Abstract:
The theory of $n$-valued groups and its applications is developed by going over from groups defined axiomatically to combinatorial groups defined by generators and relations. A wide class of cyclic $n$-valued groups is introduced on the basis of cyclically presented groups. The best-known cyclically presented groups are the Fibonacci groups introduced by Conway. The problem of the existence of the orbit space of $n$-valued groups is related to the problem of the integrability of $n$-valued dynamics. Conditions for the existence of such spaces are presented. Actions of cyclic $n$-valued groups on $\mathbb R^3$ with orbit space homeomorphic to $S^3$ are constructed. The projections $\mathbb R^3 \to S^3$ onto the orbit space are shown to be connected, by means of commutative diagrams, with coverings of the sphere $S^3$ by three-dimensional compact hyperbolic manifolds which are cyclically branched along a hyperbolic knot.
Bibliography: 54 titles.
The research by V. M. Buchstaber was carried out within the framework of the HSE project “Mirror Laboratories”. The research by A. Yu. Vesnin was carried out within the framework of the agreement on Mirror Laboratories, with the support of the Development Program of Tomsk State University (project Priority 2030 no. НУ 2.0.1.23 ОНГ).
The foundations of the theory of formal $n$-valued groups were laid by Buchstaber and Novikov [1] in connection with the theory of characteristic classes of vector bundles. The axiomatics and results concerning the theory of $n$-valued (finite, discrete, topological, or algebraic-geometric) groups were obtained by Buchstaber in the series of papers [2]–[5]. This theory was further developed by Buchstaber and Rees in [6] and [7]. Important examples of $n$-valued groups are cyclic $n$-valued groups. In [8] Buchstaber and Veselov used the natural link between presentations of cyclic $n$-valued groups and the theory of discrete-time dynamical systems; see also [9]. In contrast to single-valued groups, the problem of the construction of cyclic $n$-valued groups has turned out to be nontrivial: see [10] and Problem 1.1.
In this paper we develop the theory of $n$-valued groups by going over from groups defined axiomatically to combinatorial groups defined by generators and relations. We introduce a wide class of cyclic $n$-valued groups, and $n$-valued groups in this class correspond to single-valued cyclically presented groups, that is, groups of the form
where $w=w (x_1, \dots, x_n)$ is some word in the free group $\mathbb F_n=\langle x_1, \dots, x_n\rangle$ and $\theta$ is an automorphism permuting the generators cyclically. The class of cyclically presented groups is quite rich. Here are some examples: the Higman group ${H=G_4 (x_1^{-1} x_2 x_1 x_2^{-2})}$, constructed by Higman [11]; the Fibonacci groups $F(2,n)=G_n (x_1 x_2 x_3^{-1})$, $n \geqslant 3$ (see [12] and [13]), and their numerous generalizations (see [14]–[16]). In 1965, when the problem of the classification of finite groups was in the centre of attention of group theorists, Conway [17] posed the question of whether the Fibonacci group F(2,5) is a finite cyclic group of order 11. Several proofs of this fact were published in [18]. It was shown later that $F(2,n), n \geqslant 3$, is finite if and only if $n=3,4,5,7$; see, for example, [19]. In 1986 the Sieradski groups $G_n (x_1 x_3 x_2^{-1})$, $n \geqslant 3$, were introduced [20], and in 1998 the fractional Fibonacci groups $F^{k/\ell} (2,n)$, that is, the groups $G_n (x_1^{\ell} x_2^k x_3^{-\ell})$, were defined [21]. The study of the algebraic structure of cyclically presented groups has developed into a topical area of combinatorial group theory [12], [22], [23]. In recent years, for cyclically presented groups the dynamics corresponding to the shift $\theta(x_i)=x_{i+1}$ was actively studied; see [24] and [25].
The presence of cyclic automorphisms of the groups $G_n(w)$ opens way to constructing new families of $n$-valued groups using the coset construction. Namely, as shown in Lemma 1.4, every word $w \in \mathbb F_n=\langle x_1, \dots, x_n \rangle$ in a free group of rank $n$ defines the structure of an $n$-valued group on the set $X_n (w)=G_n (w)/A_n$, where $G_n(w)$ is a cyclically presented group and $A_n$ is the cyclic group generated by the automorphism $\theta(x_i)=x_{i+1}$, $i=1, \dots, n$. In Example 1.1 a 5-valued group is presented that corresponds to the group $F(2,5)$ introduced by Conway [17]. In Example 1.2 4-valued groups are constructed for the infinite family of groups $G_4 (x_1^{\ell} x_2 x_3^{-\ell})$, $\ell \geqslant 1$. In Lemma 1.5 it is shown that the $2$-valued groups constructed from $G_4(x_1^\ell x_2 x_3^{-\ell})$ belong to the principal series of $2$-valued groups according to the classification in [10]. Example 1.3 describes an infinite family $\Phi$ of combinatorial groups $F^{1/\ell}(2,2n)$ each of which has two cyclic presentations; one of these is $G_{2n}(u_\ell)$ with $2n$ generators, and the other is $G_n(v_\ell)$ with $n$ generators. For every group $G \in \Phi$ we define a cyclic $2n$-valued coset group $X=G/A_{2n}$ and a cyclic $n$-valued coset group $Y=G/A_n$, and also a homomorphism of the $n$-valued group $Y=G/A_n$ into the $2n$-valued group $X=G/A_{2n}$; see Lemma 1.3. Results of the theory of cyclically presented groups clarify the nontriviality of the theory of cyclic $n$-valued groups. Example 1.4 shows that a cyclically presented group and a subgroup of this group that is not isomorphic to the group can give rise to isomorphic $n$-valued groups. In this connection the following problem is of interest: describe the conditions under which $n$-valued coset groups constructed from nonisomorphic groups with automorphisms of order $n$ are isomorphic; see Problem 1.2. In Theorem 1.2 we show that the coset construction can also be applied to subgroups of the group generated by a cyclic permutation of the generators $x_1, \dots, x_n$.
This paper is focused on the coset construction of $n$-valued groups. In this connection we draw the reader’s attention to the following important problem: characterize the structures of $n$-valued coset groups in the set of all structures of finite $n$-valued groups on a fixed space $X$. Results in this direction are related to deep facts in group theory, number theory and algebraic combinatorics; see [26].
Presentations of $n$-valued cyclic groups are closely related to fundamental results in the theory of branched coverings in the sense of Dold and Smith. A natural question arises: is it possible to introduce the concept of orbit space for an action of an $n$-valued group? There are no general theoretical results in this direction as yet. Based on the goals of this paper, in Definition 2.4 we formulate conditions for the existence of the orbit space of an action of an $n$-valued group on a topological space and prove that these conditions are satisfied in the case of a coset action. Namely, the following property is established in Theorem 2.1. Let $(G, A, \rho)$, where $\rho\colon A \to \operatorname{Aut} (G)$, be a triple such that the $n$-valued coset group $X=G/\rho(A)$ acts by cosets on the topological space $V=U/A$. Let $\langle G, A \rangle$ denote the extension of $G$ by means of the automorphisms in $\rho (A)$. The action of the group $\langle G, A \rangle$ on the space $U$ is defined, and $W=U/ \langle G, A \rangle$ is the orbit space of the $n$-valued action of the coset group $X= G/\rho(A)$ on the topological space $V=U/A$, and this definition is consistent with the canonical projection $V\to W$.
In problems of integrable multivalued discrete-time dynamics, presentations of $n$-valued cyclic groups on manifolds with singularities are used. At the same time, there is a natural connection between cyclically presented groups and branched cyclic coverings of the 3-sphere. The work [13] by Helling, Kim and Mennicke, in which Fibonacci manifolds were constructed, that is, closed orientable 3-manifolds whose fundamental groups are the Fibonacci groups $F(2,2n)=G_{2n} (x_1 x_2 x_3^{-1})$ for $n \geqslant 2$, initiated the study of the following question in low-dimensional topology: is a group given by a cyclic presentation the fundamental group of some closed orientable 3-manifold? Note that if $m \geqslant 3$ is odd, then $F(2,m)$ is the fundamental group of a 3-manifold if and only if $m=3,5,7$ [27]. Many other cyclically presented groups are the fundamental groups of closed orientable 3-manifolds, for example, the Sieradski groups $G_n (x_1 x_3 x_2^{-1})$ for $n \geqslant 3$ [28] and also the groups $G_{2n} (x_1 x_2^k x_3^{-1})$ [15], $G_{2n} (x_1^{\ell} x_2^{k} x_3^{-\ell})$ [21] and $G_n((x_1^{-\ell} x_2^{\ell})^k x_2^{\pm 1} (x_2^{-\ell} x_3^{\ell})^{-k})$ [29], [30].
We use this relationship to introduce the key example of a coset action of $n$-valued groups on three-dimensional space $\mathbb R^3$ with orbit space homeomorphic to $S^3$, such that the projection $\mathbb R^3 \to S^3$ onto the orbit space is connected, by means of a commutative diagram, with a covering of $S^3$ by a compact hyperbolic 3-manifold which is cyclically branched along a hyperbolic knot. Namely, Theorem 2.4 describes the action of an $n$-valued coset group in the case when it corresponds to a cyclically presented group realized as the fundamental group of a closed orientable hyperbolic 3-manifold. In Theorems 2.5, 2.7 and 2.8 we describe the actions of $n$-valued coset groups corresponding to actions of Fibonacci groups or fractional Fibonacci groups.
§ 1. $n$-valued groups corresponding to cyclically presented groups
1.1. $n$-valued coset groups
Recall the notion of an $n$-valued product following [31]. Let $X$ be a nonempty set, and let $X^n$ be its $n$th Cartesian power, that is, the set of ordered tuples $(x_1, \dots, x_n)$, where $x_i \in X$. Let $\operatorname{Sym}^n (X)$ denote the $n$-fold symmetric power of the set $X$, that is, the quotient space $X^n / \Sigma_n$, where the symmetric group $\Sigma_n$ acts on $X^n$ by permutations of the coordinates:
Elements of $\operatorname{Sym}^n(X)$ are $n$-multisets $[x_1, x_2, \dots, x_n]$, where $x_i \in X$. Recall that an element can be repeated several times in a multiset. A multiset consisting of $n$ instances of an element $x\in X$ is denoted by $n x=[x, \dots, x]$.
We fix some $n$-valued multiplication $\mu\colon X \times X \to \operatorname{Sym}^n (X)$ on the set $X$:
$$
\begin{equation*}
e \in \operatorname{inv} (x) * x\quad\text{and} \quad e \in x * \operatorname{inv} (x).
\end{equation*}
\notag
$$
An $n$-valued group is said to be commutative if, in addition, the $n$-multisets $x*y$ and $y*x$ coincide for all $x, y \in X$.
The existence of an identity element $e$ implies its uniqueness. However, the existence of a map $\operatorname{inv}$ does not imply its uniqueness. As shown in [32], if $(X, \mu)$ is an $n$-valued group such that the map $\operatorname{inv} (x)$ is uniquely determined, then ${\operatorname{inv} (\operatorname{inv} (x))=x}$. It was shown in [26] that the set of $n$-valued groups such that $\operatorname{inv} (\operatorname{inv} (x))=x$ is equivalent to the set of combinatorial algebras; cf. [33].
Definition 1.2. Let $(X, \mu)$ be an $n$-valued group. If $Y$ is a subset of $X$ such that $\operatorname{inv} (y) \in Y$, and if $y * y'$ and $y' * y$ belong to $\operatorname{Sym}^n (Y)$ for all $y, y' \in Y$, then we say that $Y$ is an $n$-valued subgroup of $X$ and write $Y \leqslant X$.
As shown in [32], if $Y$ and $Z$ are two $n$-valued subgroups of an $n$-valued group $X$, then their intersection $Y \cap Z$ is an $n$-valued subgroup of $X$.
Definition 1.3. Let $Y$ be a nonempty subset of $X$. Then the intersection of all $n$-valued subgroups of $X$ that contain $Y$ is called the $n$-valued subgroup generated by $Y$. The $n$-valued subgroup $\langle x \rangle$ generated by a single element $x \in X$ is called an $n$-valued cyclic group.
We stress that, unlike single-valued groups, the question of the classification of $n$-valued cyclic groups is still open.
Problem 1.1. Classify $n$-valued cyclic groups.
Definition 1.4. An $n$-valued group $X$ is said to be involutive if $\operatorname{inv} (x)=x$ for all ${x \in X}$.
A complete classification of finitely generated commutative involutive two-valued groups was obtained in [10]. Recently Gaifullin proved that each involutive two-valued group is commutative [34].
Given a map $f\colon X \to Y$, consider its $n$-fold symmetric power $\operatorname{Sym}^n (f)$: $\operatorname{Sym}^n (X)$ $\to\operatorname{Sym}^n (Y)$ defined by
Following [6], we define a homomorphism of $n$-valued groups.
Definition 1.5. Let $(X, \mu_X)$ and $(Y, \mu_Y)$ be two $n$-valued groups. A map $f\colon {X \mkern-1mu\!\to\!\mkern-1mu Y}$ is called a homomorphism of $n$-valued groups if
Lemma 1.2. Let $X$ be an $n$-valued group with multiplication $x*y=[z_1,\dots, z_n]$, and let $k$ be a positive integer. Then $X$ is equipped with the structure of an $nk$-valued group with multiplication $x *_k y$ such that every value $z_i$ of the product $x*y$ is repeated precisely $k$ times.
This assertion can be verified directly.
Following [31], we recall the construction of an $n$-valued coset group. Let $G$ be a (single-valued) group with multiplication $\mu_G$, identity $e_G$ and inverse $\operatorname{inv}_G (u)=u^{-1}$, and let $\operatorname{Aut}(G)$ be the automorphism group of $G$.
Let $A < \operatorname{Aut} (G)$ be a finite subgroup of order $n$; for definiteness, let $A=\{ \alpha_1, \dots, \alpha_n\}$. Then there is an action of the group $A$ on $G$, $A \times G \to G$, by the rule $(\alpha,g) \to \alpha(g)$ for $\alpha \in A$ and $g \in G$. Let $X=G / A=\{ A (g) \mid g \in G \}$ be the set of orbits of $G$ under the action of $A$. We denote by $\pi\colon G \to X$ the quotient map assigning to an element $g$ its orbit $\pi (g)=A (g)=[ \alpha_1 (g), \dots, \alpha_n(g) ]$. We define $n$-valued multiplication $\mu\colon X \times X \to \operatorname{Sym}^n (X)$ by
where $u \in \pi^{-1} (x)$ and $v\in \pi^{-1}(y)$.
The above construction leads to important examples of $n$-valued groups.
Theorem 1.1 ([31]). Let $X = G / A$, and let multiplication $\mu\colon\! X \times X \to \operatorname{Sym}^n (X)$ be defined by (1.3). Then the pair $(X, \mu)$ is an $n$-valued group with identity ${e_X= \pi (e_G)}$ and the uniquely determined inverse map $\operatorname{inv}_X (x)=\pi (g^{-1})$, where $g \in \pi^{-1} (x)$.
Definition 1.6. The multivalued group $(X,\mu)$ in Theorem 1.1, where $X=G/A$, is called the $n$-valued coset group of the pair $(G,A)$ or, briefly, a coset group.
Given a single-valued group $G$, a subgroup $A_{kn}$ of its automorphism group and a subgroup $A_{n}$ of $A_{kn}$, the coset groups $X_1=G/A_{kn}$ and $X_2=G / A_{n}$ are defined, and also a homomorphism of $X_2=G / A_n$ into $X_1=G / A_{kn}$. This homomorphism is defined by a homomorphism of $nk$-valued groups involving $X_2=G/A_n$ with multiplication $x *_k y$ defined on the basis of multiplication $x*y$ in the $n$-valued group $X_2$.
Definition 1.7. Consider two pairs $(G_1, H_1)$ and $(G_2, H_2)$, where $H_i \subset \operatorname{Aut}(G_i)$ and the order of the group $H_i$ is $n_i$, $i=1,2$. A pair of homomorphisms $(f,h)$ such that $f\colon G_2 \to G_1$ and $h\colon H_2 \to H_1$ is called an equivariant homomorphism from $(G_2, H_2)$ to $(G_1, H_1)$ if $f(g(x))=h(g)f(x)$ for all $g \in H_2$ and $x \in G_2$.
Lemma 1.3. An equivariant homomorphism $(f, h)$ from $(G_2, H_2)$ to $(G_1, H_1)$ defines a homomorphism of the $n_2$-valued group $X_2=G_2/H_2$ into the $n_1$-valued group $X_1=G_1/H_1$.
The proof follows immediately from the construction of $n$-valued groups and the definition of their homomorphisms.
Going over from groups defined by axiomatics to combinatorial groups defined by generators and relations enables us to obtain nontrivial examples of equivariant homomorphisms from $(G_2, H_2)$ to $(G_1, H_1)$.
Example 1.3 presents an equivariant homomorphism $(f,h)$, where $f$ is an isomorphism of the Fibonacci group $G_2=G_1=F(2,2n)$ and $h$ is a homomorphism of a cyclic group $H_2$ of order $n$ into a cyclic group $H_1$ of order $2n$. For $n=2$ the equivariant homomorphism $(f,h)$ induces a homomorphism $X_2 \to X_1$, where $X_2$ is a $2$-valued coset group on a three-element set and $X_1$ is a $4$-valued coset group on a two-element set.
1.2. Cyclically presented groups
For $m \,{\geqslant}\, 2$, we denote by $\mathbb F_m$ the free group with $m$ free generators $x_1, x_2, \dots, x_m$. Let $\theta$ be the automorphism of $\mathbb F_m$ that permutes the generators cyclically, namely, $\theta(x_i)=x_{i+1}$, where $i=1, \dots, m$ and indices are taken modulo $m$. Clearly, $\theta^m=1$. Let $w=w (x_1, \dots, x_m)$ be some word in $\mathbb F_m$. We denote the image of $w$ under $\theta$ by $\theta (w)$. Let us define the group $G_m (w)$ by the following presentation:
Such a presentation is said to be cyclic, and the word $w$ is called the defining word. In a cyclic presentation of a group the number of defining relations is equal to the number of generators. A presentation with this property is said to be balanced. When it is important to indicate the number of generators, we speak of $n$-cyclically presented group.
Definition 1.8. A group $G$ is called a cyclically presented group if it is isomorphic to $G_m(w)$ for some $m \geqslant 2$ and $w \in \mathbb F_m$.
To every cyclic presentation $G_m (w)$ we assign a polynomial $f_w (t)=\sum_{i=1}^{m} a_i t^{i-1}$, where $a_i$ is the sum of the powers of the generator $x_i$ in the defining word $w$. For example, if $w=x_1 x_2 x_3^{-1}$, then $f_w (t)=1+t-t^2$. We note that a group can have different cyclic presentations, and a presentation $G_m(w)$ of the trivial group does not necessarily satisfy the equality $f_w (t)=\pm t^k$. For example, as shown in [35], the cyclic presentation $G_5 (x_1^{-1} x_2^{-1} x_4 x_3 x_2)$ defines the trivial group. The corresponding polynomial is $f_w (t)=-1+t^2+t^3$.
Lemma 1.4. Every word $w$ in a free group $\mathbb F_m=\langle x_1, \dots, x_m \rangle$ of rank $m$ defines the structure of an $m$-valued group on the set $X_m (w)=G_m (w) / A_m$, where $G_m(w)$ is a cyclically presented group and $A_m$ is the cyclic group generated by the automorphism $\theta(x_i)=x_{i+1}$, $i=1, \dots, m$, permuting the generators cyclically.
Proof. We define multiplication $\mu\colon X_m (w) \times X_m (w) \to \operatorname{Sym}^m (X_m (w))$ by (1.3). Then by Theorem 1.1, $(X_m (w), \mu)$ is an $m$-valued coset group of the pair $(G_m (w), A_m)$.
This completes the proof of the lemma.
Example 1.1. Consider the group $F(2,5)$, which was the subject of Conway’s question [17],
As is known [18], $F(2,5)$ is the cyclic group $ C_{11}=\langle a\mid a^{11}=1\rangle$. Let $x_1=a$. Then $x_2=a^4$, $x_3=a^5$, $x_4=a^9$ and $x_5=a^3$. Therefore, the cyclic automorphism $\theta(x_i )=x_{i+1}$, $i=1, \dots, 5$, acts on $C_{11}$ by raising elements to the fourth power: if $b \in C_{11}$, then $\theta (b)=b^4$. The elements of $F(2,5)=C_{11}$ are grouped into three orbits with respect to the action of $\theta$: $y_0=\{ 1 \}$, $y_1=\{ a, a^4, a^5, a^9, a^3\}$, and $y_2=\{ a^2, a^8, a^{10}, a^7, a^6 \}$. We denote the set of orbits by $Y=\{ y_0, y_1, y_2 \}$. Then we obtain the following 5-valued multiplication $\mu_5\colon Y \times Y \to \operatorname{Sym}^5(Y)$:
Thus, using $F(2,5)$, a 5-valued coset group $(Y, \mu_5)$ is constructed, where $n y_i$ means that $y_i$ is repeated $n$ times in the corresponding multiset. This group belongs to the series of $(2k+1)$-valued groups on $Y$ with multiplications $\mu_{2k+1}\colon Y \times Y \to \operatorname{Sym}^{2k+1} (Y)$ given by
It was stated in [26] that $(Y, \mu_{2k+1})$ is a coset group if and only if $4k+3=p^s$ for a prime $p$ and a positive integer $s$. As we have shown, for $k=2$ this group is related to the Conway group $F(2,5)$.
Example 1.2. Consider the group $F^{1/\ell} (2,4)=G_4 (w_{\ell})$, where $w_{\ell}=x_1^{\ell} x_2 x_3^{-\ell}$ for an integer $\ell \geqslant 1$; for $\ell=1$ this group coincides with $F(2,4)$. We write out a cyclic presentation:
Here $\theta(x_i)=x_{i+1}$, $i=1, \dots, 4,$ and $\theta$ generates the automorphism group $A_4=\langle \theta \mid \theta^4=1 \rangle$. It can readily be seen that $G_4(w_{\ell})$ is finite and isomorphic to the cyclic group $C_k=\langle a \mid a^k=1\rangle$, where $k=4\ell^2+1$. In this case $x_1=a$, $x_2=a^{-2\ell}$, $x_3= a^{-1}$ and $x_4=a^{2\ell}$. On $C_k$ the automorphism $\theta$ acts by raising elements to power $(-2\ell)$; in particular, $\theta (x_1)=\theta (a)=a^{-2\ell}=x_2$, $\theta (x_2)= a^{4\ell^2}=a^{-1}=x_3$, $\theta(x_3)=(a^{-1})^{-2\ell}=a^{2\ell}=x_4$ and $\theta(x_4)= (a^{2\ell})^{-2\ell}=a=x_1$. We obtain the set of orbits $X_4 (w_{\ell})=G_4(w_{\ell}) / A_4$ consisting of $\ell^2+1$ classes, where one class has the form $y_0=\{ 1 \}$, and each of the other classes $y_1, \dots, y_{\ell^2}$ consists of four elements.
In particular, for $\ell=2$ we have $w_2=x_1^2 x_2 x_3^{-2}$ and $G_4 (w_2) \cong C_{17}$, and we obtain the set of orbits $X_4 (w_2)$, where $a=x_1$ and $a^{17}=1$:
and $\pi(1)=y_0$. Therefore, by (1.3) the 4-valued multiplication $\mu_4\colon X_4(w_2) \times X_4(w_2) \to \operatorname{Sym}^4 (X_4(w_2))$ is defined. It can readily be seen that $(X_4(w_2), \mu_4)$ is a cyclic involutive 4-valued group.
We note that, in fact, Theorem 1.1 implies a more general statement, which enables us to consider not only the group $A_m$ but also subgroups of it.
Theorem 1.2. Let $G_m(w)$ be a cyclically presented group, and let $A$ be the subgroup of order $n$ of the cyclic group $A_m$ that is generated by an automorphism $\theta$ cyclically permuting the generators. Let $X=G_m (w) / A$. Then multiplication $\mu_A\colon X\times X \to \operatorname{Sym}^n (X)$ defined by (1.3) introduces the structure of an $n$-valued group on the set $X (w)=G_m (w) / A$.
The following problem arises in connection with Theorem 1.2: given a cyclically presented group $G_n(w)$, find cyclic subgroups of the group $\operatorname{Aut} (G_n (w))$ that yield cyclic $m$-valued groups. The following example provides infinitely many cyclically presented groups, from which pairs of cyclic $2n$-valued and $n$-valued coset groups are constructed.
Example 1.3. For an integer $\ell \geqslant 1$ we set $u_{\ell}=x_1^{\ell} x_2 x_3^{-\ell}$, and for an integer $n \geqslant 2$ we consider the group
which is the Fibonacci group $F(2,2n)$ for $\ell=1$. This cyclic presentation is related to the polynomial $f_{u_{\ell}} (t)=\ell+t-\ell t^2$. The automorphism $\theta (x_k)=x_{k+1}$, where $k=1, \dots, 2n$, generates the group $A_{2n}=\langle \theta \mid \theta^{2n}=1 \rangle $ of order $2n$. By Theorem 1.2, the quotient $X(u_{\ell})=G_{2n} (u_{\ell}) / A_{2n}$, with multiplication defined by (1.3), is a $2n$-valued cyclic coset group.
We set $y_{k}=x_{2k-1}$, where $k=1, \dots, n$. Then it follows from the relations $x_{2k-1}^{\ell} x_{2k}=x_{2k+1}^{\ell}$ that $x_{2k}=y_k^{-\ell} y_{k+1}^{\ell}$, and from the equality $x_{2k}^{\ell} x_{2k+1}=x_{2(k+1)}^{\ell}$ we obtain $v_{\ell}=1$, where $v_{\ell}=(y_{k}^{-\ell} y_{k+1}^{\ell})^{\ell} y_{k+1} (y_{k+1}^{-\ell} y_{k+2}^{\ell})^{-\ell}$. Thus,
The polynomial corresponding to this presentation is $f_{v_{\ell}} (t)=- \ell^2+(2 \ell^2+1) t-\ell^2 t^2$. The automorphism $\varphi (y_k)=y_{k+1}$, where $k=1, \dots, n$, generates the group $A_n=\langle \varphi \mid \varphi^{n}=1 \rangle $ of order $n$. By Theorem 1.2 the quotient $Y(v_{\ell})=G_n (v_{\ell}) / A_n$ with multiplication defined by (1.3) is an $n$-valued cyclic group. Thus, using the $2n$-cyclic and $n$-cyclic presentations of the group $F^{1/\ell} (2,2n)$ we obtain cyclic $2n$-valued and $n$-valued coset groups $X(u_{\ell})$ and $Y(v_{\ell})$.
Let us look at this example from the standpoint of Lemma 1.3. We write ${G_1=G_{2n} (u_{\ell})}$ and $G_2=G_n (v_{\ell})$, and also $H_1=A_{2n}$ and $H_2=A_n$. Then the pair of homomorphisms $(f,h)$, where $f\colon G_2 \to G_1$ is an isomorphism such that $f(y_i)=x_{2i-1}$, $i=1, \dots, n$, and $h\colon H_2 \to H_1$ is a homomorphism such that $h (\varphi)=\theta^2$, defines an equivariant homomorphism of the pair $(G_2, H_2)$ into the pair $(G_1, H_1)$, that is, $f(g(x))=h(g) f(x)$ for all $x \in G_2$ and $g \in H_2$. It suffices to verify that this property holds for the generators $y_i \in G_2$ and $\varphi \in H_2$: $f (\varphi (y_i))=f(y_{i+1})= x_{2(i+1)-1}$ and $h(\varphi) (f (y_i))=\theta^2 (x_{2i-1})=x_{2i+1}$.
Consider the case $n=2$. We obtain the group $G_2=G_1=F(2,4)$ from Example 1.2, which is isomorphic to the cyclic group $C_5=\langle a \mid a^5=1 \rangle$. Moreover, $x_1=a$, $x_2=a^3, x_3=a^4, x_4=a^2$ and $\theta (x_i)=x_{i+1}$. Hence $G_1/H_1$ is a 4-valued coset group on two elements. Since $\varphi (x_i)=x_{i+2}$, it follows that $G_2 / H_2$ is a 2-valued coset group on a set of three elements; see Lemma 1.5.
Under the coset construction, isomorphic $n$-valued coset groups can correspond to different groups. Such an example was given in [6], Proposition 3.2. Here we point out that this example can be obtained in terms of cyclically presented groups.
Example 1.4. Consider the cyclically presented group $G_2 (w_1)$, where $w_1=x_1^2$. That is, $G_2(w_1)=\langle x_1, x_2 \mid x_1^2=x_2^2=1 \rangle=\mathbb Z_2 * \mathbb Z_2$. The automorphism $\theta$ such that $\theta(x_1)=x_2$ and $\theta(x_2)=x_1$ generates the group $A_2= \langle \theta \mid \theta^2=1 \rangle$. We obtain the set of orbits $Y_2 (w_1)=G_2 (w_1) / A_2=\{ y_0, y_1, y_2, \dots \}$, where $y_0=\{ 1 \}$, $y_1=\{ x_1, x_2 \}$ and, next, $y_{2k}=\{ (x_1 x_2)^k, (x_2 x_1)^k \}$ and $y_{2k+1}=\{ (x_1 x_2)^k x_1, (x_2 x_1)^k x_2 \}$ for $k \geqslant 0$. Applying the quotient map $\pi\colon G_2(w_1) \to Y_2 (w_1)$ we have $\pi ((x_1 x_2)^k)=\pi ((x_2 x_1)^k )=y_{2k}$ and $\pi ((x_1 x_2)^k x_1)=\pi ((x_2 x_1)^k x_2 )=y_{2k+1}$. Following (1.3), we define $2$-valued multiplication $\mu_2\colon Y_2(w_1) \times Y_2(w_1) \to \operatorname{Sym}^2 (Y_2 (w_1))$ by
All this can be described by the single formula $\mu_2 (y_m, y_n)=[y_{m+n},y_{|m-n|}]$ for ${m,n \geqslant 0}$.
Consider the infinite cyclic subgroup $H \cong \mathbb Z$ in $G_2 (w_1)$ generated by the element $c=x_1 x_2$. The automorphism $\theta$ acts on $H$ by the rule $\theta (c)=\theta (x_1 x_2)=x_2 x_1=x^{-1}_2 x^{-1}_1=c^{-1}$. As above, let $A_2=\langle \theta \mid \theta^2=1 \rangle$. We obtain the set of orbits $Z_2 (w_1)=H / A_2=\{ z_n \}$, where $z_n= \{c^n, c^{-n} \}$, and the quotient map $\pi\colon {H \to Z_2 (w_1)}$, for which $\pi (c^n)=\pi (c^{-n})=z_n$, $n \geqslant 0$. Let us define 2-valued multiplication $\nu_2\colon Z_2(w_1) \times Z_2(w_1) \to \operatorname{Sym}^2 (Z_2(w_1))$ by
Then the 2-valued groups $G_2 (w_1) / A_2=(Y_2 (w_1), \mu_2)$ and $H / A_2=(Z_2 (w_1), \nu_2)$ coincide, although the groups $G_2 (w_1)$ and $H$ are not isomorphic.
Kornev [36] investigated the growth function for the group $G_2(w_1)$, $w_1=x_1^2$, and some generalizations of this group. Consider the group $G_m(w_1)$, $m \geqslant 2$, and the $m$-valued coset group $X_m(w_1)= G_m(w_1)/A_m$, where $A_m$ is generated by the automorphism $\theta(x_i)=x_{i+1}$, $i=1, \dots, m$. By [36], Proposition 3, the group $X_m(w_1)$ has the growth function
We formulate two open problems in connection with Example 1.4.
Problem 1.2. Describe conditions under which $n$-valued coset groups constructed from nonisomorphic cyclically presented groups are isomorphic.
Problem 1.3. Find the growth order and growth function for $n$-valued coset groups corresponding to cyclically presented groups.
In $F^{1/\ell} (2,4)=G_4 (w_{\ell})$, where $w_{\ell}=x_1^{\ell} x_2 x_3^{-\ell}$, we consider the second-order automorphism $\psi (x_i)=x_{i+2}$ for $i=1,2,3,4$.
We recall that, according to [10], if $G$ is a (single-valued) Abelian group and $\varphi\colon G \to G$ is its antipodal involution: $\varphi (g)=g^{-1}$, then the 2-valued coset group $X=G / \langle \varphi \rangle$ is an involutive commutative group. In this case $X$ is said to belong to the principal series of 2-valued groups.
Lemma 1.5. Consider the group $G_4 (w_{\ell})$, where $w_{\ell}=x_1^{\ell} x_2 x_3^{-\ell}$ for an integer $\ell \geqslant 1$. Let $B_2=\langle \psi \mid \psi^2=1 \rangle$, where $\psi (x_i)=x_{i+2}$. Then the 2-valued coset group $G(w_{\ell}) / B_2$ belongs to the principal series of 2-valued groups from [10].
Proof. As shown in Example 1.2, we have $G_4 (w_{\ell})=C_k=\langle a \mid a^k= 1\rangle$, where $k=4 \ell^2+1$, $\ell \geqslant 1$, and, moreover, $\psi=\theta^2$. Then $\psi(a)=a^{4 \ell^2}=a^{-1}$. Writing $B_2=\langle \psi \mid \psi^2=1 \rangle$ we obtain the set of orbits $Z_2(w_{\ell})=G_4 (w_{\ell}) / B_2=\{ z_0= \{ 1 \},\, z_i=\{ a^i, a^{-i} \},\text{ where }i=1, \dots, 2 \ell^2 \}$. Therefore, the quotient map $\pi\colon G_4(w_{\ell}) \to Z_2 (w_{\ell})$ has the form $\pi (1)=z_0$ and $\pi (a^i)=\pi (a^{-i})=z_i$ for $i=1, \dots, 2 \ell^2$. We obtain $2$-valued multiplication $\mu_2\colon Z_2 (w_{\ell}) \times Z_2 (w_{\ell}) \to \operatorname{Sym}^2 (Z_2(w_{\ell}))$:
where $\overline{p}=p$ if $p \leqslant 2\ell^2$, and $\overline{p}=4 \ell^2+1-p$ if $p > 2 \ell^2$. Since the group $F^{1/\ell}(2,4)=C_k$ is commutative and $\psi(a)=a^{-1}$, the $2$-valued group $(Z_2(w_{\ell}), \mu_2)$ belongs to the principal series.
This completes the proof of the lemma.
§ 2. Actions of cyclic $n$-valued groups and hyperbolic three-manifolds
2.1. An action of a multivalued group
Following [31], we introduce the concept of an action of a multivalued group. Recall that a (single-valued) group $G$ with multiplication $\mu_G (g,h)=g * h$ acts from the left on a space $U$ if there is a map $G \times U \to U$ that assigns to a pair $(g, u)$ an element $g(u) \in U$ such that the following conditions are satisfied:
Assume that an action of a group $G$ on $U$ and an action of a finite group $A$ of order $n$ on $U$ are given. If a presentation $\rho\colon A \to \operatorname{Aut} (G)$ is fixed such that
$$
\begin{equation}
\alpha (g (v))=\alpha (g) (\alpha(v)) \quad \text{for all } g \in G, \ \alpha \in A, \ v \in U,
\end{equation}
\tag{2.1}
$$
where $\alpha (g)=\rho (\alpha) (g)$, then we say that the triple $(G,A, \rho)$ acts equivariantly on $U$.
Definition 2.1. We say that an $n$-valued group $(X, \mu)$ acts on a set $V$ if a map $\varphi\colon X \times V \to \operatorname{Sym}^n (V)$,
Lemma 2.1. If a triple $(G,A, \rho)$, where $\rho\colon A \to \operatorname{Aut} (G)$, acts equivariantly on $U$, then the coset group $X=G/\rho(A)$ acts on the orbit space $V=U/A$.
Proof. For definiteness we denote the elements of $A$ by $\alpha_1, \alpha_2, \dots, \alpha_n$. The orbit space of the set $U$ under the action of the group $A$ has the form $V=U/A=\{ A(u) \mid u \in U \}=\{ [\alpha_1 (u), \dots, \alpha_n (u)] \mid u \in U \}$. Consider the map $\pi_U\colon U \to V$ assigning to a point the $n$-multiset of points in its orbit. We write $\widehat{A}=\rho(A)$. The orbit space of the group $G$ under the action of $\widehat{A}$ has the form $X=G/\widehat{A}=\{ \widehat{A}(g) \mid g \in G \}=\{ [\widehat{\alpha}_1 (g), \dots, \widehat{\alpha}_n (g)] \mid g \in G \}$. Now we define the quotient map $\pi_G\colon G \to X$ that assigns to every element of the group the $n$-multiset of elements of the orbit of this element. By Theorem 1.1 the $n$-valued multiplication $\mu\colon X \times X \to \operatorname{Sym}^n (X)$ given by
where $g \in \pi_G^{-1}(x)$ and $u \in \pi_U^{-1}(v)$. A straightforward verification shows that the above action satisfies the conditions in Definition 2.1, that is, the $n$-valued coset group $X$ acts on the orbit space $V$.
This completes the proof of the lemma.
Definition 2.2. In the notation of Lemma 2.1, if the coset group $X=G / \rho(A)$ acts on the orbit space $V=U/A$, then we say that $X$ acts coset-wise on $V$.
2.2. Branched coverings and the action of coset groups
We recall the concept of an $n$-branched covering in the sense of Dold [37] and Smith, following [7]. Let $X$ and $ Y$ be nonempty sets. Then every map $\pi\colon X \to Y$ induces a map $\operatorname{Sym}^n (\pi)\colon \operatorname{Sym}^n (X) \to \operatorname{Sym}^n (Y)$ defined by (1.2). If $X$ is a Hausdorff space, then we endow $\operatorname{Sym}^n (X)$ with the quotient topology.
Definition 2.3. Let $X$ and $Y$ be Hausdorff spaces. A continuous map $\pi\colon X \to Y$ is called an $n$-branched covering if there is a continuous map $t\colon Y \to \operatorname{Sym}^n (X)$ such that
A typical example of an $n$-branched covering is the following construction; see [7], Example 3.3. Let $G$ be a finite group of order $n$ acting continuously and effectively on a Hausdorff space $X$. Recall that an action of $G$ on $X$ is called effective if for all elements $g_1 \neq g_2$ of $G$ there is a point $x \in X$ such that $g_1 (x) \neq g_2 (x)$. Let $\pi\colon X \to Y=X / G$ be the map assigning to a point $x \in X$ its orbit $Gx \in Y$. Then $\pi$ is an $n$-branched covering. Here the map $t\colon Y \to \operatorname{Sym}^n (X)$ takes $y$ to the $n$-multiset of points in the orbit $\{ x \in X \mid x=\pi^{-1} (y) \}$, considered with multiplicities. Moreover (see [37], Example 1.4), if $H < G$ is a subgroup of finite index $n$ and $G$ acts on $X$ effectively, then the quotient map $X / H \to X / G$ is an $n$-branched covering.
Recall the construction in [7]. We write $[n]=\{ 1, \dots, n \}$. By an epimorphism of the set $[n]$ onto an $n$-multiset of elements of $X$ we mean a surjective map such that elements in its image are considered taking account of their multiplicities.
Let $\pi\colon X \to Y$ be an $n$-branched covering. We denote by $E$ the set of all maps $\psi\colon [n] \to X$ such that $\pi \,{\circ}\, \psi(1) =\pi \,{\circ}\, \psi (2)=\dots=\pi \,{\circ}\, \psi(n)=y$ and $\psi$ is an epimorphism of $[n]$ onto the set $ \pi^{-1} (y) $.
Clearly, the symmetric group $\Sigma_n$ acts on $E$ by permutations of the elements of the set $[n]$, and the quotient space by this action coincides with $Y$. Moreover, $E \times_{\Sigma_n} [n]$ is isomorphic to $X$, and the projection onto the first factor can be identified with $\pi$. The map $\psi$ can be thought of as a ‘universal’ arrangement of multiplicities of points on branches of the covering. This implies the following assertion from Dold’s paper [37], which we formulate following [7].
Lemma 2.2 (see [37], Proposition 1.9, and [7], Theorem 4). Every $n$-branched covering can be represented as a projection $p\colon E \times_{\Sigma_n} [n] \to E / \Sigma_n$ for the above $\Sigma_n$-space $E$.
Let $X$ be an $n$-valued group acting on a topological space $V$; see Definition 2.1. Then $g \,{\circ}\, v$, where $v \in V$ and $x \in X$, is an $n$-multiset of points in $V$. For $v \in V$ we consider the set $ X v=\{ x \circ v \mid x \in X \} \subset \operatorname{Sym}^n (V)$. For $v' \in V$ the inclusion $v' \in Xv$ means that there is an element $x \in X$ such that $x \circ v \ni v'$. We note that if $X$ is an $n$-valued coset group acting coset-wise, then $v' \in X v$ implies that $v \in X v'$. In fact, by Theorem 1.1, if $X=G/A$ is a coset group with quotient map $\pi\colon G \to X$, then $\operatorname{inv}_X (x)=\pi(g^{-1})$, where $g \in \pi^{-1} (x) \subset G$.
Definition 2.4. We say that an action of $X$ on a set $V$ has an orbit space if there exists a set $W$ and a surjective map $\pi\colon V \to W$ such that for any pair of points $v, v' \in V$ the condition $\pi(v)=\pi(v')$ is satisfied if and only if $v' \in X v$. Moreover, we say that a continuous action of $X$ on a topological space $V$ has an orbit space if there exists a topological space $W$ and a continuous surjective map $\pi\colon V \to W$ such that for a pair of points $v, v' \in V$ we have $\pi(v)=\pi(v')$ if and only if $v' \in X v$.
Theorem 2.1. Let $(G, A, \rho)$, where $\rho\colon A \to \operatorname{Aut} (G)$, be a triple such that the $n$-valued coset group $X=G/\rho(A)$ acts coset-wise on the topological space $V=U/A$. Denote the extension of $G$ by means of the automorphisms in $\rho (A)$ by $\langle G, A \rangle$. Then $W=U/ \langle G, A \rangle$ is the space of orbits of this action with respect to the canonical projection $V \to W$.
Proof. By definition the triple $(G,A,\rho)$ consists of the single-valued group $G$, the finite group $A$ and the representation $\rho\colon A \to \operatorname{Aut} (G)$. Moreover, the triple $(G,A, \rho)$ acts equivariantly on the space $U$, that is,
$$
\begin{equation*}
\alpha (g (v))=\alpha (g) (\alpha(u)) \quad \text{for all } g \in G, \quad\alpha \in A, \quad u \in U,
\end{equation*}
\notag
$$
where $\alpha(g)=\rho(\alpha) (g)$. Therefore, the quotient spaces $U/G$ and $V= U/A$ are defined. We denote the extension of $G$ by the automorphism group $\widehat{A}=\rho(A)$ by $\widehat{G}=\langle G, A \rangle$. Corresponding to this extension, there is a short exact sequence
We write $W=U / \widehat{G}$. Then we have the following diagram of branched coverings:
The group $A$ acts on $U/G$ since the actions of $G$ and $A$ on $U$ commute. Recall that $X$ is an $n$-valued coset group, that is, $X=G / \widehat{A}=\{ [\widehat{\alpha}_1(g), \dots, \widehat{\alpha}_n (g)] \mid g \in G\}$, where $\widehat{A}= \{ \widehat{\alpha}_1, \dots, \widehat{\alpha}_n \}$.
The space $V=U/A$ consists of the orbits $[v]=[ \alpha_1(u), \dots, \alpha_n (u) ]$ of points $u \in U$ under the action of the group $A$, and elements of the $n$-valued group $X\mkern-2mu =\mkern-2mu G/\widehat{A}$ are the orbits $[x]=[\widehat{\alpha}_1 (g), \dots, \widehat{\alpha}_n (g)]$ of elements $g \in G$ under the action of the automorphism group $\widehat{A}$. Since the triple $(G, A, \rho)$ acts equivariantly, we obtain the following componentwise action:
that is, the result is the orbit $[g(v)]$ of the point $g(v) \in U/G$ under the action of the group $A$, where $g(u)$ is in its turn the orbit of $u \in U$ under the action of $G$. Thus, the canonical projection $\pi\colon V \to W$ such that $\pi ([v])=[g(u)]$ is defined. Since the coverings $U \to V=U/A$ and $U/G \to W=U / \widehat{G}$ are compatible with the action of $A$, the projection $\pi$ is well defined and satisfies the conditions of Definition 2.4.
This completes the proof of the theorem.
Recall that a compact three-dimensional manifold $M$ is said to be hyperbolic if $M=\mathbb H^3 / \Gamma$, where $\mathbb H^3$ is a three-dimensional hyperbolic space and $\Gamma < \operatorname{Isom} (\mathbb H^3)$ is a discrete cocompact torsion-free group of isometries [38], [39]. We denote the group of orientation-preserving isometries of the space $\mathbb H^3$ by $\operatorname{Isom}^+(\mathbb H^3)$. We are interested in cyclically presented groups that are the fundamental groups of closed orientable three-dimensional hyperbolic manifolds. For example, the Fibonacci groups $F(2,2n)$ have this property for $n \geqslant 4$.
Three-dimensional hyperbolic manifolds have the following important property, which is called the rigidity theorem.
Theorem 2.2 ([38], Theorem C.5.2). Let $M_1=\mathbb H^3 / \Gamma_1$ and $M_2=\mathbb H^3 / \Gamma_2$, where $\Gamma_1, \Gamma_2 < \operatorname{Isom}^{+} (\mathbb H^3)$, be compact orientable hyperbolic $3$-manifolds. If there is an isomorphism $\phi\colon \Gamma_1 \to \Gamma_2$, then there is an isometry $q \in \operatorname{Isom} (\mathbb H^3)$ such that
which induces an isometry $\chi\colon M_1 \to M_2$ such that $\chi_*=\phi$.
Moreover, the following relationship between the group of outer automorphisms of the fundamental group of a hyperbolic manifold and the group of its isometries is known.
Theorem 2.3 ([38], Theorem C.5.6). Let $M$ be a connected compact orientable hyperbolic three-dimensional manifold. Then the outer automorphism group $\operatorname{Out} (\pi_1(M))$ is finite and isomorphic to the group of isometries $\operatorname{Isom}(M)$.
In what follows we are interested in closed orientable hyperbolic $3$-manifolds whose fundamental groups have cyclic presentations.
Theorem 2.4. Let $G_n (\mkern-1muw\mkern-1mu) \!<\! \operatorname{Isom}^{+} \mkern-1mu(\mathbb H^3\mkern-1mu)$ be a cyclically presented group such that the quotient space $M_n (w)=\mathbb H^3 / G_n(w)$ is a closed orientable hyperbolic $3$-manifold. Let $A$ be a finite cyclic subgroup of order $k$ in $\operatorname{Isom}^{+} (M_n(w))$. Then the $k$-valued coset group $G_n(w) / A$ acts on the space $\mathbb H^3 / A$.
Proof. By Lemma 2.1 it suffices to show that the pair $(G_n(w), A)$ acts equivariantly on the space $\mathbb H^3$. Let $\phi \in A$. Then by Theorems 2.2 and 2.3 there is an isometry $q \in \operatorname{Isom} (\mathbb H^3)$ such that $\phi (g) =q \,{\circ}\, g \,{\circ}\, q^{-1}$ for all $g \in G_n(w)$. Thus, $\phi \in A$ acts on $G_n (w)$ by conjugation by the isometry $q$, and it acts on $\mathbb H^3$ by the isometry $q$. Consequently, $\phi (g) (\phi(v))=q \,{\circ}\, g \,{\circ}\, q^{-1} ( q (v)) =q (g(v))=\phi (g(v))$ for all $g \in G_n(w)$ and $v \in \mathbb H^3$, and thus the pair $(G_n(w), \phi)$ acts equivariantly for every $\phi \in A$. By Lemma 2.1 the $k$-valued coset group $G_n(w) / A$ acts on the space $\mathbb H^3 / A$.
This completes the proof of the theorem.
2.3. Branched cyclic coverings with branching along knots
We recall, following Fox [40], the notion of a branched covering of a three-dimensional manifold with branching along a link.
All manifolds under consideration are assumed to be closed, connected, orientable, and with piecewise linear topology. Let $M$ and $N$ be triangulable $m$-dimensional manifolds, and let $L$ be an $(m-2)$-dimensional subcomplex of $N$. A nondegenerate piecewise linear map $f\colon M \to N$ is called an $n$-fold covering branched along $L$, where $n>1$, if
In this case $M$ is said to be a branched covering of $N$, and $L$ is called the branch set of this covering. Two branched coverings $f_1\colon M_1 \to N_1$ and $f_2\colon M_2 \to N_2$ are considered to be equivalent if there are homeomorphisms $\psi\colon M_1 \to M_2$ and $\phi\colon N_1 \to N_2$ such that $\psi \circ f_2=f_1 \circ \phi$.
As follows from [41], Proposition 2.3.2, if $f\colon M \to N$ is a Fox branched covering of three-dimensional manifolds without boundary, then it is also an $n$-branched covering in the sense of Dold and Smith, where $n$ is the maximum multiplicity of $f$. Since the map $t \circ \pi\colon M \to \operatorname{Sym}^n (M)$ in Definition 2.3 consists essentially in assigning multiplicities to points of $M$, we obtain the following assertion.
Corollary 2.1. Let $f\colon M \to N$ be a Fox $n$-fold branched covering. Then to every point $x \in M$ one can continuously assign a positive integer $k(x)$, the multiplicity of $f$ at this point.
Here continuity is understood in the following sense, used in the Dold–Smith theory. Let $f\colon Y \to X$ be an $n$-fold branched Dold–Smith covering. Then to every point $y \in Y$ we can assign an integer $k(y)$ such that the map $g\colon X \to \operatorname{Sym}^n (Y)$ defined by $g(x)=[k(y_i) y_i, i=1, \dots, m]$, where $[y_i, i=1, \dots, m]=f^{-1} (x)$ and $\sum_i k(y_i)=n$, is continuous.
In [40] Fox showed that a branched covering is uniquely determined by the ordinary covering defined on the complement to the preimage of the set of branch points.
This fact enables us to establish a one-to-one correspondence between the $n$-sheeted coverings of $N$ branched over $L$ and the equivalence classes of monodromy maps $\omega\colon \pi_1(N \setminus L, x_0)\to \Sigma_n$, where $\Sigma_n$ is the symmetric group and $x_0 \in N \setminus L$ is an arbitrary base point. If $N$ is the $m$-dimensional sphere $S^m$, then a covering of $N$ branched over $L$ is briefly called a branched covering of $L$.
Thus, two $n$-sheeted branched coverings $f_1\colon M_1 \to N_1$ and $f_2\colon M_2 \to N_2$ with branch sets $L_1 \subset N_1$ and $L_2 \subset N_2$ and monodromy maps $\omega_{f_1}$ and $\omega_{f_2}$ are equivalent if and only if there exist an inner automorphism $\lambda$ of the group $\Sigma_n$ and a homeomorphism $\phi\colon N_1 \to N_2$ such that $\phi (L_1)=L_2$ and $\lambda \circ \omega_{f_1}=\omega_{f_2}\circ \phi_*$, where the homomorphism $\phi_*\colon \pi_1(N_1 \setminus L_1, x_0) \to \pi_1(N_2 \setminus L_2,\phi(x_0))$ is induced by the map $\phi _{|N_1 \setminus L_1}$.
Fox’s result enables us to transfer the notion of a cyclic covering from ordinary coverings to branched ones. Namely, a branched covering is said to be cyclic if the corresponding ordinary covering is cyclic. The notions of regular and Abelian branched coverings are introduced similarly.
Since every cyclic covering is Abelian, it follows that this covering is determined by the epimorphism
up to equivalence, where $\mathbb Z_n$ denotes the cyclic group of order $n$ embedded in $\Sigma_n$ by a monomorphism taking $1\in \mathbb Z_n$ to the standard cyclic permutation $(1\,2\,\dots\,n) \in \Sigma_n$. If $N=S^3$ and $L=\bigcup_{i=1}^{\mu} K_i$ is a $\mu$-component link in $S^3$, then $H_1(N \setminus L) \cong \mathbb Z^{\mu}$, and the set of homology classes of meridian loops around components of $L$ forms a basis. Therefore, an $n$-sheeted cyclic branched covering $f$ of $L$ is defined by prescribing an orientation of $L$ and associating with every component $K_i$ an integer $k_i \in \mathbb Z_n \setminus \{0\}$ such that the set $\{k_1,\dots,k_{\mu}\}$ generates $\mathbb Z_n$. If $m_i$ is a meridian around a component $K_i$ oriented in agreement with the orientation of $L$, then we write $\widetilde{\omega}_f[m_i]=k_i \in \mathbb Z_n$. Consequently, $\omega_f[m_i]=(1\,2\,\dots\,n)^{k_i}$. We denote the corresponding manifold by $M_{n,k_1,\dots,k_{\mu}}(L)$. Multiplying every $k_i$ by the same invertible element $u$ of $\mathbb Z_n$ yields an equivalent covering. Namely, two $n$-sheeted cyclic branched coverings $f_1\colon M_1 \to N_1$ and $f_2\colon M_2 \to N_2$ with maps $\widetilde{\omega}_{f_1}\colon H_1(N_1 \setminus L_1)\to \mathbb Z_n$ and $\widetilde{\omega}_{f_2}\colon H_1(N_2 \setminus L_2)\to \mathbb Z_n$ corresponding to them are equivalent if and only if there is $u \in \mathbb Z_n$ and a homeomorphism $\phi\colon N_1 \to N_2$ such that $\gcd(u,n)=1$, $\phi (L_1)=L_2$, and $\widetilde{\omega}_{f_2} \circ \phi_\#=u \cdot \widetilde{\omega}_{f_1}$, where the homomorphism $\phi_\#\colon H_1(N_1 \setminus L_1)\to H_1(N_2 \setminus L_2)$ is induced by the map $\phi_{\vert N_1 \setminus L_1}$, and $u \cdot \widetilde{\omega}_{f_1}$ denotes for the product $\widetilde{\omega}_{f_1}$ by $u$.
We recall [42] the following terminology used for branched cyclic coverings of links. The manifold $M_{n,k_1,\dots,k_{\mu}}(L)$ is said to be
The implications $\text{(i)} \Rightarrow \text{(ii)} \Rightarrow\text{(iii)} \Rightarrow \text{(iv)} \Rightarrow \text{(v)}$ are obvious. Moreover, all five definitions are equivalent if $L$ is a knot or $n=2$. By choosing an appropriate orientation of the link, from an almost strictly cyclic covering we can pass to a strictly cyclic one. Given a single-cyclic covering, we can always assume, up to equivalence and reordering the components, that $k_1=1$. In particular, if $\mu= 1$, then the $n$-sheeted cyclic branched covering of $K$ is denoted by $M_n(K)$.
We recall that a link $L$ is said to be hyperbolic if its complement $S^3 \setminus L$ admits a complete hyperbolic metric of finite volume. A link is said to be $2\pi / n$-hyperbolic, $n \geqslant 2$, if $S^3$ admits a metric of constant negative curvature that is singular with an angle of $2\pi/n$ about each component of the link; for more details, see [43]. For $n=2$ we obtain the notion of a $\pi$-hyperbolic link. We denote by $\mathcal O_n (L)$ the three-dimensional orbifold whose support is the topological space $S^3$, whose singular set is the link $L$, with a singular angle of $2\pi / n$ about every component of $L$. Saying that $L$ is a $2\pi/n$-hyperbolic link is equivalent to saying that $\mathcal O_n (L)$ is a hyperbolic orbifold. As is well known ([43], Corollary 3), if $K \subset S^3$ is a hyperbolic knot and $n\geqslant 3$, then, except in the case of $n=3$ and the ‘figure-eight’ knot, $K$ is $2\pi/n$-hyperbolic.
If $L$ is a link in $S^3$ with $\mu$ components $K_i$, $i=1, \dots, \mu$, and $m_i$ is an oriented meridian of the knot $K_i$, then by the orbifold fundamental group of the orbifold $\mathcal O_n (L)$ we mean the group
Every epimorphism $\psi\colon \pi_1 (\mathcal O_n (L)) \to \mathbb Z_n$ defines an $n$-sheeted branched covering $M$ of $\mathcal O_n (L)$ such that $\pi_1 (M)= \operatorname{Ker} \psi$. We may assume that $\psi$ takes the homotopy class of every meridian of $L$ to a generator of the cyclic group $\mathbb Z_n$, that is, the branched covering is strictly cyclic. In this case $M$ is a closed 3-manifold, and the preimage of $L$ in $M$ is a link $\widetilde{L}$ with the same number of components as $L$. The complement $M \setminus \widetilde{L}$ is the regular unbranched covering of the complement $S^3 \setminus L$ that corresponds to the kernel of the composition (which we also denote by $\psi$)
We note that if the orbifold $\mathcal O_n(L)$ is hyperbolic, then all $n$-sheeted cyclic branched coverings of $L$ are hyperbolic manifolds.
Among the knots and links in $S^3$ one distinguishes the class of two-bridge knots (they are also called rational); see the definition and properties in [44]. We denote by $\mathbf{b} (p/q)$ the two-bridge knot (or link) that corresponds to the rational parameter $p/q$, where $p > 1$ is an integer and $q \in \mathbb Z_{2p}$ is such that $(p,q)=1$. As is well known, $\mathbf{b} (p/q)$ is a knot if $p$ is odd and a two-component link if $p$ is even, and its diagram is determined by the continued fraction expansion of $p/q$. In addition, a two-bridge knot is either toric (for $|q|=1$) or hyperbolic (for $|q| > 1$). The survey [42] provides descriptions of branched cyclic coverings of two-bridge knots and links in terms of fundamental polyhedra, Heegaard diagrams, link surgery and crystallization. Below we consider two-bridge knots $\mathbf{b} (p/q)$ such that $p/q= 2\ell+1/(2\ell)$ for $\ell \geqslant 1$, and we set
The knot $K_1=\mathbf{b}(5/2)$ is known as the ‘figure-eight’ knot.
2.4. Fibonacci groups and the ‘figure-eight’ knot
The interest in Fibonacci groups in the context of three-dimensional geometry and topology is related to the following result of Helling, Kim and Mennicke [13]: for every ${n \geqslant 2}$ the Fibonacci group $F(2,2n)$ is realized as the fundamental group of a closed orientable three-manifold $M_n$, which is called the Fibonacci manifold. Formulae for the volumes of the hyperbolic Fibonacci manifolds $M_n$, $n \geqslant 4$, were obtained in [45]. As is known, $M_2$ is the lens space $L(5,2)$, and $M_3$ is the Euclidean Hantzsche–Wendt manifold constructed in 1935; see [46], [47], and [48], § 3.5.
acts freely on $\mathbb H^3$, and the fundamental polyhedron for its action is the $4n$-hedron $\mathcal P_n$ represented for $n=5$ by the development shown in Figure 1, where it is assumed that left- and right-hand edges are glued together along the polygonal line $P A_{10} A_1 Q$. The polyhedron $\mathcal P_5$ has $20$ faces and has the combinatorial structure of an icosahedron. The surface of $\mathcal P_n$ consists of $4n$ identical equilateral triangular faces. The faces $x_i (F_i)=F^*_i$ are pairwise identified under the action of the generators $x_i$, $i=1, \dots, 2n$, in accordance with the following rules:
With respect to the action of the generators $x_i$ all edges of $\mathcal P_n$ (in the amount of $6n$) are divided into $2n$ equivalence classes, and all $2n+2$ vertices belong to the same equivalence class. We note that the polyhedron $\mathcal P_n$ has a symmetry of order $n$, and the axis $PQ$ of this symmetry is shown by dashed line in Figure 1. The pairwise identification of faces mentioned above maintains this symmetry.
According to [49], § 60, Theorem I, the three-dimensional complex obtained by the pairwise identification of the sides of a polyhedron is a manifold if and only if its Euler characteristic is zero, which is the case here. It follows from the geometric realization of the polyhedron $\mathcal P_n$, $n \geqslant 4$, in the Lobachevskii space, which was described in [13], that this manifold is hyperbolic.
Hilden, Lozano and Montesinos-Amilibia [50] showed that rotations about the $PQ$-axis (see Figure 1) define an action of the cyclic group of order $n$ on the Fibonacci manifold $M_n$, $n \geqslant 2$. Moreover, $M_n$ is an $n$-sheeted cyclic covering of the 3-sphere $S^3$ branched along the image of the $PQ$-axis, the ‘figure-eight’ knot $K_1$, shown in Figure 2, that is, $M_n=M_n (K_1)$ is an $n$-fold cyclic covering of the orbifold $\mathcal O_n(K_1)$. Corresponding to this covering, there is an automorphism $r_n \in \operatorname{Out} (F(2,2n))$ such that $r_n (x_i)=x_{i+2}$, $i=1, \dots, 2n$. By Theorems 2.2 and 2.3 there exists an isometry $\rho_n \in \operatorname{Isom} (\mathbb H^3)$ which induces this automorphism: $r_n (x_i)=\rho_n^{-1} x_i \rho_n=x_{i+2}$, where $i=1, \dots, 2n$. It is clear from Figure 1 that $\rho_n$ can be chosen to be a rotation around the $PQ$-axis. For $n \geqslant 4$ we write
Theorem 2.5. For $n \!\geqslant\! 4$ there is an action of the $n$-valued coset group $F(2,2n) / R_n$ on a space homeomorphic to $\mathbb R^3$, with orbit space homeomorphic to $S^3$. The projection $\mathbb R^3 \to S^3$ onto the orbit space is related by a commutative diagram to a covering of the sphere $S^3$ by the Fibonacci manifold $M_n=\mathbb H^3 / F(2,2n)$ which is cyclically branched along a ‘figure-eight’ knot.
Proof. We claim that for $n \geqslant 4$ the $n$-valued coset group $F(2,2n) / R_n$ acts on $\mathbb H^3 / \mathcal R_n$ with orbit space $\mathcal O_n(K_1)$.
Consider the group $\Omega_n\mkern-2mu=\langle F(2, 2n) , \rho_n \rangle\mkern-1mu=\pi_1 (\mathcal O_n(K_1))$. As shown in [29] and [51], it has the presentation
where the orientations of the loop $\rho$, which can be taken as a meridian of the knot, and the loop $b$ correspond to the arrows in Figure 2.
We recall [50] that every representation of the group $\pi_1 (S^3 \setminus K)$ of $K$ into the group $\operatorname{PSL}(2, \mathbb C)=\operatorname{Isom}^+(\mathbb H^3)$ lifts to $\operatorname{SL} (2, \mathbb C)$. For the ‘figure-eight’ knot $K_1$ the representations $\sigma\colon G \to \operatorname{SL} (2, \mathbb C)$ of the group $G=\pi_1 (S^3 \setminus K_{1})$ were studied in [52], where the notation $A=\sigma (\rho^{-1})$ and $B=\sigma (b)$ was used. Let $\Lambda$ be the set of all subgroups of $\operatorname{SL} (2, \mathbb C)$ that are the images of representations of the group $G$. An Abelian group $H$ belongs to $\Lambda$ if and only if $H$ is cyclic, and a non-Abelian group $H$ belongs to $\Lambda$ if and only if $H=\langle A, B \rangle$, where $\operatorname{tr} (A)=\operatorname{tr} (B)=\alpha$ for some $\alpha \in \mathbb C$ and $\operatorname{tr} (AB)=\beta$, where
The hyperbolic structure on the orbifold $\mathcal O_n(K_1)$ corresponds to the discrete representation $\sigma_n\colon \Omega_n \to \operatorname{SL} (2, \mathbb C)$ such that $\sigma_n (\rho_n)=A^{-1}$, $\sigma_n (b_n)=B$, and we have $A^n=B^n=- I$, where $I$ is the identity matrix [50]. Therefore, up to conjugation,
so that $\lambda=e^{\pi i / n}$ and $\alpha=\operatorname{tr} (A)=2 \cos \frac{\pi}{n}$. Thus, the action of the group $\mathcal R_n$ on $\mathbb H^3$ is generated by a rotation through $2 \pi / n$ about some geodesic $\Gamma$. Hence $\mathbb H^3 / \mathcal R_n$ is an orbifold with support homeomorphic to $\mathbb R^3$ and singular set equal to the geodesic curve $\Gamma$, with singularity of order $n$. The induced metric on the complement of $\mathbb H^3 / \mathcal R_n \setminus \Gamma$ is a nonsingular incomplete Riemannian metric of constant negative curvature, whose completion produces $\mathbb H^3 / \mathcal R_n$ [43]. In the cylindrical coordinates the metric on $\mathbb H^3 / \mathcal R_n \setminus \Gamma$ is given by
where $r \in (0,+\infty)$ is the distance to $\Gamma$, $\theta \in [0, 2\pi)$ is the angular parameter, and $h \in \mathbb R$ is the distance along $\Gamma$.
Next, we use Theorem 2.4. We obtain the following diagram of coverings:
Going over, as in Example 1.3, to the $n$-cyclic presentation of the group $F(2,2n)$ with generators $y_k=x_{2k-1}$, $k=1, \dots, n$, we obtain the action $r_n (y_i)=y_{i+1}$ for $i=1, \dots, n$, from which it is clear that the $n$-valued coset group $F(2,2n)/\mathcal R_n$ is cyclic. Since the support of the orbifold $\mathbb H^3 / \mathcal R_n$ is homeomorphic to $\mathbb R^3$ and the support of the orbifold $\mathcal O_n(K_1)$ is homeomorphic to $S^3$, we obtain the assertion of the theorem.
We note that the $2$-valued coset group $F(2,4) / R_2$ corresponding to the spherical case was described in Lemma 1.5 as the group $(Z_2 (w_1), \mu_2)$.
The symmetries of the ‘figure–of-eight’ knot $K_1$ were studied by Dehn [53] and Magnus [54]. Dehn established that the group $\pi_1(S^3 \setminus K_1)$ has eight outer automorphisms, forming the dihedral group $\mathbb D_8$ of order 8. Then Magnus showed that there are no other outer automorphisms. By virtue of the correspondence between the generators $\rho$ and $b$ of the group $\pi_1 (S^3 \setminus K_1)$ and the generators $\rho_n$ and $b_n$ of $\pi_1 (\mathcal O_n(K_1))$, the symmetries of the knot $K_1$ induce symmetries of the orbifold $\mathcal O_n(K_1)$ that lift to isometries of the Fibonacci manifold $M_n(K_1)$. Namely, the following result was established in [15] for $n=4, 5, 6, 8, 12$ and in [51] for $n \geqslant 4$.
Theorem 2.6 (see [15] and [51]). Every symmetry of the ‘figure-eight’ knot $K_1$ lifts to an isometry of the hyperbolic Fibonacci manifold $M_n(K_1)$, $n \geqslant 4$.
The action of the group $\mathbb D_8$ on the generators of the group $\pi_1 (\mathcal O_n(K_1))$ was described in [51]. We consider the subgroup of order four of this group, namely,
The three nontrivial elements of the group are involutions, and they act on the generators of $\pi_1 (\mathcal O_n(K_1))$ as follows, where $\tau=\sigma \lambda$:
The axes of the involutions $\tau$ and $\sigma$ are shown in Figure 2 by dashed lines, and the axis of the involution $\lambda$ is perpendicular to the plane of the diagram and is shown by a dot.
Let us describe the action of the group $T_2=\langle \tau \mid \tau^2=1 \rangle$ on the Fibonacci group $F(2, 2n)$. We recall that $x_2=b_n \rho_n$ and $x_1=\rho_n x_2^{-1} \rho_n^{-1} x_2=b_n^{-1} \rho_n^{-1} b_n \rho_n$. Since $\tau (\rho_n)=\rho_n^{-1}$ and $\tau (b_n)=b_n^{-1}$, it follows that
The images of the remaining generators $x_k$, $k=3, \dots, 2n$, of the group $F(2, 2n)$ are determined from the relations $x_i x_{i+1}=x_{i+2}$, $i=1, \dots, 2n$. For example, $\tau(x_3)=\tau(x_1) \tau(x_2)=x_2 x_1 x_2^{-1} x_1 x_2^{-1}$. We denote the group of isometries of the space $\mathbb H^3$ corresponding to the automorphism group $T_2 \in \operatorname{Out} (F(2,2n))$ by $\mathcal T_2$. It was shown in [51] that the support of the orbifold $\mathbb H^3 / \langle F(2,2n), T_2 \rangle$ is $S^3$, and its singular set was also described there.
Theorem 2.7. For $n \geqslant 4$ there exists an action of the $2$-valued coset group ${F(2,2n) / T_2}$ on a space homeomorphic to $\mathbb R^3$, with orbit space homeomorphic to $S^3$. The projection $\mathbb R^3 \to S^3$ onto the orbit space is connected by means of a commutative diagram with a two-sheeted branched covering of the sphere $S^3$ by the Fibonacci manifold $M_n=\mathbb H^3 / F(2,2n)$.
Proof. Similarly to Theorem 2.5, the proof follows from the fact that for $n \geqslant 4$ the $2$-valued coset group $F(2,2n) / T_2$ acts on $\mathbb H^3 / \mathcal T_2$ with orbit space $\mathbb H^3 / \langle F(2,2n), T_2 \rangle$, in accordance with the following diagram:
This completes the proof of the theorem.
Similar statements hold for the involutions $\sigma$ and $\lambda$, namely, these involutions lead to actions of the 2-valued coset groups $F(2,2n) / S_2$ and $F(2,2n) / L_2$. Here the group $S_2=\langle \sigma \mid \sigma^2=1 \rangle$ acts on $F(2,2n)$ in such a way that $\sigma (x_1)=x_1^{-1}$ and $\sigma(x_2)=x_2 x_1^{-1}$. The relation $x_{i} x_{i+1}=x_{i+2}$ enables us to find the images of the remaining generators; in particular, $\sigma(x_3)=x_1^{-1} x_2 x_1^{-1}$. In turn, the group $L_2=\langle \lambda \mid \lambda^2=1 \rangle$ acts on $F(2,2n)$ in such a way that $\lambda(x_1)=x_2 x_1^{-1} x_2^{-1}$ and $\lambda(x_2)=x_2^{-1}$, so that $\lambda(x_3)=x_2 x_1^{-1} x_2^{-2}$, and so on.
2.5. Fractional Fibonacci groups and a family of two-bridge knots
In [21], for positive coprime integers $k$ and $\ell$ and for $n \geqslant 2$ a family of cyclically presented groups was introduced:
The groups $F^{k/\ell} (2,2n)$ were called fractional Fibonacci groups since for $k=\ell=1$ we obtain the Fibonacci groups $F(2,2n)$. It was shown in [21] that the groups $F^{k/\ell} (2,2n)$ are the fundamental groups of the closed orientable three-dimensional manifolds $M_n^{k/\ell}$, which belong to the family of Takahashi manifolds [30]. The manifolds $M_n^{k/\ell}$ were called fractional Fibonacci manifolds. In particular, the manifolds $M_n^k$ were constructed and studied by Maclachlan and Reid [15].
As shown in [21], for $\ell \geqslant 1$ the manifold $M_n^{1/\ell}$ is an $n$-sheeted cyclic covering of $S^3$ branched along the knot $K_{\ell}$ shown in Figure 3, that is, $M_n^{1/\ell}=M_n (K_{\ell})$, and this manifold is an $n$-sheeted cyclic covering of the orbifold $\mathcal O_n(K_{\ell})$. In particular, for $\ell \geqslant 2$ the manifold $M_2 (K_{\ell})$ is a lens space, and $M_n (K_{\ell})$ for $n \geqslant 3$ is a hyperbolic manifold.
Here the covering $M_n(K_{\ell}) \to \mathcal O_n (K_{\ell})$ corresponds to an automorphism $r_n \in \operatorname{Out} (F^{1/\ell}(2,2n))$ such that $r_n (x_i)=x_{i+2}$, where $i=1, \dots, 2n$; see [21] and [30]. By Theorems 2.2 and 2.3 there exists an isometry $\rho_n \in \operatorname{Isom} (\mathbb H^3)$ inducing this automorphism: $r_n (x_i)=\rho_n^{-1} x_i \rho_n=x_{i+2}$, where $i=1, \dots, 2n$. For $n \geqslant 3$ we write $\mathcal R_n=\langle \rho_n \mid \rho_n^n=1\rangle < \operatorname{Isom} (\mathbb H^3)$ and $R_n=\langle r_n \mid r_n^n=1\rangle < \operatorname{Out}(F^{1/\ell}(2,2n))$.
Theorem 2.8. For $n \geqslant 3$ and $\ell \geqslant 2$ there exists an action of the $n$-valued coset group $F^{1/\ell}(2,2n) / R_n$ on a space homeomorphic to $\mathbb R^3$, with orbit space homeomorphic to $S^3$. The projection $\mathbb R^3 \to S^3$ onto the orbit space is connected, by means of a commutative diagram, with the covering of the sphere $S^3$ by a compact hyperbolic three-manifold $\mathbb H^3 / F^{1/\ell}(2,2n)$ which is cyclically branched along the hyperbolic two-bridge knot $K_{\ell}$.
Proof. Similarly to Theorem 2.5, the proof follows from the fact that for $n \geqslant 3$ and $\ell \geqslant 2$ the $n$-valued coset group $F^{1/\ell}(2,2n) / R_n$ acts on $\mathbb H^3 / \mathcal R_n$ with orbit space $\mathcal O_n (K_{\ell})$, and this corresponds to the diagram
This completes the proof of the theorem.
We note that the $2$-valued coset groups $F^{1/\ell}(2,4) / R_2$ corresponding to the spherical case were described in Lemma 1.5 as the groups $(Z_2 (w_\ell), \mu_2)$.
In conclusion, we stress that Theorems 2.5 and 2.8 establish a relationship between the cyclically presented groups that correspond to $n$-fold branched cyclic coverings of knots in $S^3$ and the action of $n$-valued coset groups corresponding to a cyclic automorphism. Moreover, the $n$-valued coset groups $F^{1/\ell} (2,2n) / R_n$, $\ell \geqslant 1$, act on the hyperbolic orbifolds $\mathbb H^3 / \mathcal R_n$ with support $\mathbb R^3$, and the orbit spaces are the hyperbolic orbifolds $\mathcal O_n (K_{\ell})$ with support $S^3$.
In connection with Theorems 2.5 and 2.8, the following problem is of interest.
Problem 2.1. Classify the actions of $n$-valued coset groups corresponding to cyclic coverings of the three-dimensional sphere that are branched over knots and links.
Acknowledgements. The authors express their gratitude to A. A. Gaifullin and
D. V. Gugnin for useful discussions.
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Citation:
V. M. Buchstaber, A. Yu. Vesnin, “$n$-valued groups, branched coverings and hyperbolic 3-manifolds”, Sb. Math., 215:11 (2024), 1441–1467