ be a free group of rank $r$ of the polynilpotent variety
$\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{m}},m\geq 1.$ The main statement
in [16] is computing of the test rank of any free solvable group. In fact, in
this paper a test rank of $F_{r}/V^{\prime}$ was compute. Especially, it was
shown that the test rank of a free group of the polynilpotent variety of the
form $\mathfrak{A}\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{m}}$ was
computed. Moreover, it was shown how a test set $\\{g_{1},\ldots,g_{r-1}\\}$
for $F_{r}/V^{\prime}$ can be constructed such that: if an endomorphism
$\varphi$ fixes every element of this set then $\varphi$ is an inner
automorphism that induces identity map on $V/V^{\prime}.$ In [17], it was
shown that the mentioned above inner automorphism is induced by conjugate to
an element that lies in $V/V^{\prime},$ then $\\{g_{1},\ldots,g_{r-1}\\}$ is
strong test set.

∎

######  Lemma 2.

Let
$G=F_{r}/V^{\prime}=F_{r}(\mathfrak{A}\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{m}}),\,m\geq
1,\,r\geq 2,\,$ and let $\\{z_{1},\ldots,z_{r}\\}$ be a basis of $G.$ Let
$H\leq G$ is isomorphic to
$F_{n}(\mathfrak{A}\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{m}})$ for some
$n\geq 2.$ Let
$T=\\{t_{1}(z_{1},\ldots,z_{r}),\ldots,t_{n-1}(z_{1},\ldots,z_{r})\\}$ be a
strong test set in $H.$ Suppose that

$t_{i}(h_{1},\ldots,h_{r})=t_{i}(z_{1},\ldots,z_{r}),\,\,\,\,i=1,\ldots,n-1,$

for some elements $h_{1},\ldots,h_{r}\in H.$ Then map

$z_{j}\mapsto h_{j},\,\,j=1,\ldots,r,$

is extended to a retraction $G\rightarrow H.$

###### Proof.

Extend map $z_{j}\mapsto h_{j},\,\,j=1,\ldots,r,$ to endomorphism
$\varphi:G\rightarrow H.$ Let $\varphi|_{H}$ be restriction of $\varphi$ to
$H,$ that is an endomorphism of $H.$

Then
$\varphi|_{H}(t_{i}(z_{1},\ldots,z_{r}))=\varphi(t_{i}(z_{1},\ldots,z_{r}))=t_{i}(\varphi(z_{1}),\ldots,\varphi(z_{r}))=t_{i}(h_{1},\ldots,h_{r})=t_{i}(z_{1},\ldots,z_{r}),\,i=1,\ldots,n-1.$
Hence $\varphi|_{H}$ is an inner automorphism $\alpha_{a}$ of $H$ induced by
$a\in V/V^{\prime}.$ Consider an inner automorphism
$\psi=\varphi_{a^{-1}}\varphi,$ where $\varphi_{a^{-1}}$ of $G$ corresponds to
$a^{-1}.$ For $h\in H$ we have

$\psi(h)=\varphi_{a^{-1}}\varphi(h)=\varphi_{a^{-1}}(h^{a})=h,$

and for $g\in G$

$\psi(g)=(\varphi(g))^{a^{-1}}\in H.$

Thus $\psi$ is a seeking retraction onto $H$.

∎

The lemmas above imply the following assertion.

######  Proposition 2.

Let $H$ be a subgroup of
$F_{r}(\mathfrak{A}\mathfrak{N}_{c_{1}}\ldots\mathfrak{N}_{c_{l}})$ generated
by $(m+1)$ elements and rank of $H_{ab}=H/H^{\prime}$ is equal to $m+1.$ Then,
if $H$ is $m$-verbally closed subgroup, it is a retract.

######  Lemma 3.

([8]). Let $N$ be a verbal subgroup of a group $G.$ If $H$ is a verbally
closed subgroup of $G,$ then its image $H_{N}=HN/N$ is verbally closed in
$G_{N}=G/N.$ Specifically, if $G/G^{\prime}$ is a free abelian group of a
finite rank then the image $\bar{H}$ of its verbally closed subgroup $H$ in
$G/G^{\prime}$ is a direct factor.

######  Lemma 4.

Let $G=S_{r,d}$ be a free solvable group of class $d\geq 2$ with basis
$z_{1},\ldots,z_{r},\,\,r\geq 2.$ Let $H\leq G$ be a verbally closed subgroup
generated by elements $c_{1}z_{1},\ldots,c_{r}z_{r},c_{r+1}\ldots,$ where any
$c_{i}$ lies in $G^{(d-1)}.$ Then $H^{(d-1)}$ is
$\mathbb{Z}[H/H^{(d-1)}]-$closed subgroup of $G^{(d-1)},$ i.e., if for any
$c\in G^{(d-1)}$ and $0\neq\alpha\in\mathbb{Z}[G/G^{(d-1)}]$ we have
$c^{\alpha}\in H^{(d-1)},$ then $c\in H^{(d-1)}.$

###### Proof.

Denote $A=G/G^{(d-1)},\,\,B=H/H^{(d-1)},$ then $G^{(d-1)}\cap H=H^{(d-1)}.$

Let $1\neq c^{\alpha}\in H^{(d-1)}.$ Write $c$ as $c=c(z_{1},\ldots,z_{r}).$
Then $\alpha$ has an expression of the form $\alpha=\Sigma
m_{p}\overline{h}_{p},$ where $m_{p}\in\mathbb{Z},$ $\overline{h}_{p}\in B.$
Let $a_{i}=z_{i}G^{(d-1)}.$ Then every element $\overline{h}_{p}$ lies in the
subgroup generated by $a_{1},\ldots,a_{r},$ i.e.,
$\alpha=\alpha(a_{1},\ldots,a_{r}).$

Consider an equation in unknowns $x_{1},\ldots,x_{r}:$

$c(x_{1},\ldots,x_{r})^{(1-\overline{x}_{1}^{m_{1}})(1-\overline{x}_{1}^{-m_{1}})\ldots(1-\overline{x}_{r}^{m_{r}})(1-\overline{x}_{r}^{m_{r}})\alpha(\overline{x}_{1},\ldots,\overline{x}_{r})}$
$=c(z_{1},\ldots,z_{r})^{(1-a_{1}^{m_{1}})(1-a_{1}^{-m_{1}})\ldots(1-a_{r}^{m_{r}})(1-a_{r}^{-m_{n}})\alpha(a_{1},\ldots,a_{r})},$

where $m_{i}$ are integers. This equation has a solution in $G.$ Hence, it has
a solution $x_{i}=h_{i}$ in $H.$ Then

$c(h_{1},\ldots,h_{r})^{(1-\overline{h}_{1}^{m_{1}})(1-\overline{h}_{1}^{-m_{1}})\ldots(1-\overline{h}_{r}^{m_{r}})(1-\overline{h}_{r}^{-m_{r}})\alpha(\overline{h}_{1},\ldots,\overline{h}_{r})}=$
(2)
$=c(z_{1},\ldots,z_{r})^{(1-a_{1}^{m_{1}})(1-a_{1}^{-m_{1}})\ldots(1-a_{r}^{m_{r}})(1-a_{r}^{-m_{r}})\alpha(a_{1},\ldots,a_{r})}.$

Here $\overline{g}$ denotes the image of $g\in G$ in $A.$

The group ring $\mathbb{Z}[G]$ is equipped by the left Fox derivations
$\partial_{j}(g)$ with values in $\mathbb{Z}[A]$ (see details in [18], [19],
[20], [21]). For $\alpha\in\mathbb{Z}[A]$ and $b\in G^{(d-1)}$ the following
equality is valid:

$\partial_{i}(b^{\alpha})=\alpha\partial_{i}(b).$

Computing the values of $j$th derivation of two sides of the equality (2) we
obtain

$(1-\overline{h}_{1}^{m_{1}})(1-\overline{h}_{1}^{-m_{1}})\ldots(1-\overline{h}_{r}^{m_{r}})(1-\overline{h}_{r}^{-m_{r}})\alpha(\overline{h}_{1},\ldots,\overline{h}_{r})\partial_{j}(c(h_{1},\ldots,h_{r}))=$
(3)
$=(1-a_{1}^{m_{1}})(1-a_{1}^{-m_{1}})\ldots(1-a_{r}^{m_{r}})(1-a_{n}^{-m_{r}})\alpha(a_{1},\ldots,a_{r})\partial_{j}(c).$

Let $\Delta$ be the fundamental ideal of $\mathbb{Z}[A].$ $\Delta$ is endowed
with a norm $\omega$ (see [27]). Namely, for an element $u\in\Delta$ we set
$\omega(u)=n,$ where $u\in\Delta^{n}\backslash\Delta^{n+1}.$

We’ll show that no $h_{i}$ lies in $G^{\prime}.$ Since
$c=c(z_{1},\ldots,z_{r})\not=1,$ there is $j$ for which $\partial_{j}(c)\neq
0.$ We choose $j$, for which the norm of $\partial_{j}(c)$ is minimal among
all $\omega(\partial_{1}(c)),\ldots,\omega(\partial_{r}(c)).$ We can assume
for simplicity that $j=1.$ In (3) we compare the norms of the left and the
right sides. By the known rule we have

$\partial_{1}(c(h_{1},\ldots,h_{r}))=\partial_{1}(c)[\overline{h}_{1},\ldots,\overline{h}_{r}]\partial_{1}(h_{1})+\ldots+\partial_{r}(c)[\overline{h}_{1},\ldots,\overline{h}_{r}]\partial_{1}(h_{r}),$

where $\partial_{i}(c)[\overline{h}_{1},\ldots,\overline{h}_{r}]$ means a
result of substitution to $\partial_{i}(c)$ of elements $\overline{h}_{q}$
instead $\overline{z}_{q}.$ By choice of $j$ we obtain
$\omega(\partial_{1}c(h_{1},...,h_{r}))\leq\omega(\partial_{1}(c)).$ If at
least one element $h_{i}$ belongs to $G^{\prime},$ the norm of the left side
of (3) is less than the norm of the right side, because
$\omega(1-a_{i})=1<\omega(1-\overline{h}_{i}),$ for $h_{i}\in G^{\prime}.$

In [16], the following statement was proved. Let $S$ be a free polynilpotent
group of rank $r\geq 2$ with basis $\\{s_{1},...,s_{r}\\},$ and
$0\not=\alpha\in\mathbb{Z}[S].$ Let $p_{i}$ be the least $s_{i}-$ exponent of
$\alpha,$ and $q_{i}$ be the greatest $s_{i}-$ exponent of it. Let a positive
integer $m$ is greater than

$\sum_{i=1}^{r}(|p_{i}|+|q_{i}|).$

For $y_{1}\in G\setminus S^{\prime},$ equality

$(1-s_{1}^{m})\alpha=(1-y_{1}^{m})\beta$ (4)

is possible for $\beta\in\mathbb{Z}[S]$ if and only if $y_{1}=s_{1}$ or
$y_{1}=s_{1}^{-1}.$

Now we choose an available $m_{i}$. Then $\overline{h}_{1}=a_{1}^{\pm 1}.$ The
ring $\mathbb{Z}[B]$ has no zero divisors. The expression
$(1-a_{i}^{m_{i}})(1-a_{i}^{-m_{i}})$ is not changed under substitution
$a_{i}$ to $a_{i}^{-1}.$ We cancel step by step (3) to
$(1-a_{i}^{m_{1}})(1-a_{i}^{-m_{1}})$ and obtain $\overline{h}_{i}=a_{i}^{\pm
1}$ for all $i=1,\ldots,r.$ Then we get
$\partial_{i}(c(h_{1},\ldots,h_{r}))=\partial_{i}(c),$ and

$c=c(h_{1},\ldots,h_{r})\in H^{(d-1)}.$

∎

######  Lemma 5.

Let $H=\langle h\rangle$ be a non-trivial cyclic subgroup of $G=S_{r,d}.$ Then
the following conditions are equivalent:

  1. 1.

$H$ is a verbally closed in $G;$

  2. 2.

$H$ is a retract of $G;$

  3. 3.

the image $\bar{h}$ of $h$ in $G/G^{\prime},$ i.e., in the free abelian group
$A_{r},$ is primitive.

###### Proof.

By Lemma 3 1 implies 3.

Let we show that 3 implies 2. Write $h$ as $h=au,$ where $a$ is a primitive
element of $A_{r},\,\,u\in G^{\prime}.$ Since any basis of $A_{r}$ is induced
by a basis of $G$ (moreover, it is induced by a basis of $F_{r},$ see for
example [21]). Hence there is a preimage $z$ of $a$ in $G$ that belongs to a
basis of $G.$ We take the endomorphism of $G$ that maps $z$ to , and any other
element of this basis to $1.$ We see that this endomorphism is a retraction
$G\rightarrow H.$

By Remark 1, 2 implies 1.

∎

######  Lemma 6.

Let $H=\langle g,f\rangle$ be a $2$-generated non-cyclic subgroup of a free
solvable group $G=S_{r,d}$ of rank $r\geq 2$. Then the following statement is
valid: if $H$ is a verbally closed subgroup of $G,$ then $g,f$ induce in
$A_{r}=G/G^{\prime}$ (free abelian group of rank $r$) a direct factor of rank
2, and $H$ is a free solvable group of rank 2 and class $d.$

###### Proof.

By Lemma 3, the images of $g,f$ in $A_{r}$ generate a verbally closed
subgroup, that is a direct factor of full group. It remains to prove, that
this factor is non-cyclic. By Lemma 3 and induction on $d$ we assume that the
image of $H$ in $S=G/G^{(d-1)}$ is generated by the image $\overline{g}$ of
$g$, and $f$ lies in $G^{(d-1)}.$ Let $f(z_{1},,z_{r})$ be an expression of
$f$ as a word in generators of $G.$ Equation $f(x_{1},,x_{r})=f$ is solvable
in $G.$ Hence, it has a solution $h_{1},,h_{r}$ in $H.$ The components of this
solution are written as $h_{i}=g^{t_{i}}f^{\alpha_{i}},t_{i}\in\mathbb{Z},$
where $\alpha_{i}\in\mathbb{Z}[\overline{g}],\,i=1,,r.$ Then
$f(h_{1},,h_{r})=f(g^{t_{1}},,g^{t_{r}})f^{\delta},$ where
$\delta\in\mathbb{Z}[\overline{g}]$ is easily to compute by collecting
exponents of $f$ in $f(h_{1},,h_{r}).$ It remains to note that $\delta$ lies
in the fundamental ideal of $\mathbb{Z}[\overline{g}],$ because
$f(x_{1},,x_{r})$ is a commutator word. Then $f^{1-\delta}=1$ for
$1-\delta\not=0$ that is impossible in view of well-known fact that
$G^{(d-1)}$ has no module torsion (see for example [21]). By Baumslag’s
theorem [22] $H=\langle g,f\rangle$ is a free solvable group of rank $r$ and
class $d$ with basis consisting of the elements described above.

∎

The following statement was proved in [23] (see also [21])

######  Lemma 7.

([23]). Let $M_{2}$ be the free metabelian group of rank $2$ with basis
$\\{z_{1},z_{2}\\}.$ Let $x_{1}=g_{1},x_{2}=g_{2}$ be a solution of

$[x_{1},x_{2},x_{2},x_{1}]\equiv[z_{1},z_{2},z_{2},z_{1}].$

Then $g_{i}\equiv z_{i}^{\varepsilon_{i}}(mod\
M_{2}^{\prime}),\varepsilon_{i}\in\\{\pm 1\\}.$

Here and further $[g_{1},g_{2},...,g_{k}]$ means a simple commutator that is
defined inductively:
$[g_{1},g_{2},...,g_{k}]=[[g_{1},g_{2},...,g_{k-1}],g_{k}].$

§2. PROOFS OF THE MAIN RESULTS

Proof of Theorem 1. Denote $G=S_{r,d}.$ We use induction on $d.$ The case
$d=1$ is trivial. Suppose that $r,d\geq 2.$ By induction and Lemma 3 we can
choose a basis $\\{z_{1},\ldots,z_{r}\\}$ of $G$ in such way that

$H=\langle z_{1}c_{1},c_{2},\ldots,c_{m}\rangle,$

where $c_{1},\ldots,c_{m}\in G^{(d-1)}.$ The quotient $G/G^{(d-1)}$ is
isomorphic to $S_{r,d-1}.$ Denote $C=G^{(d-1)}$ and $A=G/C.$ Let
$\\{a_{1},...,a_{r}\\}$ be a basis of $A$ corresponding to
$\\{z_{1},...,z_{r}\\}.$

We’ll show that each $c_{i}$ is equal to $1.$ We write any $c_{i}$ in the form
$c_{i}=c_{i}(z_{1},\ldots,z_{r}),$ where $c_{i}(x_{1},x_{2},\ldots,x_{r})$ is
a word that lies in the $(d-1)$th member of the derived series of the free
group $F(X_{r}),X_{r}=\\{x_{1},...,x_{r}\\}$. Since $H$ is verbally closed
there exist elements $h_{1},...,h_{r}\in H,$ such that

$c_{i}=c_{i}(h_{1},\ldots,h_{r}).$ (5)

From (5) we get the equalities:

$c_{i}={c_{1}}^{-\delta_{i1}}\ldots{c_{i}}^{-\delta_{ii}}\ldots{c_{m}}^{-\delta_{im}},i=1,...,m,$
(6)

where $\delta_{ij}(a_{1})\in\mathbb{Z}[a_{1}^{\pm 1}].$ Let
$\nu_{i}=1+\delta_{ii}.$ Each element $\delta_{ij}$ for $i\not=j$ lies in
ideal $id(a_{1}-1)$ of $\mathbb{Z}[a_{1}^{\pm 1}],$ and any $\nu_{i}=1$ modulo
this ideal.

Then (6) implies a system of equalities:

$c_{1}^{\nu_{1}}c_{2}^{\delta_{12}}\ldots c_{m}^{\delta_{1m}}=1,$
$c_{1}^{\delta_{21}}c_{2}^{\nu_{2}}\ldots c_{m}^{\delta_{2m}}=1,$
$\ldots\ldots\ldots\ldots$ $c_{1}^{\delta_{m1}}c_{2}^{\delta_{m2}}\ldots
c_{m}^{\nu_{m}}=1.$

Compute Fox derivations of the left sides that are equal to $0.$

$\nu_{1}\partial_{j}(c_{1})+\delta_{12}\partial_{j}(c_{2})+\ldots+\delta_{1m}\partial_{j}(c_{m})=0,$
$\delta_{21}\partial_{j}(c_{1})+\nu_{2}\partial_{j}(c_{2})+\ldots+\delta_{2m}\partial_{j}(c_{m})=0,$
$\ldots\ldots\ldots\ldots$ (7)
$\delta_{m1}\partial_{j}(c_{1})+\delta_{m2}\partial_{j}(c_{2})+\ldots+\nu_{m}\partial_{j}(c_{m})=0,$

where $j=1,\ldots,m.$ Determinant of each system (7) is $1$ modulo
$id(a_{1}-1),$ thus it is not $0.$ Hence the system has the only zero
solution. It follows that each $c_{i}$ is $1.$ By Lemma 5 we finish the proof.

$\square$

Proof of Theorem 2). Denote $G=S_{r,d}$. We use induction on $d.$ The case
$d=1$ is trivial. We assume that $r,d\geq 2.$ By induction and Lemma 3 we
construct a basis $\\{z_{1},\ldots,z_{r}\\}$ of $G$ such that

$H=\langle z_{1}c_{1},\ldots,z_{r}c_{r},c_{r+1},...,c_{m}\rangle,$

where $c_{1},\ldots,c_{r},c_{r+1},...,c_{m}\in C.$ It is enough to check that
each element $c\in C$ lies $H^{(n-1)}.$

Denote $C=G^{(d-1)},A=G/C\simeq S_{r,d-1}$. Abelian group $C$ is a right
$\mathbb{Z}[A]-$module in which action of $a\in A$ to $c\in C$ is defined as
$a^{-1}ca.$ Elements $a_{i}=z_{i}C,\,\,i=1,\ldots,r,$ give a basis of $A.$ Let
$T$ be a free right $\mathbb{Z}[A]-$module with basis
$\\{t_{1},\ldots,t_{r}\\}.$ We take the Magnus embedding of $G$ into split
extension

$M=\left(\begin{array}[]{ll}A&0\\\ T&1\end{array}\right).$

See details in [18], [21]. Any element $z_{i}$ maps to

$\left(\begin{array}[]{ll}a_{i}&0\\\ t_{i}&1\end{array}\right).$

The image of $C$ in $M$ is a submodule of $T,$ that is identified with $C.$ An
element $t_{1}u_{1}+\ldots t_{r}u_{r},\,\,u_{i}\in\mathbb{Z}[A],$ lies in $C$
if and only if

$u_{1}(a_{1}-1)+\ldots+u_{r}(a_{r}-1)=0.$ (8)

The image of $C$ in $M$ is a submodule of $T,$ consisting of all elements that
satisfy to (8). We keep its denotion $C.$

It is well-known that $\mathbb{Z}[A]$ satisfies to the right Ore condition
[26], i.e., for any pair of non-zero elements elements
$\alpha,\beta\in\mathbb{Z}[A]$ there are non-zero elements
$\mu,\nu\in\mathbb{Z}[A]$ such that $\alpha\mu=\beta\nu\neq 0.$ Since
$\mathbb{Z}[A]$ is also a domain it is embedded into a sfield of fractions
$P.$ Then $T$ embeds into a linear space $V$ of dimension $r$ over $P.$ By (8)
we conclude that the dimension of the linear subspace $C^{P}$ generated by $C$
in $V$ is $r-1.$

Let we consider a subset of non-trivial elements
{$w_{i}=w_{i}(z_{1},z_{i}),\,\,i=2,\ldots,r$} of $C.$ It is easily compute
that the image of $w_{i}$ under Magnus embedding in $M$ (more exactly in $T$)
has the form $t_{1}\gamma_{1}+t_{i}\gamma_{i},$ where $\gamma_{1}$ and
$\gamma_{i}$ are non-zero. Independent of
$t_{1}\gamma_{1}+t_{i}\gamma_{i},\,i=2,\ldots,r$ in $T$ is clear. We are to
check that elements $w_{i}(z_{1}c_{1},z_{i}c_{i}),\,i=2,\ldots,r,$ are
linearly independent over $\mathbb{Z}[A].$

Denote by $G_{1}$ the subgroup generated by elements
$z_{1}c_{1},\ldots,z_{r}c_{r}.$ By Baumslag’s theorem [22] the map

$\varphi:z_{i}\mapsto z_{i}c_{i},\,\,\,i=1,\ldots,r,$

extends to isomorphism $G$ and $G_{1},$ and then to isomorphism of $C$ and
$G_{1}^{(d-1)}$, and also to isomorphism of $\mathbb{Z}[A]$ and
$\mathbb{Z}[G_{1}/G_{1}^{(d-1)}].$ Moreover, $\mathbb{Z}[A]-$module $C$ is
isomorphic to $\mathbb{Z}[G_{1}^{(d-1)}]-$module $G_{1}^{(d-1)}.$ Any of these
isomorphisms is denoted by $\varphi.$

We are to check that elements $w_{i}(z_{1}c_{1},z_{i}c_{i}),\,\,i=2,\ldots,r$
are independent over $\mathbb{Z}[A].$

Suppose that it is not happen:

$w_{2}(z_{1}c_{1},z_{2}c_{2})^{\beta_{2}(z_{1},\ldots,z_{r})}\ldots
w_{r}(z_{1}c_{1},z_{r}c_{r})^{\beta_{r}(z_{1},\ldots,z_{r})}=1$

for some $\beta_{i}\in\mathbb{Z}[A],$ at least one of them is not $0$. Hence

$w_{2}(z_{1}c_{1},z_{2}c_{2})^{\beta_{2}(z_{1}c_{1},\ldots,z_{r}c_{r})}\ldots
w_{r}(z_{1}c_{1},z_{r}c_{r})^{\beta_{r}(z_{1}c_{1},\ldots,z_{r}c_{r})}=1.$

We apply $\varphi^{-1}$ and get a contradiction with independence of
$w_{2},\ldots,w_{r}$ over $\mathbb{Z}[A].$

We are to check that elements $w_{i}(z_{1}c_{1},z_{i}c_{i}),\,\,i=2,\ldots,r,$
are linearly independent not only over $\mathbb{Z}[A],$ but over $P$ too.
Elements of $P$ are equivalency classes of $\mathbb{Z}[A]\times\mathbb{Z}[A],$
that usually denoted as $\alpha\beta^{-1}$ for
$0\neq\beta,\alpha\in\mathbb{Z}[A].$ Suppose that there is a non-trivial
linear combination over $P$:

$w_{2}(z_{1}c_{1},z_{2}c_{2})\alpha_{2}\delta_{2}^{-1}+\ldots+w_{r}(z_{1}c_{1},z_{r}c_{r})\alpha_{r}\delta_{r}^{-1}=0.$

Since $\mathbb{Z}[A]$ is the Ore domain every a finite set of its elements has
a right common factor. Take such factor $\delta$ for
$\delta_{2},\ldots,\delta_{r}.$ By definition
$0\neq\delta=\delta_{i}\gamma_{i},\,\,i=2,\ldots,r,\gamma_{i}\in\mathbb{Z}[A].$
Then

$w_{2}(z_{1}c_{1},z_{2}c_{2}){\alpha_{2}\gamma_{2}}+\cdot+w_{r}(z_{1}c_{1},z_{r}c_{r}){\alpha_{r}\gamma_{r}}=0.$

This contradicts to linear independence of elements
$w_{i}(z_{1}c_{1},z_{i}c_{i}),\,\,i=2,\ldots,r,$ over $\mathbb{Z}[A].$

We proved that elements
$w_{2}(z_{1}c_{1},z_{2}c_{2}),\ldots,w_{r}(z_{1}c_{1},z_{r}c_{r})$ are
linearly independent over $P.$

There are elements $0\neq\sigma,\sigma_{1},\ldots,\sigma_{r-1}\in P$ such that

$c^{\sigma}=w_{2}(z_{1}c_{1},z_{2}c_{2})^{\sigma_{1}}\ldots
w_{r}(z_{1}c_{1},z_{r}c_{r})^{\sigma_{r-1}}.$

By multiplying all $\sigma,\sigma_{1},\ldots,\sigma_{r-1}$ to available
element $u\in\mathbb{Z}[A]$, we can assume that
$\sigma,\sigma_{1},\ldots,\sigma_{r-1}\in\mathbb{Z}[A].$ Thus $c^{\sigma}\in
H^{(d-1)}.$ Lemma 4 implies that $c\in H^{(d-1)}.$

$\square$

Proof of Theorem 3. By Lemma 6 $H$ is solvable of rank $2.$ By Baumslag’s
theorem [22] it isomorphic to $S_{2d}.$ By Lemma 1 $H$ contains a strong test
element. By Lemma 2 $H$ is a retract.

$\square$

Proof of Theorem 4. Denote $G=M_{r}.$ Let $\\{z_{1},...,z_{r}\\}$ be a basis
of $G$, and let $\\{a_{1},...,a_{r}\\}$ be the corresponding basis of
$A_{r}=G/G^{\prime}.$

By Lemma 3 we can assume that $H=\langle
h_{1},...,h_{l+1},h_{l+2},...h_{l+t}\rangle$, where $h_{i}\equiv z_{i}(mod\
G^{\prime}),\ i=1,...,l+1,$ and $h_{l+k}\in G^{\prime},k=2,...,t.$

The case $l=0$ was proved in Theorem 1. Suppose that $l\geq 1.$

For any $g\in G$ by $\bar{g}$ we denote its image in $A_{r}.$ Then
$\bar{h}_{i}=a_{i},\ i=1,...,l+1;\bar{h}_{l+k}=1,\ k=2,...,t.$ Hence
$\bar{H}=A_{l+1}=\langle a_{1},...,a_{l+1}\rangle.$ By Baumslag’s theorem [22]
subgroup $H_{l+1}=\langle h_{1},...,h_{l+1}\rangle$ is isomorphic to
$M_{l+1}$, and $\\{h_{1},...,h_{l+1}\\}$ is a basis of $H_{l+1}$. In
particular, every map of this basis to $G$ extends to a homomorphism
$H_{l+1}\rightarrow G.$

In the proof that follows we show that there is a homomorphism
$\psi:G\rightarrow H$ identical on $H_{l+1},$ and then we show that it is
identical on $H,$ i.e., it is a retraction of $G$ onto $H.$

Let $h_{i}=h_{i}(z_{1},...,z_{r}),i=1,...,l+1$ be expressions fixed generators
of $H_{l+1}$ as words in basic elements of $G$. For any $i\in\\{2,...,l+1\\}$
we consider equation

$[h_{i}(x_{1},...,x_{r}),h_{1}(x_{1},...,x_{r}),h_{1}(x_{1},...,x_{r}),h_{i}(x_{1},...,x_{r})]=[h_{i},h_{1},h_{1},h_{i}].$
(9)

This equation is solvable in $G$: $x_{j}=z_{j},j=1,...,r.$ Hence it has a
solution in $H$: $x_{j}=f_{j},f_{j}\in H,j=1,...,r.$

We have:

$[h_{i}(f_{1},...,f_{r}),h_{1}(f_{1},...,f_{r}),h_{1}(f_{1},...,f_{r}),h_{i}(f_{1},...,f_{r})]\equiv[h_{i},h_{1},h_{1},h_{i}](mod\
\gamma_{5}G),$ $i=2,...,l+1.$ (10)

Denote $\tilde{h}_{j}=h_{j}(f_{1},...,f_{r}),\ j=1,i.$ Let
$\tilde{h}_{1}\equiv\prod_{j=1}^{l+1}z_{j}^{k_{j}}(mod\,G^{\prime}),\tilde{h}_{i}\equiv\prod_{j=1}^{l+1}z_{j}^{m_{j}}(mod\,G^{\prime}),\
k_{j},m_{j}\in\mathbb{Z},\ j=1,...,l+1.$ Then (10) is equivalent to

$[\prod_{j=1}^{l+1}z_{j}^{m_{j}},\prod_{j=1}^{l+1}z_{j}^{k_{j}},\prod_{j=1}^{l+1}z_{j}^{k_{j}},\prod_{j=1}^{l+1}z_{j}^{m_{j}}]\equiv[z_{i},z_{1},z_{1},z_{i}](mod\
\gamma_{5}G).$ (11)

Quotient $G_{5}=G/\gamma_{5}G$ is a free group of variety
$\mathfrak{A}^{2}\cap\mathfrak{N}_{5}$ of all metabelian nilpotent groups of
nilpotency class $\leq 5$. Its the last non-trivial member of the low central
series $\gamma_{4}G_{5}$ is a free abelian group with basis consisting of all
basic commutators of length $4$ of the form
$[z_{i_{1}},z_{i_{2}},z_{i_{3}},z_{i_{4}}],$ where $i_{1}>i_{2},i_{2}\leq
i_{3}\leq i_{4}.$ Any metabelian group satisfies to identities
$[x_{1},x_{2},x_{3},...,x_{n}]=[x_{1},x_{2},x_{\tau(3)},...,x_{\tau(n)}]$,
where $n\geq 3$, $\tau$ is a substitution on $3,...,n.$ $G_{5}$ satisfies to
identity
$[x_{1}^{k_{1}},x_{2}^{k_{2}},x_{3}^{k_{3}},x_{4}^{k_{4}}]=[x_{1},x_{2},x_{3},x_{4}]^{k},\,k=\prod_{j=1}^{4}k_{j}.$
See details in [21].

These facts mean that after rewriting of the left side of (11) as a product of
exponents of basic commutators of length $4$ these exponents will be $0$ with
just one exception for exponent of $[z_{i},z_{1},z_{1},z_{i}]$, which in
correspondence with the right side of (11) should be $1.$ We write a product
of all basic commutators in this expression that depend of $z_{1}$ and $z_{i}$
only.

$[z_{i},z_{1},z_{1},z_{i}]^{\alpha_{1}}[z_{i},z_{1},z_{1},z_{1}]^{\alpha_{2}}[z_{i},z_{1},z_{i},z_{i}]^{\alpha_{3}},$
(12)

where

$\alpha_{1}=(k_{1}m_{i}-k_{i}m_{1})(k_{1}m_{i}+k_{i}m_{1}),\alpha_{2}=(k_{1}m_{i}-k_{i}m_{1})k_{1}m_{1},\alpha_{3}=(k_{1}m_{i}-k_{i}m_{1})k_{i}m_{i}.$
(13)

A solution of (19 corresponds to values:
$\alpha_{1}=1,\alpha_{2}=\alpha_{3}=0.$ Note that
$\delta=(k_{1}m_{i}-k_{i}m_{1})\not=0,\delta\in\\{\pm 1\\},$ thus,
$k_{1}m_{1}=k_{i}m_{i}=0.$ Consider two the cases.

1) $k_{1}=0\Rightarrow\delta=-k_{i}m_{1}\Rightarrow
k_{i}\not=0\,\&\,m_{i}=0\Rightarrow\alpha_{1}=-k_{i}^{2}m_{1}^{2}=1,$
contradiction.

2) $m_{1}=0\Rightarrow\delta=k_{1}m_{i}\not=0\Rightarrow
m_{i}\not=0\,\&\,k_{i}=0\Rightarrow\alpha_{1}=k_{1}^{2}m_{i}^{2}=1\Rightarrow
k_{1}=\varepsilon_{1}\in\\{\pm 1\\},m_{i}=\varepsilon_{i}\in\\{\pm 1\\},$
which gives a solution of (18).

Consider (10) again. It follows from our proof that the image of
$\tilde{h}_{1}$ in $A_{r}$ does not depend of $a_{i},$ and the image of
$\tilde{h}_{i}$ does not depend of $a_{1}.$ Obviously these both the images as
image of any element of $H$, does not depend of $a_{l+2},...,a_{r}.$ The image
of $\tilde{h}_{j}$ includes $a_{j}$ in exponent $\varepsilon_{j}\in\\{\pm
1\\},j=1,i.$

Consider a system of equations

$[h_{i}(x_{1},...,x_{r}),h_{1}(x_{1},...,x_{r}),h_{1}(x_{1},...,x_{r}),h_{i}(x_{1},...,x_{r})]=[h_{i},h_{1},h_{1},h_{i}],$
$i=2,...,l+1.$ (14)

Here as above $h_{i}(x_{1},...,x_{r})$ corresponds to
$h_{i}=h_{i}(z_{1},...,z_{r})$ that is an expression of $h_{i}$ by the basic
elements of $G$.

Since $G$ is $l$-verbally closed this system has a solution in $G$:
$x_{j}=z_{j},j=1,...,r.$ Hence it has a solution in $H$: $x_{j}=f_{j},f_{j}\in
H,j=1,...,r.$ Denote $u_{i}=h_{i}(f_{1},...,f_{r}),i=1,...,l+1.$ Let $\nu$ be
an endomorphism of $G$ defined by the map $\nu:z_{j}\mapsto f_{j},j=1,...,r.$
Note that $u_{j}=\nu(h_{j}(z_{1},...,z_{r})=\nu(h_{j}),\ j=1,...,r.$ Then
$\nu(G)\leq H$. The following equalities are valid:

$[u_{i},u_{1},u_{1},u_{i}]=[h_{i},h_{1},h_{1},h_{i}],i=1,...,l+1.$ (15)

In other words $\nu$ fixes every of elements
$[h_{i},h_{1},h_{1},h_{i}],i=1,...,l+1.$ The same is true for $\eta=\nu^{2}$.
By arguments similar to ones about [eq:15] applied to each of equations (20)
we get congruence

$u_{1}\equiv h_{1}^{\varepsilon_{1}}(mod\ G^{\prime}),\varepsilon_{1}\in\\{\pm
1\\}.$ (16)

By direct computation we obtain congruences $u_{i}\equiv
h_{i}^{\varepsilon_{i}}(mod\ G^{\prime}),\varepsilon_{i}\in\\{\pm
1\\},i=2,...,l+1.$ Let for example $u_{2}\equiv
h_{2}^{varepsilon_{2}}h_{3}^{\lambda_{3}}...h_{l+1}^{\lambda_{l+1},\,\varepsilon_{2}\in\\{\pm
1}$ (this was proved above), $\lambda_{i}\in\mathbb{Z}$ and at least one of
$\lambda_{j}$ is not $0$, let be $\lambda_{3}.$ Then in expression of
$[u_{2},u_{1},u_{1},u_{2}]$ as a product of exponents of basic commutators
modulo $\gamma_{5}H$ of $[u_{2},u_{1},u_{1},u_{2}]$ we have
$[h_{3},h_{1},h_{1},h_{3}]$ with exponent $\lambda_{3}^{2}\not=0,$ that
contradics to (15).

Denote $w_{j}=\nu(u_{j})=\eta(h_{j}),$ then $w_{i}\equiv z_{i}(mod\
G^{\prime})$ and $w_{i}\equiv h_{i}(mod\ H^{\prime}),i=1,...,l+1.$ We have a
system of equalities

$[w_{i},w_{1},w_{1},w_{i}]=[h_{i},h_{1},h_{1},h_{i}],i=1,...,l+1.$ (17)

We use congruences $w_{i}\equiv h_{i}(mod\ H^{\prime}),\ i=1,...,l+1.$ Let
$w_{i}=c_{i}h_{i},c_{i}\in H^{\prime},i=1,...,l+1.$ Then

$[w_{i},w_{1},w_{1},w_{i}]=[h_{i},h_{1},h_{1},h_{i}]c_{i}^{(1-a_{1})^{2}(1-a_{i})}c_{1}^{(1-a_{i})^{2}(a_{1}-1)},i=2,...,l+1.$
(18)

After cancellations in exponents we obtain

$c_{1}^{1-a_{i}}=c_{i}^{1-a_{1}},i=2,...,l+1.$ (19)

Hence there is $d\in H^{\prime}$ such that
$c_{1}=d^{1-a_{1}},c_{i}=d^{1-a_{i}},i=2,...,l+1.$ The $\eta$ is inner
automorphism corresponding to conjugation by $d\in H^{\prime}.$ Define
endomorphism $\psi$ as composition of $\eta$ with inner automorphism
corresponding to conjugation by $d^{-1}.$ Then $\psi$ maps $G$ to $H$ and is
identical on $H_{l+1}.$

It remains to prove that $\psi$ is identical on $H.$

Let $h=h_{l+k}\,(k=2,...,t)$ be one of the chosen generators of $H.$ Then
there is $0\not=\alpha_{k}\in\mathbb{Z}[a_{1}^{\pm 1},...,a_{l+1}^{\pm 1}]$,
such that $h^{\alpha_{k}}\in H_{l+1}.$

Let $h(z_{1},,z_{r})$ be an expression of $h$ as a word in basic elements of
$G.$ Then

$h(x_{1},,x_{r})=h$ (20)

is solvable in $G.$ Hence it has a solution $f_{1},,f_{r}$ in $H.$ We
substitute this solution to (20) and select in the left side
$h_{l+j},\,j=2,...,t.$ We obtain

$\tilde{h}\prod_{j=2}^{t}h_{l+j}^{\alpha_{kj}}=h,\,\tilde{h}\in
H_{l+1},\alpha_{kj}\in\mathbb{Z}[A_{l+1}]=\mathbb{Z}[a_{1}^{\pm
1},...,a_{l+1}^{\pm 1}].$ (21)

We write this for every equation in (21) for $k=1,...,t,$ and obtain a system

$\tilde{h}_{l+k}=h_{l+k}^{1-\alpha_{kk}}\prod_{j=2,j\not=k}^{t}h_{l+j}^{-\alpha_{kj}},\,\tilde{h}_{l+k}\in
H_{l},\alpha_{kj}\in\mathbb{Z}[A_{l}]=\mathbb{Z}[a_{1}^{\pm 1},...,a_{l}^{\pm
1}],\,k=2,...,t.$ (22)

It remains to note that all $\alpha_{ij}$ in this equality lie in the
fundamental ideal of the group ring $\mathbb{Z}[A_{l+1}],$ because
$h_{l+k}(x_{1},,x_{r})$ is commutator word for every $k=2,...,t.$ Then
${1-\alpha_{kk}}$ is $1$ modulo the fundamentsl ideal. It follows that
determinant of the matrix consisting of exponent of the right side in (22) is
not $0.$ If consider $h_{l+k},\,k=2,...,t,$ as unknowns the system is solvable
over the field of fractions of the ring $\mathbb{Z}[A_{l+1}].$ Applying to
(22) the Gauss elimination process we get

$h_{l+k}^{\alpha_{k}}=\tilde{h}_{k},\,\alpha_{k}\in\mathbb{Z}[A_{l}],\,\tilde{h}_{k}\in
H_{l+1},\,k=2,...,t.$ (23)

The endomorphism $\psi$ acts identically to $H_{l+1}.$ Hence it induces
identical map to $A_{l+1}.$ Then

$\eta(h_{l+k}^{\alpha_{k}})=(\eta(h_{l+k}))^{\alpha_{k}}=h_{l+k}^{\alpha_{k}},\,k=2,...,t.$
(24)

It is known that $G^{\prime}$ has no module tersion. Since
$\psi(h_{l+k})=h_{l+k}$ for any $k=2,...,t,$ then $\psi$ is identical on $H.$
This means that $\psi$ is a retraction $G\rightarrow H$.

$\square$

§3. PROBLEMS

We put a few of natural problems.

######  Problem 1.

Is it true that any finitely generated verbally closed subgroup of a free
solvable group $S_{r,d}$ is a retract? What is the answer in a specific case
$d=2$, i.e., in the case of a free metabelian group $M_{r}$?

More specific question:

######  Problem 2.

Is it true that a subgroup $H$ of a free solvable group $S_{r,d}$ of class
$d\geq 3,$ such that $r_{ab}(H)=2$ is verbally closed if and only if $H$ is a
retract of $S_{r,d}?$

A positive answer will strengthen Theorem 3.

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