First Betti numbers of orbits of Morse functions on surfaces

TL;DR

This paper investigates the algebraic structure of groups arising from the fundamental groups of Morse function orbits on surfaces, revealing a key property relating their centers and first Betti numbers.

Contribution

It establishes a novel relation between the ranks of centers and abelianizations of groups generated by products and wreath products, linked to Morse function orbit topology.

Findings

Ranks of the center and abelianization coincide for groups in class G.

First Betti number of Morse function orbit equals the rank of the group's center.

Provides algebraic insight into the topology of Morse function orbits on surfaces.

Abstract

In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of the center $Z(G)$ and the quotient by commutator subgroup $G/[G,G]$ coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.

Equations

$$A\underset{1}{\wr }\mathbb{Z}=A\times \mathbb{Z},$$

$$\left(1\underset{3}{\wr }\mathbb{Z}\right)\times \mathbb{Z}=\mathbb{Z}\times \left(1\underset{3}{\wr }\mathbb{Z}\right)=\mathbb{Z}\times \mathbb{Z}=1\times \mathbb{Z}\times \mathbb{Z}.$$

$$Z(G)\cong G/[G,G]\cong {\mathbb{Z}}^{{\beta }_1(\omega )}.$$

$$\mathcal{O}(f)=\{f\circ h|h\in \mathcal{D}(M)\}$$

$$H_1(\mathcal{O}(f),\mathbb{Z})\simeq {\mathbb{Z}}^{{\beta }_1(\omega )}.$$

$$H_{1}(X,\mathbb{Z})\simeq\pi_{1}X\bigl{/}[\pi_{1}X,\pi_{1}X].$$

$$H_1(\mathcal{O}(f),\mathbb{Z})\simeq {\pi }_1\mathcal{O}(f)/[{\pi }_1\mathcal{O}(f),{\pi }_1\mathcal{O}(f)]=G/[G,G]\simeq {\mathbb{Z}}^{{\beta }_1(\omega )}.$$

$$f\circ {\phi }_b:X⟶X⟶A.$$

$$(f_{1},b_{1})\cdot(f_{2},b_{2})=\bigl{(}(f_{1}\circ\varphi_{b_{2}})\cdot f_{2},b_{1}\cdot b_{2}\bigr{)}$$

$$\sigma (f)=\left\{x\in X|f(x)\ne e,\text{mbox}wheree\text{mbox}isaunitofA\right\},$$

$$Z(AW{r}_{X}B)\cong \tilde{D}(A),Z(Aw{r}_{X}B)\cong D(A),$$

$$Z(AW{r}_{X}B)=\tilde{D}(A)\times \left(\ker \phi \cap Z(B)\right)\cong \prod _{\lambda \in \mathcal{O}}Z(A)\times \left(\ker \phi \cap Z(B)\right),$$

$$Z(Aw{r}_{X}B)=D(A)\times \left(\ker \phi \cap Z(B)\right)\cong \underset{\lambda \in {\mathcal{O}}_{fin}}{⨁}Z(A)\times \left(\ker \phi \cap Z(B)\right).$$

$$g_{y,c}(x)=\{\begin{array}{ll}c,& \text{mbox}ifx=y;\\ e,& \text{mbox}ifx\ne y.\end{array}$$

$$\tilde{Q}\subset Z(AW{r}_{X}B)\subset \tilde{C}(S).$$

$$Z(AW{r}_{X}B)=\tilde{C}(S)=\tilde{Q}.$$

$$\left((f\circ {\phi }_p)\cdot g_{y,c},lp\right)=\left((g_{y,c}\circ {\phi }_l)\cdot f,pl\right).$$

$$(f\circ {\phi }_p(x))\cdot g_{y,c}(x)=(g_{y,c}\circ {\phi }_l(x))\cdot f(x).$$

$$f\circ {\phi }_p(x)=f(x).$$

$$f(y)g_{y,c}(y)=g_{y,c}(y)f(y),$$

$$f(y)g_{y,c}(y)=f(y).$$

$$Z(Aw{r}_{X}B)=Z(AW{r}_{X}B)\cap (Aw{r}_{X}B).$$

$$Z(Aw{r}_{X}B)\subset C(S)\subset \tilde{C}(S)\cap (Aw{r}_{X}B)\stackrel{\text{mbox}Lemma\text{leavevmode}\text{ref}lm:centerofunrestrictedproduct}{=}Z(AW{r}_{X}B)\cap (Aw{r}_{X}B).$$

$${\psi }_1:Z(AW{r}_{X}B)\to \prod _{\lambda \in \Lambda }Z(A)\times \left(\ker \phi \cap Z(B)\right),$$

$${\psi }_2:Z(Aw{r}_{X}B)\to \underset{\lambda \in {\Lambda }_{fin}}{⨁}Z(A)\times \left(\ker \phi \cap Z(B)\right)$$

$$Z(A{\wr }_{n}B)=D(A)\times \left(\ker \phi \cap Z(B)\right).$$

$$Z\left(A\underset{n}{\wr }\mathbb{Z}\right)=\left\{(a,a,\dots ,a,nk)|a\in Z(A),k\in \mathbb{Z}\right\}\cong D(A)\times n\mathbb{Z}\cong Z(A)\times \mathbb{Z}.$$

$$Z(A\times B)\cong Z(A)\times Z(B).$$

$$Z\Big{(}\big{(}(\mathbb{Z}\wr_{3}\mathbb{Z})\times(\mathbb{Z}\wr_{5}\mathbb{Z})\big{)}\wr_{7}\mathbb{Z}\Big{)}\cong Z\Big{(}(\mathbb{Z}\wr_{3}\mathbb{Z})\times(\mathbb{Z}\wr_{5}\mathbb{Z})\Big{)}\times 7\mathbb{Z}\cong$$

$$\cong Z\Big{(}(\mathbb{Z}\wr_{3}\mathbb{Z})\times(\mathbb{Z}\wr_{5}\mathbb{Z})\Big{)}\times\mathbb{Z}\cong Z\big{(}\mathbb{Z}\wr_{3}\mathbb{Z}\big{)}\times Z\big{(}\mathbb{Z}\wr_{5}\mathbb{Z}\big{)}\times\mathbb{Z}\cong$$

Full text

1. Introduction

Let $A$ be a group and $n\in\mathbb{Z}$. We will denote by $A\wr_{n}\mathbb{Z}$ a semidirect product of $A^{n}$ and $\mathbb{Z}$ with respect to the natural action of $\mathbb{Z}$ on $A^{n}$ by cyclic shifts of coordinates. More precisely, $A\wr_{n}\mathbb{Z}=A^{n}\rtimes_{\varphi}\mathbb{Z}$, where the homomorphism $\varphi\colon\mathbb{Z}\to\operatorname{Aut}A^{n}$ is defined by $\varphi(k)(a_{0},\dots,a_{n-1})=(a_{k\operatorname{mod}n},\dots,a_{n-1+k\operatorname{mod}n})$ for all $k\in\mathbb{Z}$, $(a_{0},\dots,a_{n-1})\in A^{n}$. Such semidirect product $A\wr_{n}\mathbb{Z}$ is called the wreath product of $A$ and $\mathbb{Z}$.

Note that the following groups are same:

[TABLE]

Definition 1.1**.**

Let $\mathcal{G}$ be a minimal class of groups satisfying the following conditions:

$1\in\mathcal{G}$; 2. 2)

if $A,B\in\mathcal{G}$, then $A\times B\in\mathcal{G}$; 3. 3)

if $A\in\mathcal{G}$ and $n\geq 1$, then $A\wr_{n}\mathbb{Z}\in\mathcal{G}$.

In other words a group $G$ belongs to the class $\mathcal{G}$ iff $G$ is obtained from trivial group by a finite number of operations $\times$, $\wr_{n}\mathbb{Z}$. It is easy to see that every group $G\in\mathcal{G}$ can be written as a word in the alphabet $\mathcal{A}=\left\{1,\mathbb{Z},\left(,\right),\times,\wr_{2},\wr_{3},\wr_{4},\dots\right\}$. We will call such word a presentation of the group $G$ in the alphabet $\mathcal{A}$. Evidently, the presentation of a group is not uniquely determined.

For example, there are the following presentations for the same group

[TABLE]

We will show that the number of symbols $\mathbb{Z}$ in the presentation of a group $G\in\mathcal{G}$ in the alphabet $\mathcal{A}$ is uniquely determined by $G$.

Denote by $Z(G)$ and $[G,G]$ the center and the commutator subgroup of $G$ respectively. The main result of the article is the following theorem.

Theorem 1.2**.**

Let $G\in\mathcal{G}$, $\omega$ be an arbitrary presentation of $G$ in the alphabet $\mathcal{A}$, and $\beta_{1}(\omega)$ be the number of symbols $\mathbb{Z}$ in the presentation $\omega$. Then there are the following isomorphisms:

[TABLE]

In particular, the number $\beta_{1}(\omega)$ depends only on the group $G$.

Geometric interpretation. The groups from the class $\mathcal{G}$ appear as fundamental groups of orbits of Morse functions on surfaces.

Let $M$ be a compact surface and $\mathcal{D}(M)$ be the group of $C^{\infty}$-diffeomorphisms of $M$. There is a natural right action of the group $\mathcal{D}(M)$ on the space of smooth functions $C^{\infty}(M,P)$ defined by the rule: $(f,h)\longmapsto f\circ h$, where $h\in\mathcal{D}(M)$, $f\in C^{\infty}(M,P)$ and $P$ is a real line $\mathbb{R}$ or a circle $S^{1}$.

Let

[TABLE]

be the orbit of $f$ under the above action.

Endow the spaces $\mathcal{D}(M)$, $C^{\infty}(M,P)$ with Whitney $C^{\infty}$-topologies. Let $\mathcal{O}_{f}(f)$ denote the path component of $f$ in $\mathcal{O}(f)$.

A map $f\in C^{\infty}(M,P)$ will be called Morse if all its critical points are non-degenerate. A Morse map $f$ is generic if it takes distinct values at distinct critical points.

Homotopy types of stabilizers and orbits of Morse functions were calculated in a series of papers by Sergiy Maksymenko [Mak06], [Mak12], Bohdan Feshchenko [Fes16],[Fes15], and Elena Kudryavtseva [Kud12a], [Kud12b], [Kud13], [Kud16]. In particular, E. Kudryavtseva shown that in the case $M\neq S^{2}$ for each Morse function $f$ there is a free action of some finite group $H$ on $k$-torus $(S^{1})^{k}$ such that $\mathcal{O}_{f}(f)$ is homotopy equivalent to $k$-dimensional torus $(S^{1})^{k}/H$. In fact, it was shown in [Mak06] by S. Maksymenko that if $f$ is generic, then $H$ is trivial, so $\pi_{n}\mathcal{O}_{f}(f)\simeq\pi_{n}((S^{1})^{k})$, and the general case of nontrivial $G$ was described in [Kud12a], [Kud12b], [Kud13], [Kud16] by E. Kudryavtseva. Furthermore, precise algebraic structure of such groups for the case $M\neq S^{2},T^{2}$ was described in [Mak12]. The following theorem relating $\pi_{1}\mathcal{O}(f)$ with the class $\mathcal{G}$ is a direct consequence of results of [Mak12].

Theorem 1.3**.**

[Mak12]** Let $M$ be a connected compact oriented surface except 2-sphere and 2-torus and let $f\colon M\to P$ be a Morse function. Then $\pi_{1}\mathcal{O}(f)\in\mathcal{G}$. ∎

As a consequence of our main result Theorem 1.2 and the previous theorem we get the following.

Corollary 1.4**.**

Let $M$ be a connected compact oriented surface distinct from $S^{2}$ and $T^{2}$, $f$ be a Morse function on M, $G=\pi_{1}\mathcal{O}_{f}(f)\in\mathcal{G}$, $\omega$ be an arbitrary presentation of $G$ in the alphabet $\mathcal{A}$, and $\beta_{1}(\omega)$ be the number of symbols $\mathbb{Z}$ in the presentation $\omega$. Then the first integral homology group $H_{1}(\mathcal{O}(f),\mathbb{Z})$ of $\mathcal{O}(f)$ is a free abelian group of rank $\beta_{1}(\omega)$:

[TABLE]

In particular, $\beta_{1}(\omega)$ is the first Betti number of $\mathcal{O}(f)$.

Proof.

To prove the corollary we use the well-known Hurewicz theorem, see [Hat02], stating that for every path-connected topological space $X$ there is isomorphism

[TABLE]

As a consequence of the Hurewicz theorem, Theorem 1.3 and Theorem 1.2 we get

[TABLE]

∎

Structure of the paper. In $\S 2$ we prove Theorem 2.1 about centers of wreath products $A\wr_{X}B$ of arbitrary groups $A$ and $B$ in the case of non-effective action of $B$ on the set $X$. Let $G\in\mathcal{G}$, $\omega$ be an arbitrary presentation of $G$. We also show in Theorem 2.6 that the center of a group $G$ of the class $\mathcal{G}$ is isomorphic to $\mathbb{Z}^{\beta_{1}(\omega)}$, it is the first part of Theorem 1.2. In $\S 3$ we find the commutator subgroups of any group $G$, see Theorem 3.1, and the quotient groups G\mathop{\wr}_{n}\mathbb{Z}\bigl{/}\left[G\wr_{n}\mathbb{Z},G\wr_{n}\mathbb{Z}\right] (Theorem 3.2). The second part of Theorem 1.2 that $G/[G,G]$ is also isomorphic to $\mathbb{Z}^{\beta_{1}(\omega)}$ is established in Theorem 3.3.

2. Centers of wreath products

Let $A$ and $B$ be two groups. Suppose there is an action of $B$ on the set $X$. In other words we have the homomorphism $\varphi$ from $B$ to the permutation group $\Sigma(X)$. For $b\in B$ denote by $\varphi_{b}\colon X\to X$ the corresponding permutation. Let also $Map(X,A)$ be the group of all maps $f:X\rightarrow A$ with respect to the pointwise multiplication. Then the group $B$ acts on $Map(X,A)$ by the following rule: the result of the action of $b\in B$ on $f\in Map(X,A)$ is the composition map:

[TABLE]

The semidirect product $Map(X,A)\rtimes_{\varphi}B$ corresponding to this action is called the unrestricted wreath product of $A$ and $B$ and denoted by $A\ Wr_{X}B$. Hence, it is the Cartesian product $Map(X,A)\times B$ with the multiplication given by the formula

[TABLE]

for $(f_{1},b_{1}),(f_{2},b_{2})\in Map(X,A)\rtimes_{\varphi}B$.

Denote by $\sigma(f)$ the support of the function $f\in Map(X,A)$:

[TABLE]

and by $Map_{fin}(X,A)$ the subset of $Map(X,A)$ consisting only of functions with a finite support, $\left|\sigma(f)\right|<\leavevmode\nobreak\ \infty$. The semidirect product $Map_{fin}(X,A)\rtimes_{\varphi}B$ is called the restricted wreath product, we denote it by $A\ wr_{X}B$.

Notice, that for the center of the group $A$ we use the notation $Z(A)$. Let $\tilde{D}(A)$ denote the subgroup $Map(X,Z(A))$ of functions $h:X\rightarrow Z(A)$ which are constant on each orbit of action of $B$ on $X$, and let $D(A)$ denote the subgroup $Map_{fin}(X,Z(A))$ of functions with the same property.

It follows from Theorem 4.2 [Mel95] that there are isomorphisms:

[TABLE]

where the group $B$ acts on $X$ effectively.

In the case of non-effective action of $B$ on $X$ for arbitrary groups $A,B$ we get more general situation. In this subsection we extend the Theorem 4.2 [Mel95] and consider the case of non-effective action of $B$ on $X$.

Denote the set of all the orbits of $B$ on $X$ by $\mathcal{O}$, and the set of finite ones by $\mathcal{O}_{fin}$. Recall that the direct product indexed by infinite set consists of all infinite sequences, while the direct sum consists only of sequences with finitely many elements distinct from zero.

Theorem 2.1**.**

There are the following isomorphisms:

[TABLE]

[TABLE]

Let $\tilde{Q}$ be a subgroup of $A\ Wr_{X}B$ whose elements $(f,l)$ satisfy the conditions:

1. a)

$f$ is constant on each orbit of $B$ on $X$, i.e. $f(x)=a_{\lambda}$ for any $x\in O_{\lambda}$, $O_{\lambda}\in\mathcal{O}$, and every $a_{\lambda}$ is the element of the center $Z(A)$, 2. b)

$l\in\ker\varphi\cap Z(B)$, where $Z(B)$ is the center of $B$.

Obviously, if an element of $A\ Wr_{X}B$ satisfies the conditions a) and b), then the element belongs to the center, so $\tilde{Q}\subset Z(A\ Wr_{X}B)$.

For any $y\in X$ and $c\in A$ we define the function $g_{y,c}\in Map(X,A)$ by:

[TABLE]

Let $S$ be the set of elements $(g_{y,c},p)$ of $A\ Wr_{X}B$, where $p\in B$. The set $S$ is also a subset of the restricted product $A\ wr_{X}B$. Let the centralizer of the set $S$ of $A\ Wr_{X}B$ and the centralizer of the set $S$ of $A\ wr_{X}B$ be denoted by $\tilde{C}(S)$ and $C(S)$ respectively. It is clear that $Z(A\ Wr_{X}B)\subset\tilde{C}(S)$ and $Z(A\ wr_{X}B)\subset C(S)$. Therefore, for the group $\tilde{Q}$ we have inclusions:

[TABLE]

Lemma 2.2**.**

The following identities hold:

[TABLE]

Proof.

To prove the lemma it is enough to check the inclusion $\tilde{C}(S)\subset\tilde{Q}$.

Assume the element $(g_{y,c},p)\in S$ and $(f,l)\in\tilde{C}(S)$, where $f,g_{y,c}\in Map(X,A)$, $g_{y,c}$ is defined by (3), and $l,p\in B$. Then, by definition, we get equality $(f,l)(g_{y,c},p)=\leavevmode\nobreak\ (g_{y,c},p)(f,l)$. Hence

[TABLE]

Therefore, $lp=pl$ for any $p$, so $l\in Z(B)$, and we also get

[TABLE]

For $x\neq y,x\neq\varphi_{l}^{-1}(y)$ in (4) we have:

[TABLE]

The equality (5) is true for an arbitrary $p\in B$, so $f$ takes the same value on the whole orbit.

Notice that we can choose $g_{y,c}(x)$ with another fixed element $y$. Therefore $f$ is constant on each orbit of $B$ on $X$, i.e. $f(x)=a_{\lambda}$ for any $x\in O_{\lambda}$, $O_{\lambda}\in\mathcal{O}$.

It remains to show that every $a_{\lambda}$ is the element of the center $Z(A)$ and $l\in\ker\varphi$.

To show this let us analyze the equality (4) for the case $x=y$. There are two cases:

1. (i)

if $\varphi_{l}(y)=y$ we get

[TABLE] 2. (ii)

if $\varphi_{l}(y)\neq y$ we get

[TABLE]

The second case is impossible since $g_{y,c}(y)\neq e$. Hence, $l\in\ker\varphi$. It follows from the first case that for every $y\in X$ we get $f(y)\in Z(A)$ since $g_{y,c}(y)=c$, where $c$ is an arbitrary element of $A$. So, we obtain that the conditions a) and b) hold.

∎

Lemma 2.3**.**

The center $Z(A\ wr_{X}B)$ is the intersection of $Z(A\ Wr_{X}B)$ and $A\ wr_{X}B$, i.e.

[TABLE]

Proof.

Indeed, the inclusion $\left(Z(A\ Wr_{X}B)\cap(A\ wr_{X}B)\right)\subset Z(A\ wr_{X}B)$ is obvious.

Let us check the inverse inclusion. Assume $(f,l)\in Z(A\ wr_{X}B)$. Since $S\subset A\ wr_{X}B$ we get

[TABLE]

∎

Proof of Theorem 2.1.

If we have the unrestricted product $A\ Wr_{X}B$, then the number of orbits in $Z(A\ Wr_{X}B)$ can be infinite. If we have the restricted product $A\ wr_{X}B$, then there can be only a finite number of finite orbits in $Z(A\ wr_{X}B)$.

According to Lemma 2.2 and Lemma 2.3, there are bijections

[TABLE]

[TABLE]

defined by $\psi_{1}(f,l)=(a_{1},a_{2},\dots,a_{s_{1}},l)$ and $\psi_{2}(f,l)=(a_{1},a_{2},\dots,a_{s_{2}},l)$. It is easy to check that $\psi_{1}$ and $\psi_{2}$ are homomorphisms. ∎

Remark 2.4**.**

If $X=\mathbb{Z}_{n}$ we will denote a wreath product of $A$ and $B$ by $A\wr_{n}B$. In this notation it was proved in Theorem 2.1 that $D(A)\times\left(\ker\varphi\cap Z(B)\right)$ is the subgroup of $A\wr_{n}B$ such that

[TABLE]

Corollary 2.5**.**

[TABLE]

[TABLE]

Indeed, for groups $A\mathop{\wr}\limits_{n}\mathbb{Z}$ of the class $\mathcal{G}$ we have only one orbit of the action of $B$ on $X$. According to Proposition 2.1, we obtain the first equivalence of Corollary 2.5. The second is obvious.

For example,

[TABLE]

[TABLE]

[TABLE]

Theorem 2.6**.**

Let $G\in\mathcal{G}$, $\omega$ be any presentation of $G$ in the alphabet $\mathcal{A}$, and $\beta_{1}(\omega)$ be the number of symbols $\mathbb{Z}$ in the presentation $\omega$. Then $Z(G)\simeq\mathbb{Z}^{\beta_{1}(\omega)}$.

Proof.

The proof follows from Remark 2.5 and the induction on the number of symbols 1 and $\mathbb{Z}$ in the presentation $\omega$ denoted by $l(\omega)$. It will be convenient to denote by $\widetilde{\omega}$ the group from the class $\mathcal{G}$ determined by a word $\omega$. In particular, $\omega$ is a presentation of $\widetilde{\omega}$ in the alphabet $\mathcal{A}$. Since $\widetilde{\omega}$ is a group isomorphic to $G$, we have

[TABLE]

Obviously, if $l(\omega)=1$, then $\omega$ is either 1 or $\mathbb{Z}$, so $Z(G)\simeq 1$ or $Z(G)\simeq\mathbb{Z}$ respectively. Suppose we proved that $Z(G)\simeq\mathbb{Z}^{\beta_{1}(\omega)}$ for all words with $l(\omega)\leq k$. Let us show this for $l(\omega)=k+1$. In this case the presentation $\omega$ is either

1. (1)

a direct product $\omega_{1}\times\omega_{2}$ such that $l(\omega_{1})+l(\omega_{2})=k+1$, $l(\omega_{1})\leq k$, $l(\omega_{2})\leq k$, or 2. (2)

a wreath product $\omega_{1}\wr_{n}\mathbb{Z}$, where $l(\omega_{1})=k$.

According to Remark 2.5, the induction assumption and the evident observation $\beta_{1}(\omega_{1})+\beta_{1}(\omega_{2})={\beta_{1}(\omega_{1}\times\omega_{2})}$ we get for the first case

[TABLE]

and for the second case

[TABLE]

∎

3. Commutator subgroup

Theorem 3.1**.**

For any group $G$ the commutator subgroup of $G\mathop{\wr}\limits_{n}\mathbb{Z}$ coincides with the following group

[TABLE]

Proof.

Let us first show that every $g=(g_{1},g_{2},\dots,g_{n},0)$ such that $\prod_{i=1}^{n}g_{i}\in[G,G]$ is in $\left[G\wr_{n}\mathbb{Z},G\wr_{n}\mathbb{Z}\right]$. We will prove that the elements $h_{1},h_{2}$ of the group $G\wr_{n}\mathbb{Z}$,

[TABLE]

lie in the same conjugacy class, i.e.

[TABLE]

Indeed, $f=(e,e,\dots,e,g_{n},g_{n}^{-1},0)$ satisfies the equality (6). It is easy to check that $f=\leavevmode\nobreak\ [c,d]$, where $c=(e,e,\dots,e,g_{n}^{-1},1)$, $d=(e,e,\dots,e,g_{n},0)$, and hence $f\in\left[G\wr_{n}\mathbb{Z},G\wr_{n}\mathbb{Z}\right]$.

Similarly, by induction, we can obtain that the elements

[TABLE]

lie in the same conjugacy class.

Notice that for elements $\alpha=(a,e,\dots,e,0)$, $\beta=(b,e,e,\dots,e,0)$ from $G\wr_{n}\mathbb{Z}$ we have the equality

[TABLE]

So, every $g=(g_{1},g_{2},\dots,g_{n},0)$ such that $\prod_{i=1}^{n}g_{i}\in[G,G]$ is in the same conjugacy class with the element

[TABLE]

Let now $g=(g_{1},g_{2},\dots,g_{n},k)\in\left[G\wr_{n}\mathbb{Z},G\wr_{n}\mathbb{Z}\right]$. Let also $a=(a_{1},a_{2},\dots,a_{n},l)$ and $b=(b_{1},b_{2},\dots,b_{n},p)$ be the elements in $G\wr_{n}\mathbb{Z}.$ By straightforward calculation we obtain

[TABLE]

So, evidently, $k=0$. Obviously, in the notation (7) each $a_{i}$, $b_{i}$, $a^{-1}_{i}$, $b^{-1}_{i}$ enters once for each commutator $aba^{-1}b^{-1}$ and its inverse. Each $g\in\left[G\wr_{n}\mathbb{Z},G\wr_{n}\mathbb{Z}\right]$ is generated by commutators with such property, so the product of its first $n$ coordinates has a form

[TABLE]

where $c_{i}$ may not be different, but the sum of powers of same elements is always zero.

Since $c_{i}c_{j}=c_{j}c_{i}[c_{i}^{-1},c_{j}^{-1}]$ by permutations we cancel out all $c_{i}$, so only commutators will remain. Thus, we get $\prod_{i=1}^{n}g_{i}\in[G,G]$.

∎

Theorem 3.2**.**

For any group $G$ we have the following isomorphisms of quotient groups

[TABLE]

Proof.

Let us construct a homomorphism $\varphi\colon G\wr_{n}\mathbb{Z}\to G/[G,G]\times\mathbb{Z}$ defined by

[TABLE]

To check that $\varphi$ is a homomorphism we compute

[TABLE]

Since $G/[G,G]$ is abelian we get

[TABLE]

hence $\varphi$ is a homomorphism.

It is onto since for each element $(h,n)\in G/[G,G]\times\mathbb{Z}$ there is $(h,e,e,\cdots,n)$ which satisfies

[TABLE]

The kernel of $\varphi$ is then $\ker\varphi=\{(g_{1},g_{2},\dots,g_{n},0)|\prod_{i=1}^{n}g_{i}\in[G,G]\}$, which coincides with $\left[G\wr_{n}\mathbb{Z},G\wr_{n}\mathbb{Z}\right]$. ∎

Theorem 3.3**.**

Let $G\in\mathcal{G}$, $\omega$ be any presentation of $G$ in the alphabet $\mathcal{A}$, and $\beta_{1}(\omega)$ be the number of symbols $\mathbb{Z}$ in the presentation $\omega$. Then $G/[G,G]\simeq\mathbb{Z}^{\beta_{1}(\omega)}$.

Proof.

The proof is similar to Theorem 2.6. One should only use Theorem 3.2 and the fact that for any two groups $A$ and $B$ it holds

[TABLE]

instead of Remark 2.5. To prove (8) it is enough to check that the map $\varphi\colon A\times B\to A/[A,A]\times B/[B,B]$ defined by

[TABLE]

is a surjective homomorphism with the kernel $\ker\varphi=[A\times B,A\times B]$.

Since $\varphi$ is a product of surjective homomorphisms $\varphi_{1}$, $\varphi_{2}$ defined by

[TABLE]

it is a surjective homomorphism as well.

Further, notice that

[TABLE]

Let us check that $\ker\varphi\subset[A\times B,A\times B].$ Indeed, let $(\prod[a_{i},b_{i}],\prod[c_{j},d_{j}])\in\ker\varphi$, where $a_{i},b_{i}\in A$, $c_{j},d_{j}\in B$, then

[TABLE]

Conversely, for any commutator $[(a,b),(c,d)]$ in $[A\times B,A\times B]$ we have

[TABLE]

Theorem is proved.

∎

Now we can get the evident proof of our main result, Theorem1.2. Let us recall it.

Theorem.1.2 Let $G\in\mathcal{G}$, $\omega$ be an arbitrary presentation of $G$ in the alphabet $\mathcal{A}$, and $\beta_{1}(\omega)$ be the number of symbols $\mathbb{Z}$ in the presentation $\omega$. Then there are the following isomorphisms:

[TABLE]

Proof.

Under the same assumptions, we get in Theorem 2.6 that $Z(G)\simeq\mathbb{Z}^{\beta_{1}(\omega)}$, and in Theorem 3.3 that $G/[G,G]\simeq\mathbb{Z}^{\beta_{1}(\omega)}$. Thus, evidently,

[TABLE]

∎