On finite systems of equations in acylindrically hyperbolic groups

TL;DR

This paper proves that in acylindrically hyperbolic groups without finite normal subgroups, finite systems of equations can be simplified to a single equation, revealing structural properties of algebraic sets and subgroup closures.

Contribution

It establishes the equivalence of finite systems of equations to a single equation in acylindrically hyperbolic groups and explores implications for algebraic and verbal closure properties.

Findings

Finite systems of equations are equivalent to a single equation in the specified groups.

Algebraic sets associated with systems are projections of sets from single splitted equations.

H is verbally closed in G if and only if it is algebraically closed in G.

Abstract

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system $S$ of equations with constants from $H$ is equivalent to a single equation. We also show that the algebraic set associated with $S$ is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such equation has the form $w(x_1,\dots,x_n)=h$, where $w\in F(X)$, $h\in H$). From this we deduce the following statement: Let $G$ be an arbitrary overgroup of the above group $H$. Then $H$ is verbally closed in $G$ if and only if it is algebraically closed in $G$. Another corollary: If $H$ is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from $H$ is equivalent to a single equation.

Equations

$$V_G(S)=\{(g_1,\dots ,g_n)\in G^n|f(g_1,\dots ,g_n;H)=1\text{ for all}f\in S\}.$$

$$V_H(f)=V_H(S).$$

$${\text{\bf pr}}_{n}\bigl{(}V_{H}(f)\bigr{)}=\underset{\alpha\in\mathbb{Z}}{\bigcup}V_{H}(S)^{f_{0}^{\alpha}}.$$

$${\text{\bf pr}}_{n}\bigl{(}V_{G}(f)\bigr{)}\supseteq\underset{\alpha\in\mathbb{Z}}{\bigcup}V_{G}(S)^{f_{0}^{\alpha}}.$$

$$V_{H}(S)={\text{\bf pr}}_{n}\bigl{(}V_{H}(f)\bigr{)}\cap{\text{\bf pr}}_{n}\bigl{(}V_{H}(g)\bigr{)}.$$

$$S=\{W_i(x_1,\dots ,x_n;H)=1|i=1,\dots ,m\}$$

$$d(x,gx)⩽\epsilon \text{and}d(y,gy)⩽\epsilon .$$

$$\bigl{|}\{f\in G\,|\,|f|_{X}\leqslant\varepsilon,\hskip 5.69054pt|g^{-1}fg|_{X}\leqslant\varepsilon\}\bigr{|}\leqslant N.$$

$$\mathcal{H}=\underset{\lambda \in \Lambda }{⨆}{H}_{\lambda }.$$

$$d(g^{n}x,g^{m}x)⩾\frac{1}{\varkappa }|n-m|-\epsilon .$$

$$⟨c⟩\cap ⟨g⟩=1.$$

$$W(a_1,\dots ,a_k)=W(x_1,\dots ,x_k)$$

$${\mathcal{U}}_k(x_1,\dots ,x_k):=W_k(x_1,\dots ,x_k,1^{k-2}).$$

$$V_H(S^{\prime })=V_H(S)\times \{g_1\}\times \cdots \times \{g_k\}$$

$$f:W_{k+2}(a_1,\dots ,a_{k+2},1^k)=W_{k+2}(a_1,\dots ,a_{k+2},s_1,\dots ,s_k).$$

$$a{b}^s,a{b}^t,a{b}^{k_1},\dots ,a{b}^{k_m},a{b}^{l_1},\dots ,a{b}^{l_m}.$$

$$T=(a{b}^s,a{b}^t,h_i^{-1}a{b}^{k_i}{h}_i,h_i^{-1}a{b}^{l_i}{h}_i;i=1,\dots ,m)$$

$$f_0={\mathcal{U}}_{2m+2}(a{b}^s,a{b}^t,h_i^{-1}a{b}^{k_i}{h}_i,h_i^{-1}a{b}^{l_i}{h}_i;i=1,\dots ,m).$$

$$f_1^{-1}={\mathcal{U}}_{2m+2}(y{z}^s,y{z}^t,w_i(\underline{x}{)}^{-1}y{z}^{k_i}{w}_i(\underline{x}),w_i(\underline{x}{)}^{-1}y{z}^{l_i}{w}_i(\underline{x});i=1,\dots ,m).$$

$$(x_1,\dots ,x_n,y,z)=(C_1,\dots ,C_n,A,B).$$

Full text

We also show that the algebraic set associated with $S$ is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such equation has the form $w(x_{1},\dots,x_{n})=h$, where $w\in F(X)$, $h\in H$).

From this we deduce the following statement: Let $G$ be an arbitrary overgroup of the above group $H$. Then $H$ is verbally closed in $G$ if and only if it is algebraically closed in $G$.

Another corollary: If $H$ is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from $H$ is equivalent to a single equation.

1. Introduction

Study of equations in groups is a classical subject of group theory with long history, nice results and many intriguing problems (see the inspiring paper of Neumann [12] of 1943, the survey of Roman’kov [18] of 2012 and the recent paper of Klyachko and Thom [8]).

Acylindrically hyperbolic groups (definied by Osin in [14], see Definition 3.2 below) is a large class of groups that is intensively studied in the modern geometric group theory. This class includes non-(virtually cyclic) groups that are hyperbolic relative to proper subgroups, many 3-manifold groups, groups of deficiency at least 2, many groups acting on trees, non-(virtually cyclic) groups acting properly on proper CAT(0)-spaces and containing rank-one elements, non-cyclic directly indecomposable right-angled Artin groups, all but finitely many mapping class groups, $Out(F_{n})$ for $n\geqslant 2$, and many other interesting groups; see the survey of Osin [15].

However almost nothing is known on solutions of equations and related problems in the class of acylinfrically hyperbolic groups. In [2], we described solutions of certain equations of the form $x^{n}y^{m}=a^{n}b^{m}$ in acylindrically hyperbolic groups. Using this description, we studied in [2] the verbal closedness (see Definition 2.2 below) of acylindrically hyperbolic subgroups in groups.

One of the corollaries there solves Problem 5.2 from the paper [11] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts.

More information on algebraic and verbal closedness of groups in overgroups or in classes of groups can be found in [2, Sections 1, 2, 15]. The present paper is a continuation of [2]. We formulate briefly the main results. For convenience, we call a group clean if it does not contain nontrivial finite normal subgroups.

$\bullet$ Theorem A, says that if $H$ is a clean acylindrically hyperbolic group, then any finite system of equations with constants in $H$ has the same set of solutions in $H$ as a single equation. Moreover, this set is a projection (up to conjugacy) of the set of solutions of a single splitted equation (see Definition 2.1 below).

$\bullet$ Theorem C, says that for any clean acylindrically hyperbolic group $H$ and any overgroup $G$ of $H$ the notions of verbal and algebraic closedness of $H$ in $G$ are equivalent. Some special cases of this theorem were considered earlier in [7] and [10], see Remark 7.1 below.

$\bullet$ A part of Corollary D says that if $H$ is a finitely generated clean acylindrically hyperbolic group and $G$ is a finitely presented overgroup of $H$, then the notions for $H$ to be verbally closed in $G$, to be algebraically closed in $G$, and to be a retract of $G$ are equivalent.

$\bullet$ A special case of Corollary B says that if $H$ is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with constants from $H$ and finitely many variables is equivalent to a single equation with constants from $H$, i.e., they have the same sets of solutions in $H$.

In the proof we use test words for acylindrically hyperbolic groups, see [2] and Section 5 below.

2. Full formulations of main results and definitions

Let $H$ be a group. An equation with variables $x_{1},\dots,x_{n}$ and constants from $H$ is an element of the free product $F_{n}\ast H$, where $F_{n}$ is the free group with basis $x_{1},\dots,x_{n}$. Sometimes we write an equation $f$ in the form $f(x_{1},\dots,x_{n};H)$ stressing that $f$ involves the variables $x_{1},\dots,x_{n}$ and constants from $H$. Sometimes, for convenience, we write an equation $f_{0}f_{1}$ in the form $f_{0}=f_{1}^{-1}$.

Let $S\subseteq F_{n}\ast H$ be a system of equations and let $G$ be an overgroup of $H$.A tuple $(g_{1},\dots,g_{n})$ with components from $G$ is called a solution of the system $S$ in $G$ if $f(g_{1},\dots,g_{n};H)=1$ in $G$ for every equation $f(x_{1},\dots,x_{n};H)$ from $S$. Let $V_{G}(S)$ be the set of all solutions of the system $S$ in $G$, i.e.,

[TABLE]

2.1. Systems of equations versus a single equation

Definition 2.1**.**

An equation $f\in F_{n}\ast G$ is called splitted if it has the form $wg$, where $w\in F_{n}$ and $g\in G$. **

For $m\geqslant n$, let ${\text{\bf pr}}_{n}:G^{m}\rightarrow G^{n}$ be the projection to the first $n$ coordinates, i.e., ${\text{\bf pr}}_{n}(g_{1},\dots,g_{m})=(g_{1},\dots,g_{n})$. For $g,u\in G$, we denote $g^{u}=u^{-1}gu$.For $(g_{1},\dots,g_{n})\in G^{n}$ and $u\in G$, we set $(g_{1},\dots,g_{n})^{u}=(g_{1}^{u},\dots,g_{n}^{u})$. The first main result of this paper is the following theorem.

Theorem A. Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. Let $S\subset F_{n}\ast H$ be a finite system of equations with constants from $H$. Then the following statements hold:

1. (1)

There exists a single equation $f\in F_{n}\ast H$ such that

[TABLE] 2. (2)

There exists a natural $k\geqslant n$ and a single splitted equation $f\in F_{k}\ast H$ of the form $f_{1}f_{0}$, where $f_{1}\in F_{k}$ and $f_{0}\in H$ such that the following two properties are satisfied:

1. (a)

[TABLE] 2. (b)

For any overgroup $G$ of the group $H$ we have

[TABLE] 3. (3)

There exist a natural $k\geqslant n$ and two splitted equations $f,g\in F_{k}\ast H$ such that

[TABLE]

Note that in the proof of this theorem we essentially use a proposition on test words in acylindrically hyperbolic groups obtained in [2].

Recall that a group $H$ is called equationally noetherian if every system of equations with constants from $H$ and a finite number of variables is equivalent to a finite subsystem, see [1]. Since hyperbolic groups are equationally noetherian (see [17, Corolary 6.13] for the general case and  [19, Theorem 1.22] for the torsion-free case), the following corollary follows directly from Theorem A.

Corollary B. *Let $H$ be a non-(virtually cyclic) hyperbolic group without nontrivial finite normal subgroups. Then every (possibly infinite) system of equations with constants from $H$ and finitely many variables is equivalent to a single equation with constants from $H$, i.e., they have the same sets of solutions in $H$. *

2.2. Verbal and algebraic closedness.

Let $X=\{x_{1},x_{2},\dots\}$ be a countably infinite set of variables and let $F(X)$ be the free group with basis $X$.We recall definitions of algebraically (verbally) closed subgroups and retracts.

Definition 2.2**.**

Let $H$ be a subgroup of a group $G$.

1. (a)

(see [13, 11]) The subgroup $H$ is called algebraically closed in $G$ if for any finite system of equations

[TABLE]

with constants from $H$ the following holds: if $S$ has a solution in $G$, then it has a solution in $H$. 2. (b)

(see [11, Definition 1.1]) The subgroup $H$ is called verbally closed in $G$ if for any word $W\in F(X)$ and any element $h\in H$ the following holds: if the equation $W(x_{1},\dots,x_{n})=h$ has a solution in $G$, then it has a solution in $H$. 3. (c)

The subgroup $H$ is called a retract of $G$ if there is a homomorphism $\varphi:G\rightarrow H$ such that $\varphi|_{H}={\rm id}$. The homomorphism $\varphi$ is called a retraction.

Obviously, if $H$ is a retract of $G$, then $H$ is algebraically closed in $G$. Algebraic closedness implies verbal closedness, but the converse implication is not valid in general, see example in [2, Remark 13.2].

The following proposition of Myasnikov and Roman’kov says that, under some general assumptions, the property of $H$ to be algebraically closed in $G$ is equivalent to the property of $H$ to be a retract of $G$.

Recall that a group $G$ is called finitely generated over a subgroup $H$ if there exists a finite subset $X\subseteq G$ such that $G=\langle X,H\rangle$.

Proposition 2.3**.**

([11, Proposition 2.2])* Let $H$ be a subgroup of a group $G$. Suppose that at least one of the following holds:*

1. (a)

$H$* is finitely generated and $G$ is finitely presented.* 2. (b)

$H$* is equationally noetherian and $G$ is finitely generated over $H$.*

Then $H$ is algebraically closed in $G$ if and only if $H$ is a retract of $G$.

Our second main theorem establishes the equivalence of verbal and algebraic closedness for clean acylindrically hyperbolic subgroups of arbitrary groups.

Theorem C. *Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups and let $G$ be an arbitrary overgroup of $H$. Then $H$ is verbally closed in $G$ if and only if $H$ is algebraically closed in $G$. *

The proof of this theorem is very short modulo the second statement of Theorem A, see Section 7. The following corollary follows directly from Theorem C and Proposition 2.3. This corollary was already obtained by the author in [2] without appealing to Theorem C.

Corollary D. ([2, Theorems 2.2 and 2.4]) Let $H$ be a subgroup of a group $G$ such that at least one of the following holds:

1. (a)

$H$ is finitely generated and $G$ is finitely presented. 2. (b)

$H$ is equationally noetherian and $G$ is finitely generated over $H$.

Suppose additionally that $H$ is acylindrically hyperbolic and does not have nontrivial finite normal subgroups. Then the following three statements are equivalent.

1. (1)

$H$ is algebraically closed in $G$. 2. (2)

$H$ is verbally closed in $G$. 3. (3)

$H$ is a retract of $G$.

3. Necessary definitions

3.1. Definitions of acylindrically hyperbolic groups

All actions of groups on metric spaces are assumed to be isometric in this paper.

Definition 3.1**.**

(see [3] and Introduction in [14]) An action of a group $G$ on a metric space $S$ is called acylindrical if for every $\varepsilon>0$ there exist $R,N>0$ such that for every two points $x,y$ with $d(x,y)\geqslant R$, there are at most $N$ elements $g\in G$ satisfying

[TABLE]

Given a generating set $X$ of a group $G$, we say that the right Cayley graph $\Gamma(G,X)$ is acylindrical if the left action of $G$ on $\Gamma(G,X)$ is acylindrical. For Cayley graphs, the acylindricity condition can be rewritten as follows: for every $\varepsilon>0$ there exist $R,N>0$ such that for any $g\in G$ of length $|g|_{X}\geqslant R$ we have

[TABLE]

Recall that an action of a group $G$ on a hyperbolic space $S$ is called elementary if the limit set of $G$ on the Gromov boundary $\partial S$ contains at most 2 points.

Definition 3.2**.**

(see [14, Definition 1.3]) A group $G$ is called acylindrically hyperbolic if it satisfies one of the following equivalent conditions:

1. (${\rm AH}_{1}$)

There exists a generating set $X$ of $G$ such that the corresponding Cayley graph $\Gamma(G,X)$ is hyperbolic, $|\partial\Gamma(G,X)|>2$, and the natural action of $G$ on $\Gamma(G,X)$ is acylindrical. 2. (${\rm AH}_{2}$)

$G$ admits a non-elementary acylindrical action on a hyperbolic space.

In the case (AH1), we also write that $G$ is acylindrically hyperbolic with respect to $X$.

3.2. Hyperbolically embedded subgroups

We use this notion only in formulations of lemmas in Section 4. For convenience of the reader, we recall relevant definitions from [4]; see also [14].

Let $G$ be a group with a fixed collection of subgroups $\{H_{\lambda}\}_{\lambda\in\Lambda}$. Given a symmetric subset $X\subseteq G$ such that $G$ is generated by $X$ together with the union of all $H_{\lambda}$, we denote by $\Gamma(G,X\sqcup\mathcal{H})$ the right Cayley graph of $G$ whose edges are labelled by letters from the alphabet $X\sqcup\mathcal{H}$, where

[TABLE]

We consider the Cayley graph $\Gamma(H_{\lambda},H_{\lambda})$ as a complete subgraph of $\Gamma(G,X\sqcup~{}\mathcal{H})$.

Definition 3.3**.**

(see [4, Definition 4.2])* For every $\lambda\in\Lambda$, we introduce a relative metric $\widehat{d}_{\lambda}:H_{\lambda}\times H_{\lambda}\rightarrow[0,+\infty]$ as follows. Let $a,b\in H_{\lambda}$. A path in $\Gamma(G,X\sqcup\mathcal{H})$ from $a$ to $b$ is called $H_{\lambda}$-admissible* if it has no edges in the subgraph $\Gamma(H_{\lambda},H_{\lambda})$. The distance $\widehat{d}_{\lambda}(a,b)$ is defined to be the length of a shortest $H_{\lambda}$-admissible path connecting $a$ to $b$ if such exists. If no such path exists, we set $\widehat{d}_{\lambda}(a,b)=\!\infty$. **

Definition 3.4**.**

(see [4, Definition 4.25]) Let $G$ be a group, $X$ a symmetric subset of $G$. A collection of subgroups $\{H_{\lambda}\}_{\lambda\in\Lambda}$ of $G$ is called hyperbolically embedded in $G$ with respect to $X$ (we write $\{H_{\lambda}\}_{\lambda\in\Lambda}\hookrightarrow_{h}(G,X)$) if the following hold.

1. (a)

The group $G$ is generated by $X$ together with the union of all $H_{\lambda}$ and the Cayley graph $\Gamma(G,X\sqcup\mathcal{H})$ is hyperbolic. 2. (b)

For every $\lambda\in\Lambda$, the metric space $(H_{\lambda},\widehat{d}_{\lambda})$ is proper. That is, any ball of finite radius in $H_{\lambda}$ contains finitely many elements.

Further, we say that $\{H_{\lambda}\}_{\lambda\in\Lambda}$ is hyperbolically embedded in $G$ if $\{H_{\lambda}\}_{\lambda\in\Lambda}\hookrightarrow_{h}(G,X)$ for some $X\subseteq G$. **

It was proved in [14, Theorem 1.2] that a group $G$ is acylindrically hyperbolic if and only if it contains a proper infinite hyperbolically embedded subgroup.

3.3. Elliptic and loxodromic elements

The following definition is standard.

Definition 3.5**.**

Given a group $G$ acting on a metric space $S$, an element $g\in G$ is called elliptic if some (equivalently, any) orbit of $g$ is bounded, and loxodromic if the map $\mathbb{Z}\rightarrow S$ defined by $n\mapsto g^{n}x$ is a quasi-isometric embedding for some (equivalently, any) $x\in S$. That is, for $x\in S$, there exist $\varkappa\geqslant 1$ and $\varepsilon\geqslant 0$ such that for any $n,m\in\mathbb{Z}$ we have

[TABLE]

Let $X$ be a generating set of $G$. We say that $g\in G$ is elliptic (respectively, loxodromic) with respect to $X$ if $g$ is elliptic (respectively, loxodromic) for the canonical left action of $G$ on the Cayley graph $\Gamma(G,X)$.

Note that even in the case of groups acting on hyperbolic spaces, there may be other types of actions (see [5, Section 8.2] and [14, Section 3]). However, if $G$ is a group acting acylindrically on a hyperbolic space, then every element of $G$ is either elliptic or loxodromic. This was first proved Bowditch [3, Lemma 2.2]; a more general statement was proved by Osin in [14, Theorem 1.1]).

Recall that if $G$ is an acylindrically hyperbolic group with respect to a generating set $X$, then every loxodromic, with respect to $X$, element $g\in G$ is contained in a unique maximal virtually cyclic subgroup $E_{G}(g)$ of $G$, see[4, Lemma 6.5]. This subgroup is called the elementary subgroup associated with $g$; it can be described as follows (see equivalent definitions in [4, Corollary 6.6]):

[TABLE]

Clearly, the centralizer $C_{G}(g)$ of $g$ is contained in $E_{G}(g)$.

Recall that two elements $a,b\in G$ of infinite order are called commensurable if there exist $g\in G$ and $s,t\in\mathbb{Z}\setminus\{0\}$ such that $a^{s}=g^{-1}b^{t}g$.

Preparing to the proof of Theorem A, we need to produce many non-commensurable loxodromic elements $g$ with respect to the same generating set of $G$ and with the additional property that $E_{G}(g)=\langle g\rangle$. Moreover, we need that these elements have certain form. This technical part of the proof is the subject of Section 4.

4. Special elements

We reproduce the following definition from [2].

Definition 4.1**.**

Suppose that $G$ is an acylindrically hyperbolic group.

1. (a)

An element $g\in G$ is called special if there exists a generating set $X$ of $G$ such that

• $G$ is acylindrically hyperbolic with respect to $X$,

• $g$ is loxodromic with respect to $X$, and

• $E_{G}(g)=\langle g\rangle$.

In this case $g$ is called special with respect to $X$. 2. (b)

Elements $g_{1},\dots,g_{k}\in G$ are called jointly special if there exists a generating set $X$ of $G$ such that each $g_{i}$ is special with respect to $X$.

Note that point (a) of this definition was already used in the case of relatively hyperbolic groups (see comments in [16, Section 3]).

The purpose of this section is to prove Proposition 4.5; it will be used in Section 5 to construct certain test words in acylindrically hyperbolic groups. We deduce this proposition formally from the following three lemmas.

Lemma 4.2**.**

(see [2, Lemma 10.3])* Let $G$ be a group, $X\subseteq G$, $H\hookrightarrow_{h}(G,X)$ a finitely generated infinite subgroup. Then for any finite collection of elements $a_{1},\dots,a_{s}\in G\setminus H$ and any infinite subset $\widetilde{H}\subseteq H$, there exist elements $h_{1},\dots,h_{s}\in\widetilde{H}$ such that $a_{1}h_{1},\dots,a_{s}h_{s}$ are paarwise non-commensurable loxodromic elements with respect to the action of $G$ on $\Gamma(G,X\sqcup H)$.*

Remark. We will use a special case of this lemma where $a_{1}=\dots=a_{s}$. In this case this lemma is very similar to [4, Corollary 6.12] and, actually, can be deduced from the proof of this corollary.

Lemma 4.3**.**

([2, Lemma 10.4])* Let $G$ be an acylindrically hyperbolic group with respect to a generating set $Y$ and let $a,b\in G$ be two non-commensurable loxodromic with respect to $Y$ elements, where, additionally, $a$ is special. Then there exists a positive integer $n_{0}$ such that for any $n,m\geqslant n_{0}$ the element $g=a^{n}b^{m}$ is special with respect to some generating set, in particular $E_{G}(g)=\langle g\rangle$.*

Lemma 4.4**.**

([2, Lemma 10.5])* Suppose that $G$ is an acylindrically hyperbolic group without nontrivial finite normal subgroups. Then there exist an element $g\in G$ and a generating set $Y$ of $G$ such that $g$ is special with respect to $Y$ and $\langle g\rangle\hookrightarrow_{h}(G,Y)$.*

Proposition 4.5**.**

Suppose that $G$ is an acylindrically hyperbolic group without nontrivial finite normal subgroups. Then, there are special loxodromic elements $a,g\in G$ such that for any integer $k>0$, the coset $a\langle g\rangle$ contains $k$ pairwise non-commensurable and jointly special elements.

Proof. By Lemma 4.4, there exist an element $g\in G$ and a generating set $Y$ of $G$ such that $g$ is special with respect to $Y$ and $H\hookrightarrow_{h}(G,Y)$, where $H=\langle g\rangle$. It follows from Definition 4.1 that $G$ is acylindrically hyperbolic with respect to $Y$.

Let $b\in G\setminus H$ be an arbitrary element. By Lemma 4.2, there exist two non-commensurable loxodromic elements $bg^{s}$, $bg^{t}$ with respect to $Y\sqcup H$.It follows that they are loxodromic with respect to $Y$. At least one of them, say $c:=bg^{s}$ is non-commensurable with $g$. In particular,

[TABLE]

By Lemma 4.3, there exists a positive integer $n_{0}$ such that for any $n,m\geqslant n_{0}$ the element $c^{n}g^{m}$ is special with respect to some generating set, in particular

[TABLE]

By Lemma 4.2, for any $k\in\mathbb{N}$, there exist natural numbers $m_{1}<m_{2}<\dots<m_{k+1}$ such that $m_{1}\geqslant n_{0}$ and the elements $c^{n_{0}}g^{m_{1}},\dots,c^{n_{0}}g^{m_{k+1}}$ are pairwise non-commensurable and loxodromic with respect to $Y\sqcup H$. Then they are loxodromic with respect to $Y$. Moreover, by (4.1), we have $E_{G}(c^{n_{0}}g^{m_{i}})=\langle c^{n_{0}}g^{m_{i}}\rangle$ for $i=1,\dots,k+1$. Thus, these elements are pairwise non-commensurable and jointly special with respect to $Y$.

We set $a=c^{n_{0}}g^{m_{1}}$. Then $a$, $g$ and $k$ elements $ag^{m_{2}-m_{1}},\dots,ag^{m_{k+1}-m_{1}}$ satisfy the conclusion of proposition. $\Box$

Remark. One can prove a stronger version of this lemma, saying that for any infinite subset $I\subseteq a\langle g\rangle$, there exists an infinite subset of $I$ consisting of pairwise non-commensurable and jointly special elements.

5. Test words in acylindrically hyperbolic groups

The history of test words in free groups is illuminated in [6]. In this paper Ivanov constructed the so-called $C$-test words in free groups. In [9] Lee constructed $C$-test words with some additional property. In this section we construct certain test words in acylindrically hyperbolic groups.

Definition 5.1**.**

(see [2, Definition 12.1])* Let $H$ be a group and let $a_{1},\dots,a_{k}$ be some elements of $H$. A word $W(x_{1},\dots,x_{k})$ from $F_{k}$ is called an $(a_{1},\dots,a_{k})$-test word* if for every solution $(b_{1},\dots,b_{k})$ of the equation

[TABLE]

in $H$, there exists a number $\alpha\in\mathbb{Z}$ such that $b_{i}=a_{i}^{U^{\alpha}}$ for $i=1,\dots,k$, where $U=W(a_{1},\dots,a_{k})$. **

Notation. We write $\mathbb{1}^{k}$ for the tuple $(\underbrace{1,\dots,1}_{k})$.

The following proposition is a special case of Proposition 12.4 from [2] .

Proposition 5.2**.**

(see [2, Proposition 12.4])* Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups and let $a_{1},\dots,a_{k}\in H$ (where $k\geqslant 3$) be jointly special and pairwise non-commensurable elements. Then there is an $(a_{1},\dots,a_{k},\mathbb{1}^{k-2})$-test word $W_{k}(x_{1},\dots,x_{k},y_{3},\dots,y_{k})$.*

Moreover, one can choose this test word so that the elements $a_{1},\dots,a_{k}$ together with $W_{k}(a_{1},\dots,a_{k},\mathbb{1}^{k-2})$ are jointly special and pairwise non-commensurable.

In the following section we need a weaker version of Proposition 5.2, which will also simplify notations. We set

[TABLE]

Then we obtain the following corollary.

Corollary 5.3**.**

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups and let $a_{1},\dots,a_{k}\in H$ (where $k\geqslant 3$) be jointly special and pairwise non-commensurable elements. Then there is an $(a_{1},\dots,a_{k})$-test word $\mathcal{U}_{k}(x_{1},\dots,x_{k})$ such that the elements $a_{1},\dots,a_{k}$ together with $\mathcal{U}_{k}(a_{1},\dots,a_{k})$ are jointly special and pairwise non-commensurable.

6. Proof of Theorem A

Lemma 6.1**.**

Let $H$ be a group. For any finite system of equations $S\subset F_{n}\ast H$ there exists a finite system $S^{\prime}\subset F_{n}\ast H$ consisting of only splitted equations such that $|S^{\prime}|\geqslant|S|$ and

[TABLE]

for some elements $g_{1},\dots,g_{k}\in H$.

Proof. To define $S^{\prime}$, one should replace each constant $h$ in $S$ by a new variable $x_{h}$ and add the equation $x_{h}h^{-1}$. $\Box$

Notation. To shorten notation, we write $\underline{x}$ instead of the tuple $(x_{1},\dots,x_{n})$.

Proof of Theorem A.

(1) Let $S=\{s_{1},\dots,s_{k}\}$, where $s_{i}\in F_{n}\ast H$. We take arbitrary $(k+2)$ jointly special and pairwise non-commensurable elements $a_{1},\dots,a_{k+2}\in H$. The existence of such elements is guaranteed by Proposition 4.5. By Proposition 5.2, there exists an $(a_{1},\dots,a_{k+2},\mathbb{1}^{k})$-test word $W_{k+2}(z_{1},\dots,z_{k+2},y_{1},\dots,y_{k})$. Then the desired equation is

[TABLE]

(2) By Lemma 6.1, we may assume that $S$ consists of splitted equations:

[TABLE]

By Proposition 4.5, there exist special elements $a,b$ such that the coset $a\langle b\rangle$ contains $(2m+2)$ pairwise non-commensurable jointly special elements, say

[TABLE]

Then the elements of the tuple

[TABLE]

are also pairwise non-commensurable and jointly special. Let $\mathcal{U}_{2m+2}$ be the $T$-test word, see Corollary 5.3. We set

[TABLE]

Now we introduce $2$ new variables $y,z$ and set

[TABLE]

We show that the splitted equation $f$ written in the form

[TABLE]

satisfies the statements (a) and (b) of Theorem A.

(a) Suppose that $f$ has a solution in $H$, say

[TABLE]

We shall show that there exists $\alpha\in\mathbb{Z}$ such that $(C_{1},\dots,C_{n})^{f_{0}^{-\alpha}}$ is a solution of the system $S$.

Set $H_{i}=w_{i}(\underline{C})$. Then we have

[TABLE]

By definition of the test word (see Definition 5.1), applied to $\mathcal{U}_{2m+2}$ and (6.3), there exists $\alpha\in\mathbb{Z}$ such that the formulas (6.4) – (6.7) are valid:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

It follows from (6.4) and (6.5) that $(b^{s-t})^{f_{0}^{\alpha}}=B^{s-t}$. Since $b$ is special, we have

[TABLE]

From (6.4) and (6.8), we obtain that

[TABLE]

Substituting (6.8) and (6.9) in (6.6) and (6.7), we deduce that

[TABLE]

The latter equation holds since the elements $ab^{k_{i}}$ and $ab^{l_{i}}$ are special. This intersection is trivial since $ab^{k_{i}}$ and $ab^{l_{i}}$ are non-commensurable. Therefore

[TABLE]

for $i=1,\dots,m$. Hence $\underline{C}^{f_{0}^{-\alpha}}$ is a solution of $S$.

(b) Since $V_{G}(f)$ is invariant under conjugation by the element $f_{0}$, it suffices to check that

[TABLE]

The latter is trivial: If $(x_{1},\dots,x_{n})=(c_{1},\dots,c_{n})$ is a solution of the system $S$ in $G$, then

[TABLE]

is a solution of the equation (6.2) in $G$.

(3) We may again assume that $S$ has the form (6.1). We may additionally assume that the set $\{h_{1},\dots,h_{m}\}$ of right sides of equations from $S$ contains two non-commensurable special elements from $H$; otherwise we could take two non-commensurable special elements $u,v\in H$ and add two equations $x_{n+1}=u$ and $x_{n+2}=v$ to $S$. Obviously, the set of solutions of the old system $S$ is a projection of the set of solutions of the new system $S$.

In the following we will use the tuple $T$, the element $f_{0}$ and the equation $f$ defined in (a). Thus, we have

[TABLE]

By Corollary 5.3, all components of $T$ together with $f_{0}$ are pairwise non-commensurable and jointly special. Let $T^{\prime}$ be the tuple obtained from $T$ by adding the component $f_{0}$:

[TABLE]

Let $\mathcal{U}_{2m+3}$ be the $T^{\prime}$-test word from Corollary 5.2. We set

[TABLE]

Now we introduce new variables $t,y,z$ and define the word

[TABLE]

Let $g$ be the equation $g_{1}g_{0}$. Using the same arguments as in the proof of (a), we obtain

[TABLE]

Claim 1. Suppose that for some $\alpha,\beta\in\mathbb{Z}$ we have

[TABLE]

Then $\alpha=\beta=0$.

Proof. By assumption, the set $\{h_{1},\dots,h_{m}\}$ of right sides of equations from $S$ contains two non-commensurable special elements, say $u,v$. Then

[TABLE]

and we deduce

[TABLE]

The penultimate equation holds since $u$ and $b$ are special, and the latter equation holds since $u$ and $v$ are non-commensurable.

By Corollary 5.3, all components of $T^{\prime}$ together with $g_{0}$ are pairwise non-commensurable and jointly special. In particular, $g_{0}$ and $f_{0}$ are non-commensurable and have infinite orders. From this and $f_{0}^{\alpha}=g_{0}^{\beta}$, we obtain $\alpha=\beta=0$. $\Box$

This claim and equations (6.10) and (6.11) imply that

[TABLE]

$\Box$

Remark 6.2**.**

In the general case, one splitted equation in statement (3) of Theorem A is not sufficient. Indeed, let $H$ be an arbitrary nontrivial group and let $S=\{w_{i}(x_{1},\dots,x_{n})=h_{i}\,|\,i=1,\dots,m\},$ $m\geqslant 2$, be a finite system of splitted equations with constants $h_{i}$ from $H$ such that $C_{H}(h_{1})\cap C_{H}(h_{2})=1$ and $V_{H}(S)\neq\emptyset$. Then for any splitted equation $f\in F_{k}\ast H$, where $k\geqslant n$, we have

[TABLE]

This follows from the following obsevations:

1. (a)

If $f$ is a splitted equation of the form $f_{1}f_{0}$, where $f_{1}\in F_{k}$ and $f_{0}\in H$, then \bigl{(}V_{H}(f)\bigr{)}^{f_{0}}=V_{H}(f). Moreover, we have \bigl{(}V_{H}(f)\bigr{)}^{g}=V_{H}(f) for any $g\in H$ if $f_{0}=1$. 2. (b)

$V_{H}(S)^{g}\cap V_{H}(S)=\emptyset$ for every nontrivial $g\in H$. This can be proved similarly to the proof of Claim 1.

7. Proof of Theorem C

Proof. Suppose that $H$ is verbally closed in $G$. We show that $H$ is algebraically closed in $G$. Let $S$ be a finite system of equations with constants from $H$ such that $V_{G}(S)\neq\emptyset$. We shall show that $V_{H}(S)\neq\emptyset$.

Let $f$ be a splitted equation as in statement (2) of Theorem A. By part (b) of this statement, we have V_{G}(S)\subseteq{\text{\bf pr}}_{n}\bigl{(}V_{G}(f)\bigr{)}, hence $V_{G}(f)\neq\emptyset$. Since $H$ is verbally closed in $G$, we have $V_{H}(f)\neq\emptyset$. By part (a) of statement (2) of Theorem A, we have $V_{H}(S)\neq\emptyset$. Thus, $H$ is algebraically closed in $G$. The converse implication is obvious. $\Box$

Remark 7.1**.**

Consider the free product

[TABLE]

where $\mathfrak{A}$ is an arbitrary set of cardinal larger than 1 and each $H_{\alpha}$ is nontrivial. In [10] Mazhuga showed that if $H$ is a verbally closed subgroup of a group $G$, then $H$ is algebraically closed in $G$. This result (except the very special case $H=\mathbb{Z}_{2}\ast\mathbb{Z}_{2}$, which was first considered in [7]) follows from our Theorem C.

Indeed, $H$ can be splitted as $H=A\ast B$, where $A$ and $B$ are nontrivial; hence $H$ is relatively hyperbolic with respect to $\{A,B\}$. It is well known that if a non-(virtually cyclic) group is relatively hyperbolic with respect to a collection of proper subgroups, then it is acylindrically hyperbolic. Therefore $H$ is acylindricaly hyperbolic except the case, where $H\cong\mathbb{Z}_{2}\ast\mathbb{Z}_{2}$. **