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*Symmetry*
**2012**,
*4*(1),
26-38;
https://doi.org/10.3390/sym4010026

Article

Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori

Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi-Shi, Kochi 780-8520, Japan

Received: 14 November 2011; in revised form: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012

## Abstract

**:**

A symmetry group of a spatial graph $\Gamma $ in ${S}^{3}$ is a finite group consisting of orientation-preserving self-diffeomorphisms of ${S}^{3}$ which leave $\Gamma $ setwise invariant. In this paper, we show that in many cases symmetry groups of $\Gamma $ which agree on a regular neighborhood of $\Gamma $ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of $\Gamma $.

Keywords:

3-manifold; geometric topology; symmetry; finite group action; spatial graph; rational twist## 1. Introduction

There are several approaches to the theory of graphs embedded in the 3-sphere, which are often motivated by molecular chemistry, since the chemical properties of a molecule depend on the symmetries of its molecular bond graph (see, for example, [1]). The symmetries of an abstract graph $\Gamma $ are described by automorphisms. If $\Gamma $ is embedded in ${S}^{3}$, some of these automorphisms are induced from self-diffeomorphisms of ${S}^{3}$. For example, [2,3,4,5,6] studied the extendabilities of the automorphisms of $\Gamma $, mainly in the case of Möbius ladders, complete graphs, and 3-connected graphs.

Even if the automorphisms of $\Gamma $ extend to self-diffeomorphisms of ${S}^{3}$, we face the problem of the uniqueness of the extensions. In this situation, it is enough to consider $\Gamma $ to be a topological space, since we need to study self-diffeomorphisms of ${S}^{3}$ which agree on $\Gamma $. In the case of a non-torus knot in ${S}^{3}$, there are only finitely many conjugacy classes of symmetries (see [7,8]). For a cyclic period or a free period of a knot in ${S}^{3}$, it is shown in [9,10] that the cyclic group generated by the periodic self-diffeomorphism of ${S}^{3}$ defining the symmetry is unique up to conjugate in some cases. Moreover, the author [11] generalized this result to the case of links in ${S}^{3}$. In this paper, we generalize these results to the case of symmetries of spatial graphs in ${S}^{3}$.

Suppose that any component of $\Gamma $ is a non-trivial graph with no leaf. We see $\Gamma $ as a geometric simplicial complex, and denote by $\left|\Gamma \right|$ the underlying topological space of $\Gamma $. A tame embedding of $\left|\Gamma \right|$ into ${S}^{3}$ is called a spatial embedding of $\Gamma $ into ${S}^{3}$, or simply a spatial graph $\Gamma $ in ${S}^{3}$. We say that $\Gamma $ is splittable if there exists a sphere in ${S}^{3}$ disjoint from $\Gamma $ that separates the components of $\Gamma $. We say that $\Gamma $ is non-splittable if it is not splittable. Suppose that an incompressible torus in ${S}^{3}-\Gamma $ bounds a solid torus V in ${S}^{3}$ containing $\Gamma $. The core of V is called a companion knot of $\Gamma $ if it is not ambient isotopic to $\Gamma $ in V. If there is no companion knot of $\Gamma $, every incompressible torus in ${S}^{3}-\Gamma $ separates the components of $\Gamma $.

Let M be a 3-manifold, and X a submanifold of M. Denote by $N\left(X\right)$ a regular neighborhood of X, and by $E\left(X\right)=M-\mathrm{int}N\left(X\right)$ the exterior of X. We refer to a finite subgroup G of the diffeomorphism group $\mathrm{Diff}\left(M\right)$ as a finite group action on M. Finite group actions ${G}_{1}$ and ${G}_{2}$ on M are equivalent (relative to X) if some $h\in \mathrm{Diff}\left(M\right)$ conjugates ${G}_{1}$ to ${G}_{2}$ (and restricts to the identity map on X). A symmetry group G of a spatial graph $\Gamma $ in ${S}^{3}$ is a finite group action on the pair $({S}^{3},\Gamma )$ which preserves the orientation of ${S}^{3}$.

Let ${S}^{2}$ be the unit sphere in ${\mathbb{R}}^{3}$, and ${S}^{1}$ the unit circle in the $xy$-plane in ${\mathbb{R}}^{3}$. Denote by ${\mathrm{Rot}}_{\theta}\in \mathrm{Diff}\left({\mathbb{R}}^{3}\right)$ the rotation about the z-axis through angle $\theta $. Suppose that ${\sigma}_{n}\in \mathrm{Diff}({S}^{2}\times I)$ and ${\tau}_{n}\in \mathrm{Diff}({S}^{1}\times {S}^{1}\times I)$, where $n\in \mathbb{R}$, is given by ${\sigma}_{n}(x,t)=({\mathrm{Rot}}_{2\pi nt}\left(x\right),t)$ and ${\tau}_{n}(x,y,t)=({\mathrm{Rot}}_{2\pi nt}\left(x\right),y,t)$. Let F be a 2-sided sphere or torus embedded in a 3-manifold M. Split M open along F into a (possibly disconnected) 3-manifold ${M}_{F}$. Denote by ${F}_{-}$ and ${F}_{+}$ the boundary components of ${M}_{F}$ originated from F. An n-twist along F is a discontinuous map on M induced from a diffeomorphism on ${M}_{F}-{F}_{-}$ which restricts to the identity map on $E\left({F}_{+}\right)$ and the map on $N\left({F}_{+}\right)$ conjugate to ${\sigma}_{n}$ or ${\tau}_{n}$ according as F is a sphere or not. We say that the n-twist is rational if $n\in \mathbb{Q}$. Figure 1 illustrates a rotational symmetry of ${S}^{3}$ with a setwise invariant sphere S, and its conjugate by a $1/2$-twist along S.

Our main theorem is the following:

**Theorem**

**1.1.**

Let Γ be a spatial graph in ${S}^{3}$ with no companion knot. Suppose that ${G}_{1}$ and ${G}_{2}$ are symmetry groups of Γ such that

- (1)
- ${G}_{1}\left(\gamma \right)={G}_{2}\left(\gamma \right)=\gamma $ for at least one component γ of Γ,
- (2)
- either Γ is non-splittable, or ${G}_{1}$ and ${G}_{2}$ are cyclic groups acting on Γ freely, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on $N\left(\Gamma \right)$.

Then there is a finite sequence of rational twists along incompressible spheres and tori in $E\left(\Gamma \right)$ whose composition conjugates ${G}_{2}$ to a symmetry group of Γ equivalent to ${G}_{1}$ relative to $N\left(\Gamma \right)$.

This paper is arranged as follows. In Section 2, we study symmetry groups of non-splittable spatial graph in terms of the equivariant JSJ decomposition of the exteriors. In Section 3, we establish a canonical version of the equivariant sphere theorem for the exteriors of spatial graphs with cyclic symmetry groups, and prove Theorem 1.1.

## 2. Non-splittable Case

For a non-splittable spatial graph $\Gamma $ in ${S}^{3}$ with a non-trivial symmetry group, there is a canonical method for splitting $E\left(\Gamma \right)$ equivariantly into geometric pieces by the loop theorem, the Dehn’s lemma, and the JSJ decomposition theorem (see [12,13,14]).

Let M be a Haken 3-manifold with incompressible boundary. The JSJ decomposition theorem and Thurston’s uniformization theorem [15] assert that there is a canonical way of splitting the pair $(M,\partial M)$ along a disjoint, non-parallel, essential annuli and tori into pieces $({M}_{i},{F}_{i})$ each of which is one of the following four types:

- (1)
- ${M}_{i}$ is an I-bundle over a compact surface and ${F}_{i}$ is the $\partial I$-subbundle,
- (2)
- ${M}_{i}$ admits a Seifert fibration in which ${F}_{i}$ is fibered,
- (3)
- $\mathrm{int}{M}_{i}$ admits a complete hyperbolic structure of finite volume, and
- (4)
- the double of $({M}_{i},{F}_{i}-\mathrm{int}{\Phi}_{i})$ along a non-empty compact submanifold ${\Phi}_{i}$ of ${F}_{i}$ is of type (3).

For a finite group action G on M, the fixed point set $\mathrm{Fix}\left(G\right)$ of G is the set of points in M each of which has the stabilizer G. The singular set $\mathrm{Sing}\left(G\right)$ of G is the set of points in M each of which has a non-trivial stabilizer.

**Lemma**

**2.1.**

Let T be a torus embedded in ${S}^{3}$. Suppose that ${G}_{1}$ and ${G}_{2}$ are orientation-preserving finite group actions on ${S}^{3}$ such that

- (1)
- ${G}_{1}\left(N\left(T\right)\right)={G}_{2}\left(N\left(T\right)\right)=N\left(T\right)$,
- (2)
- ${G}_{1}$ and ${G}_{2}$ do not interchange the components of $\partial N\left(T\right)$, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on $\partial N\left(T\right)$.

Then a rational twist along a component of $\partial N\left(T\right)$ conjugates ${G}_{2}$ to a finite group action ${\widehat{G}}_{2}$ on ${S}^{3}$ such that the actions of ${G}_{1}$ and ${\widehat{G}}_{2}$ on $N\left(T\right)$ are equivalent relative to $\partial N\left(T\right)$.

Proof.

It is enough by Lemma 2.4 of [11] to consider the case where the actions of ${G}_{1}$ and ${G}_{2}$ on $N\left(T\right)$ are not free. For each ${G}_{i}$, Theorem 2.1 of [16] implies that $N\left(T\right)\cong T\times I$ admits a ${G}_{i}$-invariant product structure ${\mathcal{P}}_{i}$, in which $\mathrm{Sing}\left({G}_{i}\right)\cap N\left(T\right)$ consists of I-fibers. Since each element of ${G}_{i}$ takes a meridian of T to a meridian of T, the setwise stabilizer of each I-fiber is a trivial group or a 2-fold cyclic group. Therefore, the quotient space $N\left(T\right)/{G}_{i}$ admits the induced I-bundle structure over a 2-orbifold B with underlying surface F and n cone points of index two. Since T is a torus, the orbifold Euler characteristic ${\chi}_{\mathrm{orb}}\left(B\right)$ of B is calculated as follows (see [17]):
Since $n>0$, F is a sphere and $n=4$ holds.

$${\chi}_{\mathrm{orb}}\left(B\right)=\chi \left(F\right)-n/2=0.$$

Denote by ${p}_{i}:N\left(T\right)\to N\left(T\right)/{G}_{i}$ the projection map onto the quotient space for each i, and by ${T}_{t}$ the T-fiber $T\times \left\{t\right\}$ in ${\mathcal{P}}_{1}$. Connect the four cone points on ${p}_{1}\left({T}_{0}\right)$ cyclically by a collection of arcs ${\overline{a}}_{1}$, ${\overline{a}}_{2}$, ${\overline{a}}_{3}$, and ${\overline{a}}_{4}$ with disjoint interiors. Each ${\overline{a}}_{i}$ lifts to an essential loop ${a}_{i}$ on ${T}_{0}$ such that ${a}_{i}$ and ${a}_{j}$ with $i\ne j$ are disjoint if $|j-i|=2$, and otherwise ${a}_{i}$ meets ${a}_{j}$ transversally in a point. Suppose that each ${a}_{i}$ is isotopic to a loop ${b}_{i}$ on ${T}_{1}$ along an annulus ${B}_{i}$ saturated by I-fibers in ${\mathcal{P}}_{1}$, and to a loop ${c}_{i}$ on ${T}_{1}$ along an annulus ${C}_{i}$ saturated by I-fibers in ${\mathcal{P}}_{2}$. Then the endpoints of each ${p}_{2}\left({c}_{i}\right)$ is connected by ${p}_{2}\left({b}_{j}\right)$ with $|i-j|=0$ or 2. Since the underlying surface of ${p}_{2}\left({T}_{1}\right)$ is a sphere, ${\bigcup}_{i=1}^{4}{p}_{2}\left({c}_{i}\right)$ is isotopic to ${\bigcup}_{i=1}^{4}{p}_{2}\left({b}_{i}\right)$ relative to the cone points. Therefore, ${G}_{2}({\bigcup}_{i=1}^{4}{C}_{i})$ is moved by an ${G}_{2}$-equivariant isotopy relative to ${T}_{0}$ so as to agree with ${G}_{1}({\bigcup}_{i=1}^{4}{B}_{i})$ on ${T}_{1}$.

The I-bundle structures in ${\mathcal{P}}_{2}$ and ${\mathcal{P}}_{1}$ respectively induce orbifold isomorphisms ${\phi}_{1}:{p}_{2}\left({T}_{1}\right)\to {p}_{2}\left({T}_{0}\right)$ and ${\phi}_{2}:{p}_{2}\left({T}_{0}\right)\to {p}_{2}\left({T}_{1}\right)$ such that $\overline{h}={\phi}_{2}\circ {\phi}_{1}$ setwise preserves the loop ${\bigcup}_{i=1}^{4}{p}_{2}\left({b}_{i}\right)$. The restriction of $\overline{h}$ on ${\bigcup}_{i=1}^{4}{p}_{2}\left({b}_{i}\right)$ is isotopic relative to the cone points to the identity map or an involution. Since ${\bigcup}_{i=1}^{4}{p}_{2}\left({b}_{i}\right)$ splits ${p}_{2}\left({T}_{1}\right)$ into two disks with no cone point, ${\mathcal{P}}_{2}$ is deformed by a ${G}_{2}$-equivariant isotopy so that afterwards $\overline{h}$ is the identity map or an involution.

Take an $\overline{h}$-invariant ${S}^{1}$-bundle structure ${\mathcal{S}}_{1}$ on ${p}_{2}\left({T}_{1}\right)-{p}_{2}({b}_{1}\cup {b}_{3})$ with respect to which ${p}_{2}\left({b}_{2}\right)$ and ${p}_{2}\left({b}_{4}\right)$ are cross sectional, and an $\overline{h}$-invariant ${S}^{1}$-bundle structure ${\mathcal{S}}_{2}$ on ${p}_{2}\left({T}_{1}\right)-{p}_{2}({b}_{2}\cup {b}_{4})$ with respect to which every fiber in ${\mathcal{S}}_{1}$ splits into two cross sections. Then ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ induce a ${G}_{2}$-invariant product structure ${S}^{1}\times {S}^{1}$ on ${T}_{1}$. Let $h:{T}_{1}\to {T}_{1}$ be the lift of $\overline{h}$ which takes each ${c}_{i}$ to ${b}_{i}$. Then we have $h={\mathrm{Rot}}_{2\pi m}\times {\mathrm{Rot}}_{2\pi n}$ for some rational numbers m and n.

Assume $(m,n)\ne (0,0)$. Take a rational number $\gamma $ so that $\gamma m$ and $\gamma n$ are coprime integers. Then $\alpha \gamma m+\beta \gamma n=1$ holds for some integers $\alpha $ and $\beta $. Let $\rho :{\mathbb{R}}^{2}\to {S}^{1}\times {S}^{1}$ be the covering map given by $\rho (x,y)=({\mathrm{Rot}}_{2\pi x}(1,0),{\mathrm{Rot}}_{2\pi y}(1,0))$. Denote by $\phi $ the linear transformation on ${\mathbb{R}}^{2}$ represented by $\left(\begin{array}{cc}\alpha & \beta \\ -\gamma m& \gamma n\end{array}\right)$. Then the map $\rho \circ \phi \circ {\rho}^{-1}\in \mathrm{Diff}({S}^{1}\times {S}^{1})$ conjugates h to ${\mathrm{Rot}}_{2\pi /\gamma}\times {\mathrm{id}}_{{S}^{1}}$. Thus, h extends to $1/\gamma $-twist $\tau $ along ${T}_{1}$. Since h conjugates the action of ${G}_{2}$ on ${T}_{1}$ to itself, $\tau $ conjugates ${G}_{2}$ to a finite subgroup of $\mathrm{Diff}\left({S}^{3}\right)$. Therefore, it is enough to consider the case $(m,n)=(0,0)$.

It is obvious that $h={\mathrm{Rot}}_{2\pi k}\times {\mathrm{Rot}}_{2\pi l}$ holds for any integers k and l. By verifying that, for some choice of k and l, the above argument applied to ${\mathrm{Rot}}_{2\pi k}\times {\mathrm{Rot}}_{2\pi l}$ makes ${G}_{2}({\bigcup}_{i=1}^{4}{C}_{i})$ isotopic to ${G}_{1}({\bigcup}_{i=1}^{4}{B}_{i})$ relative to $\partial N\left(T\right)$, we may assume that they agree.

By considering an isotopy of $N(T)$ relative to $\partial N(T)$ which takes ${\mathcal{P}}_{2}$ to ${\mathcal{P}}_{1}$ on $\mathrm{Sing}({G}_{1})\cap N(T)$, we may assume that ${G}_{1}$ and ${G}_{2}$ agree on $\mathrm{Sing}({G}_{1})\cap N(T)$. Note that $\mathrm{Sing}({G}_{1})\cap N(T)$ splits ${G}_{1}({\bigcup}_{i=1}^{4}{B}_{i})$ into disks, and that ${G}_{1}({\bigcup}_{i=1}^{4}{B}_{i})$ splits $N(T)$ into balls. Then the identity map on ${p}_{2}(\mathrm{Sing}({G}_{2})\cap N(T))$ extends to an orbifold isomorphism $\psi :{p}_{2}({\bigcup}_{i=1}^{4}{C}_{i})\to {p}_{1}({\bigcup}_{i=1}^{4}{B}_{i})$. Since the quotient space of any finite group action on ${D}^{3}$ is isomorphic to one of the orbifolds listed on page 191 of [15], $\psi $ and the identity map on ${p}_{2}(\partial N(T))$ extend to an orbifold isomorphism ${p}_{2}(N(T))\to {p}_{1}(N(T))$. Thus, ${G}_{1}$ and ${G}_{2}$ are equivalent relative to $\partial N(T)$. Hence, the conclusion follows. □

**Lemma**

**2.2.**

Let M be a Seifert manifold in ${S}^{3}$ with non-empty boundary, and F a non-empty closed submanifold of $\partial M$. Suppose that ${G}_{1}$ and ${G}_{2}$ are finite group actions on ${S}^{3}$ such that

- (1)
- ${G}_{1}\left(M\right)={G}_{2}\left(M\right)=M$ and ${G}_{1}\left(F\right)={G}_{2}\left(F\right)=F$,
- (2)
- ${G}_{1}\left(T\right)={G}_{2}\left(T\right)=T$ for at least one component T of F,
- (3)
- ${G}_{1}$ and ${G}_{2}$ induce the same permutation on the set of the components of $\partial M$, and
- (4)
- ${G}_{1}$ and ${G}_{2}$ agree on F.

Then there is a finite sequence of rational twists along incompressible tori in M whose composition conjugates ${G}_{2}$ to a finite group action ${\widehat{G}}_{2}$ on ${S}^{3}$ such that the actions of ${G}_{1}$ and ${\widehat{G}}_{2}$ on M are equivalent relative to F.

Proof.

The case $M={D}^{2}\times {S}^{1}$ and $F=\partial M$, the case $M={S}^{1}\times {S}^{1}\times I$ and $F=\partial M$, and the case $M={S}^{1}\times {S}^{1}\times I$ and $F\ne \partial M$ respectively follow from Lemma 2.1 of [11], Lemma 2.1 of this paper, and Theorem 8.1 of [16]. We therefore exclude these cases.

Denote by ${\bigcup}_{k}{\xi}_{k}$ the system of the exceptional fibers ${\xi}_{k}$ in M. Let $N\left({\xi}_{k}\right)$ be a fibered regular neighborhood of each ${\xi}_{k}$. It follows from Theorem 2.2 of [16] that each ${G}_{i}$ preserves some Seifert fibration ${\mathcal{S}}_{i}$ of M. Then the uniqueness of a Seifert fibration of M (see VI.18.Theorem of [12]) implies that ${\bigcup}_{k}N\left({\xi}_{k}\right)$ is isotopic to a setwise ${G}_{i}$-invariant fibered regular neighborhood of the system of exceptional fibers in ${\mathcal{S}}_{i}$. Since Lemma 3.1 of [11] implies that the orders of the exceptional fibers are pairwise coprime, we may assume that ${G}_{1}\left(N\left({\xi}_{k}\right)\right)={G}_{2}\left(N\left({\xi}_{k}\right)\right)=N\left({\xi}_{k}\right)$ for each k. Therefore, it is enough by Lemma 2.1 of [11] to consider the case where M is a product ${S}^{1}$-bundle.

It follows from Theorem 2.1 of [16] that M admits a ${G}_{1}$-invariant product structure ${\mathcal{P}}_{1}$. If $F=\partial M$, M admits a ${G}_{2}$-invariant product structure ${\mathcal{P}}_{2}$ which agrees with ${\mathcal{P}}_{1}$ on F (see Theorem 2.3 of [16]). If $F\ne \partial M$, we see M as the quotient of the double $\overline{M}$ of M along $\partial M-F$ by ${\mathbb{Z}}_{2}$ generated by an orientation-reversing involution, and apply the same argument to the finite group action on $\overline{M}$, which is the extension of ${\mathbb{Z}}_{2}$ by ${G}_{2}$. Then we obtain a ${G}_{2}$-invariant product structure ${\mathcal{P}}_{2}$ of M which agrees with ${\mathcal{P}}_{1}$ on F.

By the uniqueness of the ${S}^{1}$-bundle structure of M (see VI.18.Theorem of [12]), there is a map $\phi \in \mathrm{Diff}\left(M\right)$ isotopic to the identity which takes the ${S}^{1}$-bundle structure induced by ${\mathcal{P}}_{1}$ to the ${S}^{1}$-bundle structure induced by ${\mathcal{P}}_{2}$. Modify $\phi $ in ${\mathcal{P}}_{2}$ by a fiber preserving isotopy in a fibered regular neighborhood of F so as to restrict to the identity map on F. By conjugating ${G}_{2}$ by $\phi $, we may therefore assume that ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ induce the same ${S}^{1}$-bundle structure of M.

Let $p:M\to B$ be the projection map onto the base surface B. Each ${G}_{i}$ induces a finite group action ${\overline{G}}_{i}$ on B. We consider B to be lying on ${S}^{2}$. Then each ${\overline{G}}_{i}$ extends to an action on ${S}^{2}$. Since ${\overline{G}}_{1}$ and ${\overline{G}}_{2}$ agree on $p\left(F\right)$, the quotient spaces $B/{\overline{G}}_{1}$ and $B/{\overline{G}}_{2}$ are orbifold isomorphic to suborbifolds of the same spherical orbifold listed on page 188 of [15]. We may assume that ${\overline{G}}_{1}$ and ${\overline{G}}_{2}$ are not orientation-preserving, otherwise the conclusion follows from Lemma 3.2 and Remark 3.3 of [11]. Then the assumption ${G}_{1}\left(T\right)={G}_{2}\left(T\right)=T$ implies that each ${\overline{G}}_{i}$ is generated by the reflection of ${S}^{2}$ in a loop. Since ${G}_{1}$ and ${G}_{2}$ permute the components of $\partial M$ similarly, $\partial B$ consists of loops ${\ell}_{1},\dots ,{\ell}_{2k},{\ell}_{1}^{\prime},\dots ,{\ell}_{n}^{\prime}$ such that

- (1)
- ${\overline{G}}_{1}$ and ${\overline{G}}_{2}$ interchange ${\ell}_{2i-1}$ and ${\ell}_{2i}$ for $1\le i\le k$, and
- (2)
- ${\overline{G}}_{1}$ and ${\overline{G}}_{2}$ setwise preserve ${\ell}_{i}^{\prime}$ for $1\le i\le n$.

Without loss of generality, ${\ell}_{1}^{\prime}=p\left(T\right)$. Denote by $\mathrm{Fix}\left({\overline{G}}_{i}\right)$ the fixed point circle of the action of each ${\overline{G}}_{i}$ on ${S}^{2}$. Suppose that each $\mathrm{Fix}\left({\overline{G}}_{i}\right)$ is equipped with an orientation, and splits B into two pieces ${B}_{i,1}$ and ${B}_{i,2}$ so that ${\ell}_{1}^{\prime}\cap {B}_{1,1}={\ell}_{1}^{\prime}\cap {B}_{2,1}$ and ${\ell}_{1}^{\prime}\cap {B}_{1,2}={\ell}_{1}^{\prime}\cap {B}_{2,2}$. We may assume without loss of generality that ${\ell}_{2i-1}\subset {B}_{1,1}$ and ${\ell}_{2i}\subset {B}_{1,2}$ for $1\le i\le k$, and that we meets ${\ell}_{1}^{\prime},\dots ,{\ell}_{n}^{\prime}$ in order as we go along $\mathrm{Fix}\left({\overline{G}}_{1}\right)$.

Suppose ${\ell}_{2i-1}\subset {B}_{2,2}$ and ${\ell}_{2i}\subset {B}_{2,1}$ for some i. By taking a proper arc on $B/{\overline{G}}_{2}$ connecting ${\ell}_{2i}/{\overline{G}}_{2}$ and $\mathrm{Fix}\left({\overline{G}}_{2}\right)/{\overline{G}}_{2}$, we obtain a setwise ${\overline{G}}_{2}$-invariant arc $\alpha $ on B which meets $\mathrm{Fix}\left({\overline{G}}_{2}\right)$ in a point and connects ${\ell}_{2i-1}$ and ${\ell}_{2i}$. Then $\mathrm{Fix}\left({\overline{G}}_{2}\right)$ is modified by the half twist along the loop $\partial N({\ell}_{2i-1}\cup {\ell}_{2i}\cup \alpha )\cap \mathrm{int}B$, denoted by $\lambda $, so that afterwards ${\ell}_{2i-1}\subset {B}_{2,1}$ and ${\ell}_{2i}\subset {B}_{2,2}$, as illustrated in Figure 2. The argument presented for the proof of Lemma 2.1 implies that this modification is realized by a $1/2$-twist along the torus ${p}^{-1}\left(\lambda \right)$ which conjugates ${G}_{2}$ to a subgroup of $\mathrm{Diff}\left({S}^{3}\right)$. We may therefore assume ${\ell}_{2i-1}\subset {B}_{2,1}$ and ${\ell}_{2i}\subset {B}_{2,2}$ for $1\le i\le k$.

Suppose that ${\ell}_{i}^{\prime}$ and ${\ell}_{j}^{\prime}$ are connected by an arc ${\alpha}^{\prime}$ in $\mathrm{Fix}\left({\overline{G}}_{2}\right)\cap B$. Then $\mathrm{Fix}\left({\overline{G}}_{2}\right)$ is modified by the half twists along the loop ${\lambda}^{\prime}=\partial N({\ell}_{i}^{\prime}\cup {\ell}_{j}^{\prime}\cup {\alpha}^{\prime})\cap \mathrm{int}B$ so as to meet ${\ell}_{i}^{\prime}$ and ${\ell}_{j}^{\prime}$ in the reverse order, as illustrated in Figure 3, which is realized by the conjugation of ${G}_{2}$ by a $1/2$-twist along the torus ${p}^{-1}\left({\lambda}^{\prime}\right)$, as before. Since every permutation on the set $\{{\ell}_{2}^{\prime},\dots ,{\ell}_{n}^{\prime}\}$ is a product of transpositions, we may assume that $\mathrm{Fix}\left({\overline{G}}_{2}\right)$ meets ${\ell}_{1}^{\prime},\dots ,{\ell}_{n}^{\prime}$ in order. Moreover, we can change the order in which $\mathrm{Fix}\left({\overline{G}}_{2}\right)$ meets the two points in ${\ell}_{i}^{\prime}\cap \mathrm{Fix}\left({\overline{G}}_{2}\right)$ by the half twists along ${\ell}_{i}^{\prime}$, which is also realized by a $1/2$-twist along the torus ${p}^{-1}\left({\ell}_{i}^{\prime}\right)$. We may therefore assume that ${\overline{G}}_{2}$ is equivalent to ${\overline{G}}_{1}$ relative to $\partial B$.

Now we may assume ${\overline{G}}_{1}={\overline{G}}_{2}$. Take a map $h\in \mathrm{Diff}\left(M\right)$ which restricts to the identity map on F and takes ${\mathcal{P}}_{2}$ to ${\mathcal{P}}_{1}$ setwise preserving every ${S}^{1}$-fiber. It is easy to verify that h is extendable to a map in $\mathrm{Diff}\left({S}^{3}\right)$. Hence, the conclusion follows by conjugating ${G}_{2}$ by h. □

**Lemma**

**2.3.**

Let M be a compact connected 3-manifold in ${S}^{3}$ with non-empty boundary whose interior admits a complete hyperbolic structure of finite volume, and F a non-empty closed submanifold of $\partial M$. Suppose that ${G}_{1}$ and ${G}_{2}$ are finite group actions on ${S}^{3}$ such that

- (1)
- ${G}_{1}\left(M\right)={G}_{2}\left(M\right)=M$ and ${G}_{1}\left(F\right)={G}_{2}\left(F\right)=F$,
- (2)
- ${G}_{1}$ and ${G}_{2}$ induce the same permutation on the set of the components of $\partial M$, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on F.

Then there is a sequence of rational twists along tori in F whose composition conjugates ${G}_{2}$ to a finite group action ${\widehat{G}}_{2}$ such that the actions of ${G}_{1}$ and ${\widehat{G}}_{2}$ on M is equivalent relative to F.

Proof.

It follows from Theorem 5.5 of [18] that $\mathrm{int}M$ admits two complete hyperbolic structures of finite volume, one is ${G}_{1}$-invariant and the other is ${G}_{2}$-invariant. Mostow’s rigidity theorem [15] implies that complete hyperbolic structures of finite volume on $\mathrm{int}M$ are unique up to isometry representing the identity map on $\mathrm{Out}\left({\pi}_{1}\left(M\right)\right)$. We may therefore assume that $\mathrm{int}M$ is endowed with the ${G}_{1}$-invariant hyperbolic structure, and that ${G}_{2}$ is conjugate to an isometric action ${G}_{2}^{\prime}$ by $h\in \mathrm{Diff}\left(M\right)$ which is isotopic to the identity map.

Next, we are going to modify h in a regular neighborhood of F so as to restrict to the identity map on F. It follows from Propostition D.3.18 of [19] that F consists of tori. Let ${h}_{t}$ be an isotopy from h to the identity map. Denote by ${\overline{G}}_{2}$ the finite group action on $F\times I$ whose restriction on $F\times \left\{t\right\}$ is induced from the finite group action on F given by the conjugate of ${G}_{2}$ by ${h}_{t}$. In particular, the actions of ${\overline{G}}_{2}$ on $F\times \left\{0\right\}$ and $F\times \left\{1\right\}$ are respectively given by ${G}_{2}^{\prime}$ and ${G}_{2}$. Note that ${\overline{G}}_{2}$ preserves the product structure $F\times \partial I$, and that we can embed $F\times I$ in ${S}^{3}$ so that ${\overline{G}}_{2}$ extends to a finite group action on ${S}^{3}$.

We consider the partition of the set of the components of F into the orbits under the permutation induced by ${G}_{2}$. Suppose that the orbits are represented by ${T}_{1},\dots ,{T}_{n}$. Lemma 2.1 implies that a rational twist along ${T}_{i}\times \left\{1\right\}$ conjugates the setwise stabilizer of ${T}_{i}\times I$ in ${\overline{G}}_{2}$ so that the action on ${T}_{i}\times I$ is equivalent relative to ${T}_{i}\times \partial I$ to the action which preserves the product structure. Suppose that the rational twists along the tori in $F\times \left\{1\right\}$ are equivariantly induced from those along ${T}_{1}\times \left\{1\right\},\dots ,{T}_{n}\times \left\{1\right\}$. By conjugating ${\overline{G}}_{2}$ by their composition, it is equivalent relative to $F\times \partial I$ to the action which preserves the product structure. This implies that h is modified equivariantly so as to restrict to the identity map on F.

Suppose that ${g}_{1}\in {G}_{1}$ and ${g}_{2}\in {G}_{2}$ agree on F. Then ${g}_{1}\circ {g}_{2}^{-1}$ restricts to the identity map on F. Since the isometry group of $\mathrm{int}M$ is finite (see [15]), Newman’s theorem [20] implies ${g}_{1}={g}_{2}$. Hence, ${G}_{1}$ and ${G}_{2}^{\prime}$ agree on M. This completes the proof. □

**Lemma**

**2.4.**

Let M be a compact connected 3-manifold in ${S}^{3}$ with non-empty boundary such that the double $\overline{M}$ of M along a non-empty compact submanifold Φ of $\partial M$ admits a complete hyperbolic structure of finite volume in its interior. Let F be a closed submanifold of $\partial M$ containing Φ. Suppose that ${G}_{1}$ and ${G}_{2}$ are finite group actions on ${S}^{3}$ such that

- (1)
- ${G}_{1}\left(M\right)={G}_{2}\left(M\right)=M$ and ${G}_{1}\left(F\right)={G}_{2}\left(F\right)=F$,
- (2)
- ${G}_{1}$ and ${G}_{2}$ induce the same permutation on the set of the components of $\partial M$, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on F.

Then there is a finite sequence of rational twists along tori in F whose composition conjugates ${G}_{2}$ to a finite group action ${\widehat{G}}_{2}$ such that the actions of ${G}_{1}$ and ${\widehat{G}}_{2}$ on M are equivalent relative to F.

Proof.

We see M as the quotient of $\overline{M}$ by ${\mathbb{Z}}_{2}$ generated by an orientation-reversing involution. Each ${G}_{i}$ induces a finite group action ${\overline{G}}_{i}$ on $\overline{M}$ which is an extension of ${\mathbb{Z}}_{2}$ by ${G}_{i}$. As in the proof of Lemma 2.3, we consider $\mathrm{int}\overline{M}$ endowed with a ${\overline{G}}_{1}$-invariant hyperbolic structure. Then some $\overline{h}\in \mathrm{Diff}\left(\overline{M}\right)$, which is isotopic to the identity map, conjugates ${\overline{G}}_{2}$ to an isometric action ${\overline{G}}_{2}^{\prime}$. Clearly, $\Phi $ meets $\mathrm{int}\overline{M}$ in a totally geodesic surface, and therefore $\overline{h}\left(\Phi \right)=\Phi $ holds.

Suppose that ${\overline{g}}_{1}\in {\overline{G}}_{1}$ and ${\overline{g}}_{2}\in {\overline{G}}_{2}^{\prime}$ respectively induce ${g}_{1}\in {G}_{1}$ and ${g}_{2}\in {G}_{2}$ which agree on F. Then ${\overline{g}}_{1}^{-1}\circ {\overline{g}}_{2}$ restricts to an isometry on each component ${\Phi}_{i}$ of $\Phi $, which is a compact surface of negative Euler characteristic (see Propostition D.3.18 of [19]). Since ${\overline{g}}_{1}^{-1}\circ {\overline{g}}_{2}$ is trivial in $\mathrm{Out}\left({\pi}_{1}\left({\Phi}_{i}\right)\right)$, ${\overline{g}}_{1}$ and ${\overline{g}}_{2}$ agree on ${\Phi}_{i}$. Therefore, [20] implies ${g}_{1}={g}_{2}$. Hence, some $h\in \mathrm{Diff}\left(M\right)$, which setwise preserves $\Phi $ and is isotopic to the identity map, conjugates the action of ${G}_{1}$ on M to ${G}_{2}$.

It follows from Proposition D.3.18 of [19] that $F-\Phi $ consists of tori. As in the proof of Lemma 2.3, modify h in $N(F-\Phi )$ by rational twists along tori in $F-\Phi $ so that afterwards h restricts to the identity map on $F-\Phi $ and conjugates the action of ${G}_{1}$ on M to ${G}_{2}$. Moreover, we may assume by Lemma 2.3 of [11] that h restricts to the identity map on $\Phi $. Since h extends to an automorphism of ${S}^{3}$ which is diffeomorphic outside M, the conclusion follows. □

**Proposition**

**2.5.**

Theorem 1.1 is true, if Γ is non-splittable.

Proof.

The equivariant loop theorem (see Chapter VII of [15] and [21]) implies that there is a ${G}_{1}$-invariant system ${\mathcal{D}}_{1}$ of disjoint disks properly embedded in $E\left(\Gamma \right)$ which splits $E\left(\Gamma \right)$ into pieces with incompressible boundary. The equivariant Dehn’s lemma [21,22] implies that the boundary loops of ${\mathcal{D}}_{1}$ bound a ${G}_{2}$-invariant system ${\mathcal{D}}_{2}$ of disjoint disks properly embedded in $E\left(\Gamma \right)$. Since $\Gamma $ is non-splittable, $E\left(\Gamma \right)$ is irreducible. Therefore, there is an isotopy of $E\left(\Gamma \right)$ relative to $\partial E\left(\Gamma \right)$ which takes ${\mathcal{D}}_{2}$ to ${\mathcal{D}}_{1}$. Since any finite group action on ${D}^{2}$ is orthogonal [15], we may assume that ${G}_{1}$ and ${G}_{2}$ agree on ${\mathcal{D}}_{1}$. Moreover, the induced actions on the balls obtained by splitting $E\left(\Gamma \right)$ along ${\mathcal{D}}_{1}$ are equivalent relative to the boundary (see [15]). Therefore, it is enough to consider the case where $E\left(\Gamma \right)$ is a Haken manifold with incompressible boundary.

We may assume by the equivariant JSJ decomposition theorem (see Theorem 8.6 of [16]) and by the uniqueness of the JSJ decomposition [13,14] that there is a ${G}_{1}$-invariant and ${G}_{2}$-invariant system $\mathcal{T}$ of essential annuli and tori in $E\left(\Gamma \right)$ realizing the canonical JSJ decomposition of the pair $\left(E\right(\Gamma ),\partial E(\Gamma \left)\right)$.

The argument presented for the proof of Proposition 3.10 of [11] implies that some $h\in \mathrm{Diff}\left({S}^{3}\right)$, which is isotopic to the identity map relative to $N\left(\Gamma \right)$, conjugates ${G}_{2}$ to a finite group action which agree with ${G}_{1}$ on the annuli in $\mathcal{T}$. We may therefore assume that $\mathcal{T}$ contains no annuli.

The rest of the proof proceeds by induction on the number of tori in $\mathcal{T}$. Take a piece ${M}_{k}$ attaching $\partial E\left(\Gamma \right)$. By Lemmas 2.2, 2.3 and 2.4, it is enough to consider the case where ${G}_{2}$ agrees with ${G}_{1}$ on ${G}_{1}\left({M}_{k}\right)$. Moreover, we may assume by Lemma 2.1 that ${G}_{1}$ and ${G}_{2}$ agree on the components of $\mathrm{cl}(E\left(\Gamma \right)-{G}_{1}\left({M}_{k}\right))$ each of which is a product I-bundle over a torus. Hence, the conclusion follows by the induction hypothesis. □

## 3. Possibly Splittable Case

For a symmetry group G of a splittable spatial graph $\Gamma $ in ${S}^{3}$, there is a setwise G-invariant system $\mathcal{S}$ of spheres realizing the prime factorization of $E\left(\Gamma \right)$ (see [23]). However, $\mathcal{S}$ is not unique in contrast to the JSJ decomposition of a Haken 3-manifold. If some component of $\Gamma $ is setwise invariant and every essential sphere in $E\left(\Gamma \right)$ has a trivial stabilizer, there is a canonical choice of $\mathcal{S}$ (see [11]). We first prove that this is possible also in the setting of Theorem 1.1.

**Lemma**

**3.1.**

Let Γ be a splittable spatial graph in ${S}^{3}$. Suppose that ${G}_{1}$ and ${G}_{2}$ are symmetry groups of Γ such that

- (1)
- ${G}_{1}\left(\gamma \right)={G}_{2}\left(\gamma \right)=\gamma $ for at least one component γ of Γ,
- (2)
- ${G}_{1}$ and ${G}_{2}$ are cyclic groups acting on Γ freely, and
- (3)
- ${G}_{1}$ and ${G}_{2}$ agree on $N\left(\Gamma \right)$.

Then each ${G}_{i}$ admits a setwise ${G}_{i}$-invariant system ${\mathcal{B}}_{i}$ of disjoint balls in ${S}^{3}$ not containing γ such that each $\partial {\mathcal{B}}_{i}$ realizes the prime factorization of $E\left(\Gamma \right)$. Moreover, for some choice of ${\mathcal{B}}_{1}$ and ${\mathcal{B}}_{2}$, there is a finite sequence of rational twists along incompressible tori in $E\left(\Gamma \right)-\mathrm{int}{\mathcal{B}}_{2}$ and a map in $\mathrm{Diff}\left({S}^{3}\right)$ which restricts to the identity map on $N\left(\Gamma \right)$ whose composition conjugates the action of ${G}_{2}$ on ${S}^{3}-\mathrm{int}{\mathcal{B}}_{2}$ to the action of ${G}_{1}$ on ${S}^{3}-\mathrm{int}{\mathcal{B}}_{1}$.

Proof.

Denote by ${\Gamma}_{\gamma}$ the non-splittable spatial subgraph of $\Gamma $ containing $\gamma $ which is obtained by the prime factorization of $E\left(\Gamma \right)$. It follows from the equivariant sphere theorem [23] that each ${G}_{i}$ admits a setwise ${G}_{i}$-invariant system ${\mathcal{S}}_{i}={S}_{i,1}\cup \cdots \cup {S}_{i,n}$ of disjoint, non-parallel, essential spheres in $E\left(\Gamma \right)$ realizing the prime factorization. Suppose that each ${S}_{i,j}$ bounds a ball ${B}_{i,j}$ disjoint from $\gamma $. Note that $\mathrm{Sing}\left({G}_{i}\right)$ avoids ${B}_{i,j}$ or meets ${B}_{i,j}$ in a trivial 1-string tangle (see [15]).

Suppose ${B}_{i,j}\subset {B}_{i,k}$ for some distinct j and k. Denote by ${p}_{i}:{S}^{3}\to {S}^{3}/{G}_{i}={S}^{3}$ the projection map onto the quotient space. Take an arc $\alpha $ properly embedded in ${p}_{i}({B}_{i,k}-\mathrm{int}{B}_{i,j})$ which connects ${p}_{i}(\partial {B}_{i,k})$ and ${p}_{i}(\partial {B}_{i,j})$. Suppose that $\alpha $ lies on ${p}_{i}\left(\mathrm{Sing}\left({G}_{i}\right)\right)$ if $\mathrm{Sing}\left({G}_{i}\right)$ connects $\partial {B}_{i,j}$ and $\partial {B}_{i,k}$. By replacing ${B}_{i,j}$ with another ball in $\mathrm{int}{B}_{i,k}$ if necessary, $\alpha $ meets ${\mathcal{S}}_{i}$ in its endpoints. By drilling into ${p}_{i}\left({B}_{i,k}\right)$ along $\alpha \cup {p}_{i}\left({B}_{i,j}\right)$, ${B}_{i,k}$ is deformed to a ball disjoint from ${B}_{i,j}$, as illustrated in Figure 4 in which the result of the deformation is presented in a cross-sectional view. By a finite repetition of this operation, we obtain a system ${\mathcal{B}}_{i}={B}_{i,1}\cup \cdots \cup {B}_{i,n}$ of disjoint balls. This proves the first half of the lemma. Without loss of generality, $\Gamma \cap {B}_{2,j}=\Gamma \cap {B}_{1,j}$ for $1\le j\le n$.

Proposition 2.5 implies that there is a finite sequence of rational twists along incompressible tori in $E\left({\Gamma}_{\gamma}\right)$ whose composition h conjugates ${G}_{2}$ to a symmetry group ${\widehat{G}}_{2}$ of ${\Gamma}_{\gamma}$ equivalent to ${G}_{1}$ relative to $N\left({\Gamma}_{\gamma}\right)$. By a ${G}_{2}$-equivariant isotopy, we may assume that these incompressible tori are disjoint from ${\mathcal{B}}_{2}$. Then h restricts to the identity map on ${\mathcal{B}}_{2}$. Suppose that $H\in \mathrm{Diff}({S}^{3},{\Gamma}_{\gamma})$ realizes the above equivalence of ${\widehat{G}}_{2}$ and ${G}_{1}$. Then H takes $\mathrm{Sing}({\widehat{G}}_{2})$ to $\mathrm{Sing}({G}_{1})$. As a consequence of the affirmative answer to the Smith conjecture [15], $\mathrm{Sing}\left({G}_{1}\right)$ is either an empty set, a trivial knot, or a Hopf link whose components have different indices. Suppose that the orientation of $\mathrm{Sing}\left({G}_{1}\right)$ is induced from the orientation of $\mathrm{Sing}({\widehat{G}}_{2})$ by H.

Suppose that ${B}_{2,j}$ and ${B}_{2,k}$ are connected by an arc $\beta $ in $\mathrm{Sing}({\widehat{G}}_{2})-\mathrm{int}{\mathcal{B}}_{2}$, and that $\mathrm{Sing}({\widehat{G}}_{2})$ meets ${B}_{2,k}$ in an arc $\delta $. Then ${B}_{2,k}$ can be modified by a ${\widehat{G}}_{2}$-equivariant deformation along $\beta \cup {B}_{2,j}$ similar to the inverse of that mentioned above so as to contain ${B}_{2,j}$. Moreover, it can be deformed along $\beta \cup \delta \cup {B}_{2,j}$ so as to avoid ${B}_{2,j}$ again, as illustrated in Figure 5. Note that this operation changes the order in which the circle in $\mathrm{Sing}({\widehat{G}}_{2})$ containing $\beta $ meets the balls in $\{{B}_{2,1},\dots ,{B}_{2,n}\}$.

Let C be a component of $\mathrm{Sing}\left({G}_{1}\right)$. Without loss of generality, C meets ${B}_{1,1},\dots ,{B}_{1,r}$ in order, and avoids ${B}_{1,r+1},\dots ,{B}_{1,n}$. Since ${G}_{1}$ and ${G}_{2}$ agree on $N\left(\Gamma \right)$, the component ${H}^{-1}\left(C\right)$ of $\mathrm{Sing}({\widehat{G}}_{2})$ meets ${B}_{2,1},\dots ,{B}_{2,r}$ possibly not in order. Since every permutation on the set $\{{B}_{2,1},\dots ,{B}_{2,r}\}$ is a product of transpositions realized by the above operation, we may assume that ${H}^{-1}\left(C\right)$ meets ${B}_{2,1},\dots ,{B}_{2,r}$ in order. Apply this argument to each component of $\mathrm{Sing}\left({G}_{1}\right)$. Since each $\partial {\mathcal{B}}_{i}$ realizes the prime factorization of $E\left(\Gamma \right)$, we can modify H by a ${G}_{1}$-equivariant isotopy relative to $N\left({\Gamma}_{\gamma}\right)$ so that we have $H(\mathrm{Sing}({\widehat{G}}_{2}))=\mathrm{Sing}({G}_{1})$ and $H\left({B}_{2,j}\right)={B}_{1,j}$ for each j. Thus, H is modified so as to conjugate the action of ${\widehat{G}}_{2}$ on ${S}^{3}-\mathrm{int}{\mathcal{B}}_{2}$ to the action of ${G}_{1}$ on ${S}^{3}-\mathrm{int}{\mathcal{B}}_{1}$.

After this modification, H restricts to an orientation-preserving homeomorphism on ${\mathcal{B}}_{1}$. Therefore, $H(\Gamma \cap {\mathcal{B}}_{2})$ is ambient isotopic to $\Gamma \cap {\mathcal{B}}_{1}$ in ${\mathcal{B}}_{1}$. Hence, H can be modified in ${\mathcal{B}}_{1}$ so as to restrict to the identity map on $N\left(\Gamma \right)$. This completes the proof. □

**Lemma**

**3.2.**

Suppose that ${G}_{1}$ and ${G}_{2}$ are orientation-preserving finite cyclic group actions on ${S}^{2}\times I$ such that

- (1)
- ${G}_{1}$ and ${G}_{2}$ do not interchange the components of ${S}^{2}\times \partial I$, and
- (2)
- ${G}_{1}$ and ${G}_{2}$ agree on ${S}^{2}\times \partial I$.

Then a rational twist along ${S}^{2}\times \left\{1\right\}$ conjugates ${G}_{2}$ to a finite group action equivalent to ${G}_{1}$ relative to ${S}^{2}\times \partial I$.

Proof.

It is enough to consider the case where ${G}_{1}$ is not trivial. It follows from the remark after Theorem 8.1 of [16] that ${S}^{2}\times I$ admits a ${G}_{1}$-invariant product structure ${\mathcal{P}}_{1}$ and a ${G}_{2}$-invariant product structure ${\mathcal{P}}_{2}$. Since the actions of ${G}_{1}$ and ${G}_{2}$ on ${S}^{2}\times \left\{0\right\}$ are conjugate to a rotation of ${S}^{2}$ (see [15]), each $\mathrm{Fix}\left({G}_{i}\right)$ consists of two I-fibers in ${\mathcal{P}}_{i}$. Since ${G}_{1}$ and ${G}_{2}$ agree on ${S}^{2}\times \partial I$, we have $\partial \mathrm{Fix}\left({G}_{1}\right)=\partial \mathrm{Fix}\left({G}_{2}\right)$.

Denote by ${p}_{i}:{S}^{2}\times I\to {S}^{2}\times I/{G}_{i}$ the projection map onto the quotient space for each i, and by ${S}_{t}$ the ${S}^{2}$-fiber ${S}^{2}\times \left\{t\right\}$ in ${\mathcal{P}}_{1}$. Connect the two cone points of ${p}_{1}\left({S}_{0}\right)$ by an arc $\overline{a}$ embedded in ${p}_{1}\left({S}_{0}\right)$. Then ${p}_{1}^{-1}\left(\overline{a}\right)$ is a spatial ${\theta}_{n}$-curve consisting of two vertices on the fixed points and $n>1$ edges each connecting them. Denote by ${A}_{i}$ the branched surface consisting of I-fibers in ${\mathcal{P}}_{i}$ attaching ${p}_{1}^{-1}\left(\overline{a}\right)$ for each i. Then each ${p}_{1}({A}_{i}\cap {S}_{1})$ is an arc connecting the two cone points on ${p}_{1}\left({S}_{1}\right)$. Since the underlying space of ${p}_{1}\left({S}_{1}\right)$ is a sphere, ${p}_{1}({A}_{2}\cap {S}_{1})$ is isotopic to ${p}_{1}({A}_{1}\cap {S}_{1})$ relative to the cone points. Therefore, ${A}_{2}$ is deformed by a ${G}_{2}$-equivariant isotopy relative to ${S}_{0}$ so that ${A}_{1}\cap {S}_{1}={A}_{2}\cap {S}_{1}$. There are two cases depending on whether $\mathrm{Fix}\left({G}_{1}\right)$ and $\mathrm{Fix}\left({G}_{2}\right)$ are isotopic relative to the endpoints or not.

Assume that $\mathrm{Fix}\left({G}_{1}\right)$ and $\mathrm{Fix}\left({G}_{2}\right)$ are isotopic relative to the endpoints. Then ${\mathcal{P}}_{2}$ is deformed by an isotopy relative to ${S}^{2}\times \partial I$ so as to agree with ${\mathcal{P}}_{1}$ on a setwise ${G}_{1}$-invariant tubular neighborhood $N\left(\mathrm{Fix}\left({G}_{1}\right)\right)$ saturated in the I-bundle structure induced from ${\mathcal{P}}_{1}$. Since each ${A}_{i}$ meets the solid torus $({S}^{2}\times I)-\mathrm{int}N\left(\mathrm{Fix}\left({G}_{1}\right)\right)$ in the system of meridian disks, ${A}_{2}$ is moved to ${A}_{1}$ by an isotopy relative to $({S}^{2}\times \partial I)\cup N\left(\mathrm{Fix}\left({G}_{1}\right)\right)$. We may therefore assume ${A}_{1}={A}_{2}$, and that ${G}_{1}$ and ${G}_{2}$ agree on $N\left(\mathrm{Fix}\left({G}_{1}\right)\right)$. Then the I-bundle structures in ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ respectively induce the orbifold isomorphisms ${\phi}_{1}:{p}_{2}\left({S}_{1}\right)\to {p}_{2}\left({S}_{0}\right)$ and ${\phi}_{2}:{p}_{2}\left({S}_{0}\right)\to {p}_{1}\left({S}_{1}\right)$ such that ${\phi}_{2}\circ {\phi}_{1}$ is isotopic to the identity map by an isotopy relative to the cone points which setwise preserves ${p}_{2}({A}_{2}\cap {S}_{1})$. Then we can deform ${\mathcal{P}}_{2}$ by an isotopy on ${p}_{2}({S}^{2}\times I)$ relative to ${p}_{2}\left({S}_{0}\right)$ which setwise preserves ${p}_{2}({A}_{2}\cap {S}_{1})$ so that ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ induce the same $\partial I$-bundle structure on ${S}^{1}\times \partial I$. Hence, the diffeomorphism of ${S}^{2}\times I$ which takes ${\mathcal{P}}_{2}$ to ${\mathcal{P}}_{1}$ induces the equivalence of ${G}_{1}$ and ${G}_{2}$ relative to ${S}^{2}\times \partial I$, as required.

Assume that $\mathrm{Fix}\left({G}_{1}\right)$ and $\mathrm{Fix}\left({G}_{2}\right)$ are not isotopic relative to the endpoints. Let $h:{S}_{1}\to {S}_{1}$ be a lift of an orientation-preserving involution on ${p}_{1}\left({S}_{1}\right)$ which interchanges the cone points. Then h is a diffeomorphism isotopic to the identity map which conjugates the action of ${G}_{2}$ on ${S}_{1}$ to itself and is realized by a $1/2$-twist along the sphere ${S}_{1}$. We may therefore assume that $\mathrm{Fix}\left({G}_{1}\right)$ and $\mathrm{Fix}\left({G}_{2}\right)$ are isotopic relative to the endpoints. Hence, the conclusion follows by the argument presented for the previous case. □

Proof of Theorem 1.1.

It is enough by Proposition 2.5 to prove the theorem in the case where $\Gamma $ is splittable. Then ${G}_{1}$ and ${G}_{2}$ are cyclic groups acting on $\Gamma $ freely. We may assume by Lemma 3.1 that there is a setwise ${G}_{1}$-invariant and setwise ${G}_{2}$-invariant system $\mathcal{B}$ of disjoint balls in ${S}^{3}$ not containing $\gamma $ such that $\partial \mathcal{B}$ realizes the prime factorization of $E\left(\Gamma \right)$, and that ${G}_{1}$ and ${G}_{2}$ agree on $E\left(\mathcal{B}\right)$.

Suppose that $\mathcal{B}$ consists of balls ${B}_{1},\dots ,{B}_{n}$. Each $\Gamma \cap {B}_{i}$ is a non-empty, non-splittable, spatial subgraph of $\Gamma $. By applying Proposition 2.5 to the actions of the setwise stabilisers of ${B}_{i}$ in ${G}_{1}$ and ${G}_{2}$ on ${B}_{i}$, we may assume that ${G}_{1}$ and ${G}_{2}$ agree on $E(\partial \mathcal{B})$. Hence the conclusion follows by applying Lemma 3.2 to the actions of ${G}_{1}$ and ${G}_{2}$ on $N(\partial \mathcal{B})$ equivariantly. □

**Remark**

**3.3.**

Theorem 1.1 requires the spatial graph $\Gamma $ to have no companion knot, and the symmetry groups ${G}_{1}$ and ${G}_{2}$ of $\Gamma $ to act on $\Gamma $ freely if $\Gamma $ is splittable. These requirements are needed because of the following examples.

- (1)
- Suppose that $\Gamma $ is a granny knot. Then $\Gamma $ has two companion knots ${K}_{1}$ and ${K}_{2}$, both of which are trefoil knots. We obtain $E\left({K}_{1}\right)$, $E\left({K}_{2}\right)$, and a 2-fold composing space by the JSJ decomposition of $E\left(\Gamma \right)$. Figure 6 illustrates ${\mathbb{Z}}_{2}$-symmetries ${G}_{1}$ and ${G}_{2}$ of $\Gamma $ such that ${G}_{2}$ interchanges $E\left({K}_{1}\right)$ and $E\left({K}_{2}\right)$ but ${G}_{1}$ does not. By conjugating ${G}_{1}$ by a map in $\mathrm{Diff}\left({S}^{3}\right)$ which moves $N\left(\Gamma \right)$ in the longitudinal direction, ${G}_{1}$ and ${G}_{2}$ are not equivalent but agree on $\partial N\left(\Gamma \right)$. Moreover, any rational twists along incompressible tori in $E\left(\Gamma \right)$ cannot change the induced symmetries of $E\left({K}_{1}\right)$ and $E\left({K}_{2}\right)$, since the trefoil knot exterior is atoroidal.
- (2)
- Suppose that $\Gamma $ is a spatial graph which splits into non-splittable spatial graphs ${\gamma}_{1}$, ${\gamma}_{2}$ and ${\gamma}_{3}$, as illustrated in Figure 7, where ${\gamma}_{1}$ is a spatial $\theta $-curve. According to the choice of two edges of ${\gamma}_{1}$, we obtain a trefoil knot ${K}_{1}$, a figure-eight knot ${K}_{2}$, or their connected sum ${K}_{1}\#{K}_{2}$. Then any map in $\mathrm{Diff}({S}^{3},\Gamma )$ does not permute these edges. The ${\mathbb{Z}}_{2}$-symmetries ${G}_{1}$ and ${G}_{2}$ of $\Gamma $ illustrated in Figure 7 are not equivalent, since there is no map in $\mathrm{Diff}({S}^{3},\Gamma )$ which takes $\mathrm{Sing}\left({G}_{1}\right)$ to $\mathrm{Sing}\left({G}_{2}\right)$ and interchanges ${\gamma}_{2}$ and ${\gamma}_{3}$. Moreover, we cannot perform rational twists along incompressible spheres and tori in $E\left(\Gamma \right)$ to make ${G}_{2}$ equivalent to ${G}_{1}$, since any setwise ${G}_{2}$-invariant incompressible sphere in $E\left(\Gamma \right)$ separates ${\gamma}_{2}$ and ${\gamma}_{3}$.

## Acknowledgements

The author would like to thank the referees for helpful comments which improved this paper.

## References

- Simon, J. Topological chirality of certain molecules. Topology
**1986**, 25, 229–235. [Google Scholar] [CrossRef] - Flapan, E. Symmetries of Möbius ladders. Math. Ann.
**1989**, 283, 271–283. [Google Scholar] [CrossRef] - Flapan, E. Rigidity of graph symmetries in the 3-sphere. J. Knot Theor. Ramif.
**1995**, 4, 373–388. [Google Scholar] [CrossRef] - Flapan, E.; Naimi, R.; Pommersheim, J.; Tamvakis, H. Topological symmetry groups of graphs embedded in the 3-sphere. Comment. Math. Helv.
**2005**, 80, 317–354. [Google Scholar] [CrossRef] - Flapan, E.; Naimi, R.; Tamvakis, H. Topological symmetry groups of complete graphs in the 3-sphere. J. London Math. Soc.
**2006**, 73, 237–251. [Google Scholar] [CrossRef] - Noda, C. The topological symmetry group of a canonically embedded complete graph in S
^{3}. Tokyo J. Math.**1997**, 20, 45–50. [Google Scholar] [CrossRef] - Flapan, E. Infinitely periodic knots. Cana. J. Math.
**1985**, 37, 17–28. [Google Scholar] [CrossRef] - Flapan, E. The finiteness theorem for symmetries of knots and 3-manifolds with nontrivial characteristic decompositions. Topol. Appl.
**1986**, 24, 123–131. [Google Scholar] [CrossRef] - Boileau, M.; Flapan, E. Uniqueness of free actions on S
^{3}respecting a knot. Can. J. Math.**1987**, 39, 969–982. [Google Scholar] [CrossRef] - Sakuma, M. Uniqueness of symmetries of knots. Math. Z.
**1986**, 192, 225–242. [Google Scholar] [CrossRef] - Ikeda, T. Finite group actions on homologically peripheral 3-manifolds. Math. Proc. Cambridge Philos. Soc.
**2011**, 151, 319–337. [Google Scholar] [CrossRef] - Jaco, W. Lectures on three manifold topology; CBMS Regional Conference Series in Mathematic 43; American Mathematical Society: Providence, RI, USA, 1980. [Google Scholar]
- Jaco, W.; Shalen, P. Seifert fibered spaces in 3-manifolds; Memoirs of the American Mathematical Society 220: Providence, RI, USA, 1979. [Google Scholar]
- Johannson, K. Homotopy equivalences of 3-manifolds with boundaries; Lecture Notes in Mathmatics 761; Springer: Berlin, Germany, 1979. [Google Scholar]
- Morgan, J.W.; Bass, H. (Eds.) The Smith conjecture; Pure and Applied Mathematics 112; Academic Press Inc.: Orlando, FL, USA, 1984. [Google Scholar]
- Meeks, W.H.; Scott, P. Finite group actions on 3-manifolds. Invent. Math.
**1986**, 86, 287–346. [Google Scholar] [CrossRef] - Scott, P. The geometries of 3-manifolds. Bull. Lond. Math. Soc.
**1984**, 15, 401–487. [Google Scholar] [CrossRef] - Dinkelbach, J.; Leeb, B. Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom. Topol.
**2009**, 13, 1129–1173. [Google Scholar] [CrossRef] - Benedetti, R.; Petronio, C. Lectures on hyperbolic geometry; Universitext; Springer-Verlag: Berlin, Germany, 1992. [Google Scholar]
- Newman, W.H.A. A theorem on periodic transformations of spaces. Quart. J. Math.
**1931**, 2, 1–8. [Google Scholar] [CrossRef] - Meeks, W.H.; Yau, S.T. The equivariant Dehn’s lemma and loop theorem. Comment. Math. Helvetici
**1981**, 56, 225–239. [Google Scholar] [CrossRef] - Edmonds, A.L. A topological proof of the equivariant Dehn lemma. Trans. Am. Math. Soc.
**1986**, 297, 605–615. [Google Scholar] [CrossRef] - Plotnick, S.P. Finite group actions and nonseparating 2-spheres. Proc. Am. Math. Soc.
**1984**, 90, 430–432. [Google Scholar]

**Figure 4.**Modification of ${p}_{i}\left({B}_{i,k}\right)$ which makes ${p}_{i}\left({B}_{i,k}\right)$ disjoint from ${p}_{i}\left({B}_{i,j}\right)$.

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