3·1. Convention on the fundamental groups

Throughout this paper, we choose a point p on $T = \partial E(K)$ as the base-point of $\pi_1(E(K))$ .

For each $E_i$ and $T_i$ we take a point $p_i \in T_i \subset E_i$ . Since each $T_i$ is incompressible in E(K), the Loop Theorem says that

\[\varphi \colon \pi_1(T, p) \to \pi_1(X, p),\; \varphi_i \colon \pi_1(T_i, p_i) \to \pi_1(X, p_i)textrm{and}\ \psi_i \colon \pi_1(T_i, p_i) \to \pi_1(E_i, p_i)\]

are injective, and thus

\[\Phi \colon \pi_1(X, p) \to \pi_1(E(K), p)\quad \textrm{and}\quad \Phi _i \colon \pi_1(E_i, p_i) \to \pi_1(E(K), p_i)\]

are also injective, where $\varphi, \varphi_i, \psi_i, \Phi, \Phi_i$ denote monomorphisms induced by the inclusions. For notational simplicity, we will denote $\pi_1(E(K), p)$ by $\pi_1(E(K)) = G(K)$ . We also use $\pi_1(X)$ to denote $\pi_1(X, p)$ .

Let us describe how to regard $\pi_1(T_i, p_i)$ as a subgroup of $\pi_1(X)$ . Choose and fix a path $u_i$ in X from the base point p to $p_i$ , which gives an isomorphism $u_{i, \ast} \colon \pi_1(X, p_i) \to \pi_1(X, p)$ sending $[c_i] \in \pi_1(T_i, p_i)$ to $u_{i, \ast}([c_i]) = [u_i \ast c_i \ast \overline{u_i}] \in \pi_1(T_i, p)$ , where $\ast$ denotes the concatenation and $\overline{u_i}$ denotes $u_i$ with opposite orientation. Now we put $\pi_1(T_i) = u_{i, \ast}(\varphi_i(\pi_1(T_i, p_i))) \subset u_{i, \ast}(\pi_1(X, p_i)) = \pi_1(X)$ . Also we identify $\pi_1(X)$ with the subgroup $\Phi(\pi_1(X)) \subset \pi_1(E(K))$ . This identification also specifies the subgroup $\pi_1(T_i) = \Phi(u_{i, \ast}(\varphi_i(\pi_1(T_i, p_i))))\subset \pi_1(E(K))$ . For $\pi_1(E_i, p_i)$ , using the pre-specified path $u_i$ in X, we set $\pi_1(E_i) = u_{i, \ast}(\Phi_i(\pi_1(E_i, p_i))) \subset \pi_1(E(K))$ . Note that for $T_i \subset E_i$ , $u_{i, \ast}(\Phi_i(\psi_i(\pi_1(T_i, p_i)))) \subset \pi_1(E(K), p)$ coincides with $\pi_1(T_i) \subset \pi_1(E(K))$ . The above convention gives precise specification of subgroups $\pi_1(X), \pi_1(E_i)$ and $\pi_1(T_i)$ in $\pi_1(E(K)) = G(K)$ . We call $\pi_1(X)$ and $\pi_1(E_i)$ the factor groups of G(K).

It should be noted here that for another path $w_i$ in X from p to $p_i$ , we have

\[w_i \ast \alpha_i \ast \bar{w_i} = w_i \ast \bar{u_i} \ast u_i* \alpha_i \ast \bar{u_i} \ast u_i \ast \bar{w_i}= (w_i \ast \bar{u_i} ) \ast ( u_i *\alpha_i \ast \bar{u_i}) \ast \overline{(w_i \ast \bar{u_i})},\]

where $w_i \ast \bar{u_i} \in \pi_1(X) \subset G(K)$ and $u_i \ast \alpha_i \ast \bar{u_i} \in \pi_1(E_i) \subset G(K)$ . See Figure 1.

Fig. 1.

The subgroups $\pi_1(E_i)$ .

3·2. Decomposition of elements of G(K)

In this subsection, we will observe the following.

Fig. 2.

Decomposition of elements as products of elements in factor groups.

Example 3·2. Suppose that $E(K) = X \cup E_1 \cup\cdots \cup E_n$ and $g \in G(K)$ is represented by a closed curve c as in Figure 2(right). Then g is expressed as

\begin{align*} (a_1 \ast \overline{\tau_1} \ast \overline{u_1}) & \ast (u_1 \ast \tau_1 \ast b_1 \ast \overline{\tau_2} \ast \overline{u_1}) \ast(u_1 \ast \tau_2 \ast a_2 \ast \overline{\tau_3}\ast \overline{u_2}) \\& \ast (u_2 \ast \tau_3 \ast b_2 \ast \overline{\tau_4} \ast \overline{u_2}) \ast (u_2 \ast \tau_4 \ast a_3), \end{align*}

where $a_1 \ast \overline{\tau_1} \ast \overline{u_1},\ u_1 \ast \tau_2 \ast a_2 \ast \overline{\tau_3} \ast \overline{u_2}, \ u_2 \ast \tau_4 \ast a_3$ represent elements in $\pi_1(X)$ , $u_1 \ast \tau_1 \ast b_1 \ast \overline{\tau_2} \ast \overline{u_1}$ represents an element in $\pi_1(E_1)$ , and $u_2 \ast \tau_3 \ast b_2 \ast \overline{\tau_4} \ast \overline{u_2}$ represents an element in $\pi_1(E_2)$ .

Finally if two successive elements $g_i$ and $g_{i+1}$ belong to the same factor group, then we combine them according to the following convention to get an expression satisfying conditions (A) and (B).

Convention. Suppose that $g_i$ belongs to $\pi_1(T_j)$ for some j. Then there are two possible choices: $g_i \in \pi_1(X)$ or $g_i \in \pi_1(E_j)$ . We regard $g_i \in \pi_1(X)$ or $g_i \in \pi_1(E_j)$ so that we successively combine as many $g_i$ as possible so that $g_i$ and $g_{i+1}$ do not belong to the same factor group. After this combining procedure, if we still have $g_i \in \pi_1(T_j)$ for some j, i.e. $g_{i-1} \in \pi_1(E_{j'}), g_{i+1} \in \pi_1(E_{j''})$ for some $j', j'' \ne j$ , then we regard $g_i \in \pi_1(X)$ so that $g_i \in \pi_1(E_j)$ means that $g_i \in \pi_1(E_j) - \pi_1(T_j)$ .

Then we obtain the desired expression of $g \in G(K)$ .

We call an expression of g satisfying conditions (A) and (B) a minimal expression of $g \in G(K)$ . In the following, any expression of g is always assumed to be minimal. Note that a minimal expression for the given element may not be unique.

4·1. Dehn fillings of a hyperbolic manifold

Before proving this proposition, for the convenience of readers, we briefly recall Thurston’s hyperbolic Dehn surgery theory [ Reference Thurston18, Reference Thurston19 ]. By the assumption we have a holonomy (faithful and discrete) representation

\[\rho \colon \pi_1(X) \to \mathrm{Isom}^+(\mathbb{H}^3) = \mathrm{PSL}(2, \mathbb{C}).\]

We regard $\rho = \rho_{\infty}$ and denote its image by $\Gamma_{\infty} \cong \pi_1(X)$ . Recall that $\rho(\mu)$ and $\rho(\lambda)$ are parabolic elements. By small deformation of the holonomy representation $\rho$ which keeps $\partial X - T$ as cusps (up to conjugation), we obtain representations $\rho_{x, y} \colon \pi_1(X) \to \mathrm{Isom}^+(\mathbb{H}^3) = \mathrm{PSL}(2, \mathbb{C})$ whose image is $\Gamma_{x, y}$ .

Deforming $\rho$ continuously, (x, y) varies over an open set $U_{\infty}$ in $S^2 = \mathbb{R}^2 \cup \{ \infty \}$ ; see the proof of [ Reference Thurston18 , theorem 5·8·2]. Note that under deformation, the trace of $\rho_{x, y}$ is also deformed continuously. By Mostow–Prasad rigidity theorem [ Reference Mostow14, Reference Prasad16 ] $\rho_{x, y}(\mu)$ and $\rho_{x, y}(\lambda)$ are not parabolic, i.e. loxodromic when $(x, y) \ne \infty$ . Note that for any element $\gamma \in \pi_1(T_j)$ , $\rho(\gamma)$ is parabolic and $\rho_{x, y}(\gamma)$ remains parabolic for all $1 \le j \le n$ . When (x, y) is (p, q) for coprime integers p and q with $|p| + |q|$ sufficiently large, $\rho_{p, q}$ gives an incomplete hyperbolic metric of $\mathrm{int}X$ so that its completion is $X(p/q)$ in which $K_{p/q}^*$ , the image of the filled solid torus, is the shortest closed geodesic in the hyperbolic 3–manifold $X(p/q)$ . For the non-faithful representation $\rho_{p, q}$ of $\pi_1(X)$ , its image $\Gamma_{p, q}$ is regarded as $\pi_1(X(p/q))$ .

Note that for an outer-peripheral element $\gamma$ , $\rho(\gamma)$ is parabolic, while $\rho_{p, q}(\gamma)$ is loxodromic for $p/q$ with $|p| + |q|$ sufficiently large. If $\delta$ is inner-peripheral, then $\rho(\delta)$ and $\rho_{p, q}(\delta)$ are both parabolic. Let g be a non-peripheral element of $\pi_1(X)$ , then $\rho(g)$ and $\rho_{p, q}(g)$ are loxodromic.

In the following Subsections 4·2, 4·3 and 4·4, for our purpose we restrict our attention to $1/n$ –Dehn fillings.

4·2. Dehn fillings of a composing space

We say that X is a k–fold composing space if it is homeomorphic to $[\textrm{disk with}\ \textit{k}\ \textrm{holes}] \times S^1$ , which arise in the torus decomposition of the exterior of a composite knot $K = K_1\; \#\; K_2\; \# \cdots \#\; K_k$ , where each $K_i$ is prime. For our purpose it is convenient to regard $K = K_1\; \#\; K'_2$ , where $K'_2 = K_2\; \# \cdots \#\; K_k$ , and correspondingly decompose $E(K) = X \cup E(K_1) \cup E(K'_2)$ , where X is the 2–fold composing space. Note that this does not give the torus decomposition. For non-prime knots K, in the following we consider such a decomposition of E(K).

Assume first that $g \in \pi_1(T) = \pi_1(\partial E(K))$ . Then g is written as $\mu^a \lambda^b = t^a c^b$ for some integers a, b which may not be coprime. If $b = 0$ , then $g = t^a$ belongs to $\pi_1(T_i)$ $(i = 1, 2)$ , contradicting the assumption. So in the following we assume that $b \ne 0$ .

Since $t = \mu$ and $c = \lambda$ , in $X(1/n)$ , a meridian of the filled solid torus represents $t c^n$ , and $\pi_1(X(1/n))$ has a presentation:

\[\langle c, d, t \mid [c, t]= [d, t] = 1,\ tc^{n} = 1 \rangle.\]

Taking the quotient by the normal subgroup $\langle t \rangle$ , we have a projection

\[\pi \colon \pi_1(X(1/n)) \to \pi_1(X(1/n))/\langle t \rangle,\]

where $\pi_1(X(1/n))/\langle t \rangle$ has the presentation:

\[\langle c, d \mid c^{n} = 1 \rangle \cong \langle c \mid c^{n} = 1 \rangle \ast \langle d \rangle \cong \mathbb{Z}_{n} \ast \mathbb{Z}.\]

Now let us take n so that $n \gt |b|$ . Then $\pi(p_{1/n}(g)) = c^b$ is non-trivial and cannot be any power of d or cd in $\pi_1(X(1/n))/\langle t \rangle$ . This means that $p_{1/n}(g) \not\in \pi_1(T_i)$ for $i = 1, 2$ .

Let us suppose that $g \not\in \pi_1(T)$ . Thus $g \in \pi_1(X) - \left(\pi_1(T) \cup \pi_1(T_1) \cup \pi_1(T_2)\right)$ by the assumption. We show that there exists a constant $N _g \gt 1$ such that $p_{X, 1/n}(g)$ is non-trivial and $p_{X, 1/n}(g) \in \pi_1(X(1/n)) - \left(\pi_1(T_1)\cup \pi_1(T_2)\right)$ for all $n \ge N_g$ .

Since t is central, $g \in \pi_1(X)$ is written as $t^k w(c, d)$ for some integer k and for some word w(c, d) of c and d. Furthermore, since $g \in \pi_1(X) - \left(\pi_1(T) \cup \pi_1(T_1)\cup \pi_1(T_2)\right)$ , the word w(c, d) is non-trivial, and both c and d appear in w(c, d), and g is not a power of c, d or cd. (Note that t belongs to $\pi_1(T), \pi_1(T_1)$ and $\pi_1(T_2)$ .) Let us continue to write $p_{X,1/n}(g) = t^k w(c, d) \in \pi_1(X(1/n))$ . To see that it remains non-trivial and non-peripheral, we look at $\pi(p_{X,1/n}(g)) = \pi(w(c, d))$ in $\pi_1(X(1/n))/\langle t \rangle$ .

Recall that in the given reduced word w(c, d), both c and d appear and w(c, d) is none of a power of c, d or cd. Write $w(c, d) = c^{p_1} d^{q_1} \cdots c^{p_k} d^{q_k} c^{p_{k+1}}$ for some integers $p_1, q_1 \dots p_{k+1}$ , where $p_1$ , $p_{k+1}$ may be 0, while others are not 0.

Set $p_c = \mathrm{max}\{ |p_1|, \dots , | p_k |, | p_{k+1}| \}$ and take n so that $n \gt p_c + 1$ . Then $c^{p_i} \ne 1 \in \mathbb{Z}_n$ whenever $p_i \ne 0$ . Hence, $\pi(w(c, d))$ is non-trivial and not a power of d in $\pi_1(X(1/n))/\langle t \rangle$ , which implies that $p_{X, 1/n}(g) \ne 1 \in \pi_1(X(1/n))$ and $p_{X, 1/n}(g) \not\in \pi_1(T_1)$ . Let us see if $\pi(w(c, d))$ is a power of cd in $\pi_1(X(1/n))/\langle t \rangle$ . If $\pi(w(c, d))$ is a power of cd, then it is either $c^{-p}d c^{-p}d \cdots c^{-p}d$ or $d^{-1}c^p d^{-1}c^p \cdots d^{-1}c^p$ and $n = p+1$ . In the former case $\pi(w(c, d)) = cd \cdots cd \in \mathbb{Z}_{p+1} \ast \mathbb{Z}$ and in the latter case $\pi(w(c, d)) = d^{-1}c^{-1}\cdots d^{-1}c^{-1} \in \mathbb{Z}_{p+1} \ast \mathbb{Z}$ , which are powers of cd. However, since $n \gt p_c + 1$ , such situations cannot happen. (This situation was pointed by the referee.)

Hence, $p_{X, 1/n}(g)$ is non-trivial and not contained in $\pi_1(T_j)$ if $n \ge N_g =$ $\mathrm{max}\{|b| + 1,\ p_c + 2\}$ .

4·3. Dehn fillings of a cable space

We say that K is a (p, q)–cable of a non-trivial knot k if K is embedded in the boundary of a small tubular neighbourhood of k such that it wraps p time in meridional direction and q times in longitudinal direction; for non-triviality we assume $q \ge 2$ .

Proof. Since $n \ge 2$ , $|pqn-1| \ge 3 \gt 2$ . Then, after $1/n$ –Dehn filling, $X(1/n)$ is a Seifert fiber space over the disk with two exceptional fibers of indices q and $|pqn-1| \gt 2$ .

1. (i) The above observation shows that $X(1/n)$ is boundary-irreducible.

2. (ii) Assume that $g \in \pi_1(T_1)$ . Following (i), g remains non-trivial in $\pi_1(X(1/n))$ .

3. (iii) Using a Seifert structure of X, we have the following presentation of $\pi_1(X)$ .

   \[\langle c_1, c_2, t \mid [c_1, t]= [c_2, t] = 1,\ c_2^qt^\alpha = 1 \rangle,\]

where $t = \mu^{pq} \lambda \in \pi_1(T) = \pi_1(\partial E(K))$ is represented by a regular fiber, which belongs to $\pi_1(T_1)$ as well, $c_1 \in \pi_1(T)$ and t generate $\pi_1(T)$ , and $c_1c_2 \in \pi_1(T_1)$ and t generate $\pi_1(T_1)$ . Here we choose a specific conjugacy representatives, i.e. a specific path $u_i$ introduced in Section 3 to obtain the above presentation.

Suppose that $g \in \pi_1(T)$ . Then g is expressed as $c_1^a t^b$ for some integers a, b, which may not be coprime. If $a = 0$ , then $g = t^b \in \pi_1(T_1)$ , contradicting the assumption. So in the following, we assume that $a \ne 0$ .

To get a presentation of $\pi_1(X(1/n))$ , write $c_1 = \mu^x\lambda^y$ for some coprime integers x, y such that $|pqy - x| =1$ . Then, $1/n$ –Dehn filling gives a relation $\mu \lambda^n$ , which can be re-written by $c_1^{pqn-1} t^\beta = 1$ for some integer $\beta$ . Thus we obtain the following presentation of $\pi_1(X(1/n))$ :

\[\langle c_1, c_2, t \mid [c_1, t]= [c_2, t] = 1,\ c_2^qt^\alpha = 1,\ c_1^{pqn-1} t^\beta = 1 \rangle.\]

Taking the quotient by the normal subgroup $\langle t \rangle$ , we have a projection

\[\pi \colon \pi_1(X(1/n)) \to \pi_1(X(1/n))/\langle t \rangle,\]

where $\pi_1(X(1/n))/\langle t \rangle$ has the presentation:

\[\langle c_1, c_2 \mid c_2^q = 1, c_1^{|pqn-1|} = 1 \rangle = \mathbb{Z}_q \ast \mathbb{Z}_{|pqn-1|}.\]

Now let us take n so that $|pqn -1 | \gt |a|$ .

If $p_{X, 1/n}(g) \in \pi_1(T_1)$ , then $\pi(p_{X, 1/n}(g)) = \pi(c_1^a t^b) = c_1^a$ is non-trivial and cannot be a power of $c_1c_2$ in $\pi_1(X(1/n))/\langle t \rangle$ . This means that $p_{X, 1/n}(g) \not\in \pi_1(T_1)$ .

Assume that $g \not\in \pi_1(T)$ . Then $g \in \pi_1(X) - \left(\pi_1(T) \cup \pi_1(T_1)\right)$ .

For a given $g \in \pi_1(X)$ , since t is central, we may write $g = t^k w(c_1, c_2)$ for some integer k and for some word $w(c_1, c_2)$ of $c_1$ and $c_2$ . Further, we write $w(c_1, c_2) = c_1^{p_1} c_2^{q_1} \cdots c_1^{p_{\ell}} c_2^{q_{\ell}} c_1^{p_{\ell+1}} \in \pi_1(X)$ for some integers $p_1, q_1 \dots p_{\ell}, q_{\ell}, p_{\ell+1}$ , where $p_1$ , $p_{\ell+1}$ may be 0, while others are not 0. Set $N_{g, c_1} = \mathrm{max}\{ |p_1|, \dots , |p_{\ell}|, |p_{\ell+1}| \}$ , and $N_{g, c_2} = \mathrm{max}\{ |q_1|, \dots , |q_{\ell}| \}$ . Applying the relations $[c_1, t]= [c_2, t] = 1$ and $c_2^q = t^{-\alpha}$ , we may assume $N_{g, c_2} \lt q$ .

In the expression $g = t^k w(c_1, c_2)= t^k c_1^{p_1} c_2^{q_1} \cdots c_1^{p_{\ell}} c_2^{q_{\ell}} c_1^{ p_{\ell+1}} \in \pi_1(X)$ , since $g \in \pi_1(X) - \left(\pi_1(T) \cup \pi_1(T_1)\right)$ , the word $w(c_1, c_2)$ is non-trivial, and not a power of $c_1$ or $c_1c_2$ . Note that $c_2$ appears in $w(c_1, c_2)$ , and as noted above, we may also assume $|q_i| \lt q$ .

Let us show that $p_{X,1/n}(g)$ is non-trivial and not contained in $\pi_1(T_1)$ provided $|pqn - 1| \gt N_{g, c_1} + 1$ . We continue to write $p_{X,1/n}(g) = t^k w(c_1, c_2) \in \pi_1(X(1/n))$ and consider $\pi(p_{X,1/n}(g)) = \pi(w(c_1, c_2)) \in \pi_1(X(1/n))/\langle t \rangle$ .

Since $|p_i| \lt N_{g, c_1} + 1 \lt |pqn - 1|$ , both $c_1$ and $c_2$ appear in $\pi(w(c_1, c_2)) \in \mathbb{Z}_q \ast \mathbb{Z}_{|pqn-1|}$ , thus $p_{X,1/n}(g)$ is non-trivial and not contained in $\pi_1(T)$ .

It remains to see that $p_{X,1/n}(g) \not\in \pi_1(T_1)$ . Suppose for a contradiction that $p_{X,1/n}(g) \in \pi_1(T_1)$ . Then $\pi(w(c_1, c_2))$ is a power of $c_1c_2$ in $\pi_1(X(1/n)) / \langle t \rangle$ . Recall that $w(c_1, c_2)$ is not a power of $c_1c_2$ in $\pi_1(X)$ . So this may occur only if $w(c_1, c_2) = c_1^{-N}c_2 c_1^{-N}c_2 \cdots c_1^{-N}c_2$ or $c_2^{-1}c_1^N c_2^{-1}c_1^N \cdots c_2^{-1}c_1^N$ and $|pqn-1| = N + 1$ . In the former case $\pi(w(c_1, c_2)) = c_1c_2 \cdots c_1c_2 \in \mathbb{Z}_q \ast \mathbb{Z}_{|pqn-1|}$ , and in the latter case $\pi(w(c_1, c_2)) = c_2^{-1}c_1^{-1} \cdots c_2^{-1}c_1^{-1} \in \mathbb{Z}_q \ast \mathbb{Z}_{|pqn-1|}$ , which are powers of $c_1c_2$ . However, since n satisfies $|pqn-1| \gt N_{g, c_1} + 1$ , this is impossible. Let us take a constant $N_g \gt0$ so that if $n \ge N_g$ , then $|pqn-1| \gt |a|$ and $|pqn-1| \gt N_{g, c_1}$ . Then $N_g$ is a desired constant.

4·4. Dehn filling of a torus knot space

Proof. Let K be the torus knot $T_{p, q}$ , where we assume $2 \le p \lt q$ . The case where q is negative is treated similarly. The exterior $E(T_{p,q})$ is a Seifert fiber space over $D^{2}$ with two exceptional fibers of indices $2\le p \lt q$ , and its $1/n$ -filling $T_{p, q}(1/n)$ is a Seifert fiber space over $S^2$ with three exceptional fibers of indices $2 \le p \lt q \lt pqn -1$ .

Using Seifert structures, we have the following presentation of $\pi_1(E(T_{p, q}))$ :

\[\langle c_1, c_2, t \mid [c_1, t]= [c_2, t] = 1, c_1^p t^\alpha = c_2^q t^{\beta} = 1 \rangle,\]

for some integers $\alpha$ and $\beta$ , where $t = \mu^{pq} \lambda \in \pi_1(\partial E(K))$ is represented by a regular fiber and $c_1c_2 \in \pi_1(\partial E(K))$ . Similarly, we obtain a presentation of $\pi_1(K(r))$ by adding an extra relation corresponding to $1/n$ –Dehn filling.

\[\langle c_1, c_2, t \mid [c_1, t]= [c_2, t] = 1, c_1^p t^\alpha = c_2^q t^{\beta} = (c_1c_2)^{pqn-1} t^{\gamma} = 1 \rangle.\]

for some integer $\gamma$ . The last relation arises from $1/n$ –Dehn filling.

If $n\gt1$ , then $pqn-1\geq 11\gt6$ , so ${1}/{p}+{1}/{q}+{1}/({pqn-1})\lt1$ . Then $\pi_1(T_{p, q}(1/n))$ is infinite. In particular, an element $p_{1/n}(t) \in \pi_1(T_{p, q}(1/n))$ , denoted by the same symbol t, represented by the image of a regular fiber has infinite order.

In the following, we always assume that $n\gt1$ so that we have the following commutative diagram.

where $f \colon G(T_{p, q}) \to G(T_{p, q})/ \langle t \rangle$ and $f_n \colon \pi_1(T_{p, q}(1/n)) \to \pi_1(T_{p, q}(1/n) / \langle t \rangle$ denote the natural projections.

It follows that for $g\in G(T_{p,q})$ , if $f(g)=1$ , then $g=t^{m}$ for some $m\neq 0$ . Consequently, $p_{1/n}(g) = p_{1/n}(t)^m \neq 1$ for all $n\gt1$ .

Thus we may assume that $f(g) \neq 1$ . By the commutative diagram above, to show $p_{1/n}(g) \neq 1$ , it is sufficient to show that $P_{1/n}(f(g)) \neq 1$ , which is done in the following lemma.

Therefore by lemma, by taking $N_g=\max\{2, ({\ell(f(g))+2})/{pq} \}$ , $p_{1/n}(g) \neq 1$ whenever $n \geq N_g$ .

5·1. Before Dehn fillings

Let us put $G_{v} = \pi_1(X) \subset G(K)$ and $G_{v_j} = \pi_1(E_j) \subset G(K)$ , each of which is called a vertex group. Similarly put $G_{e_i} = \pi_1(T_i) \subset G(K)$ , which we call an edge group.

Let $Y_{E(K)}$ be a graph consisting of $n+1$ vertices $v, v_1, \dots, v_n$ , together with oriented edges $e_1, \dots, e_n$ ; $e_i$ connects v and $v_i$ directed from v to $v_i$ . By definition, $Y_{E(K)}$ is just a tree as depicted by Figure 3.

Fig. 3.

Graph of groups.

Denote by $o(e_i)$ the vertex v which is called the origin of the oriented edge $e_i$ , and denote by $t(e_i)$ the vertex $v_i$ which is called the terminus of $e_i$ . Denoting $\overline{e_i}$ the edge $e_i$ with opposite orientation, $o(\overline{e_i}) = t(e_i)$ and $t(\overline{e_i}) = o(e_i)$ .

To the distinguished vertex v, assign a vertex group $G_{v} = \pi_1(X)$ , and to each $v_i$ assign a vertex group $G_{v_i} = \pi_1(E_i)$ for $1 \le i \le n$ . The middle of $e_i$ corresponds to an edge group $G_{e_i}= \pi_1(T_i)$ . Note that $G_{e_i} \subset G_{v_i}$ and $G_{\overline{e_i}} \subset G_{v}$ . Then the tree $Y_{E(K)}$ , together with a family of groups $G = \{ G_v,\ G_{v_i},\ G_{e_i}\}$ , gives a graph of groups $(G, Y_{E(K)})$ .

Now let us introduce a graphical word of the graph of groups $(G, Y_{E(K)})$ . Let c be an oriented closed path in $Y_{E(K)}$ based at the vertex v, which is expressed by a sequence of the oriented edges $(y_1, \dots, y_{\ell})$ ; each $y_i$ is one of $e_1, \overline{e_1}, \dots, e_n, \overline{e_n}$ such that $o(y_{i+1}) = t(y_i)$ for each i. Then a graphical word is a sequence

\[x_0y_1 x_1y_2 \cdots y_i x_i y_{i+1} \cdots y_{\ell} x_{\ell},\]

$x_0, x_1, \dots, x_{\ell}$ of $G_v$ or $G_{v_i} (1 \le i \le n)$ such that $x_0, x_{\ell} \in G_v$ , and $x_i \in G_{t(y_i)} = G_{o(y_{i+1})}$ . Note that $x_i$ may be the trivial element 1.

Since $Y_{E(K)}$ is a tree, for the graph (tree) of groups $(G, Y_{E(K)})$ , putting $y_j =1$ in the above graphical word, we obtain an element $x_0\cdots x_{\ell}$ in the fundamental group of $(G, Y_{E(K)})$ , which is $\pi_1(E(K))$ , the amalgams of the vertex groups along the edge groups; see [ Reference Serre17 , pp.42–43].

For a non-trivial element $g \in G(K)$ with minimal expression $g = g_1\cdots g_m$ , we may assign a graphical word in the following manner.

We start from the word $1 g_1 g_2 \cdots g_m$ if $g_1 \not \in \pi_1(X)=G_{v}$ and $g_1 g_2\cdots g_m$ if $g_1 \in \pi_1(X)=G_v$ . Here the first symbol 1 represents the trivial element of $G_v = \pi_1(X)$ . For each $i=1,\ldots,m$ , we replace each $g_i$ with sequence according to the following rule.

If $g_i \in \pi_1(X)=G_{v}$ then we remain $g_i$ unchanged. We emphasise that $g_i \in \pi_1(E_j)-\pi_1(X) = \pi_1(E_j) - \pi_1(T_j)$ in the cases where modifications are needed. Assume that $g_i \in \pi_1(E_{j(i)})= G_{v_{j(i)}}$ for some terminus vertex $v_{j(i)}$ . If $g_{i+1} \in \pi_1(X)=G_v$ , then we replace $g_i$ with $e_{j(i)}g_i \overline{e_{j(i)}}$ . If $g_{i+1} \in \pi_1(E_{j(i+1)})$ , or, $i=m$ , then then we replace $g_i$ with $e_{j(i)}g_i \overline{e_{j(i)}}1$ , where the last 1 denotes the trivial element of $G_v$ .

The result is the alternating sequence

\[x_0y_1 x_1y_2 \cdots y_i x_i y_{i+1} \cdots y_{m} x_{m+1}\]

consisting of elements $x_0, x_1, \dots, x_{m+1}$ of $G_v$ or $G_{v_i} (1 \le i \le n)$ and the sequence of oriented edges $(y_1, \dots, y_m)$ which forms a closed oriented path c based at v.

We call $x_0y_1 x_1y_2 \cdots y_i x_i y_{i+1} \cdots y_{m} x_{m+1}$ the graphical word associated to g and denote it by $|c, g|$ .

The example below illustrates how we obtain a graphical word from an element with minimal expression.

Example 5·1. Assume that $G(K) = \pi_1(E(K))$ consists of four vertices $v, v_1, v_2, v_3$ and three edges; see Figure 4. Let $g = g_1g_2g_3g_4$ , where $g_1 \in G_{v_2}= \pi_1(E_2), g_2 \in G_{v_1}=\pi_1(E_1), g_3 \in G_v = \pi_1(X)$ and $g_4 \in G_{v_2}= \pi_1(E_2)$ . Then the graphical word of g is given by

\[x_0 y_1 x_1 y_2 x_2 y_3 x_3 y_4 x_4 y_5 x_5 y_6 x_6 = 1 e_2 g_1 \overline{e_2} 1 e_1 g_2 \overline{e_1} g_3 e_2 g_4 \overline{e_2} 1.\]

Fig. 4.

A graph of groups $(G, Y_{E(K)})$ and an oriented closed path in $Y_{E(K)}$ .

Fig. 5.

backtrack condition: $x_i \not\in G_{y_i}^{t(y_i)} \subset G_{t(y_i)}$ for a backtrack $y_i x_i y_{i+1} = y_i x_i \overline{y_i}$ .

5·2. After Dehn fillings

In what follows we assume X(r) is boundary-irreducible. Then for each component $T_j \subset \partial X(r)$ , we may regard that $\pi_1(T_j)$ is a subgroup of $\pi_1(X(r)) \subset \pi_1(K(r))$ .

For simplicity we write $G_{\widehat{v}} = \pi_1(X(r))$ and $G_{v_j} = \pi_1(E_j)$ and $G_{e_i} = \pi_1(T_i)$ as in 5.1. Let $\widehat{G}$ be a family of groups $\{ G_{\widehat{v}},\ G_{v_j},\ G_{e_i} \}$ . Then we obtain a graph of groups $(\widehat{G}, Y_{K(r)})$ consisting of $n+1$ vertices $\widehat{v}, v_1, \dots, v_n$ , together with oriented edges $y_1, \dots, y_n$ ; $y_i$ connects v and $v_i$ directed from v to $v_i$ ; $\overline{y_i}$ denotes $y_i$ with the opposite direction. Note that the graph $Y_{K(r)}$ is exactly the same as $Y_{E(K)}$ except for the vertex group $G_{\widehat{v}} = \pi_1(X(r))$ .

Recall that r–Dehn filling of E(K) induces a natural epimorphism $p_r \colon G(K) \to \pi_1(K(r))$ . Let g be a non-trivial element of G(K) with a minimal expression $g_1 \cdots g_m$ with the associated graphical word

\[x_0 y_1 x_1 \cdots y_{i-1} x_{i-1} y_i x_{i} y_{i+1} \cdots x_{\ell}.\]

Then $p_r(g)$ is $p_r(g_1) \cdots p_r(g_n) \in \pi_1(K(r))$ with the associated graphical word

\[p_r(x_0) y_1 p_r(x_1) \cdots y_{i-1} p_r(x_{i-1}) y_i p_r(x_{i}) y_{i+1} \cdots p_r(x_{\ell}).\]

Note that $p_r(g_i)$ remains non-trivial if $g_i \in G_{v_j} = \pi_1(E_j)$ , while if $g_i \in G_v = \pi_1(X)$ , then $p_r(g_i) = p_{X, r}(g_i) \in G_{\hat{v}} = \pi_1(X(r))$ may be trivial, and hence $p_r(g)$ may be also trivial in $\pi_1(K(r))$ for some $r \in \mathbb{Q}$ .

We recall the following definition from [ Reference Serre17 ]. For our purpose we restrict our attention to the graph Y, which will be either $Y_{E(K)}$ or $Y_{K(r)}$ described in 5.1 and 5.2; $Y = Y_{E(K)}$ has vertex groups $G_v = \pi_1(X)$ , $G_{v_j} = \pi_1(E_j)$ and edge groups $G_{y_i} = \pi_1(T_j)$ , and $Y = Y_{K(r)}$ has vertex groups $G_{\widehat{v}} = \pi_1(X(r))$ , $G_{v_j} = \pi_1(E_j)$ and edge groups $G_{y_i} = \pi_1(T_j)$ . In the following definition, each $x_k$ denotes $g_{k'} \in G(K)$ or $p_r(g_{k'}) \in \pi_1(K(r))$ , and $y_k$ is an oriented edge in Y.

Definition 5·2 (reduced graphical word). Let us consider a graphical word

\[x_0 y_1 x_1 \cdots y_{i-1} x_{i-1} y_i x_{i} y_{i+1} \cdots x_{\ell}.\]

Let $G_{y_i}^{t(y_i)}$ denote the image of the edge group $G_{y_i}$ in the vertex group $G_{t(y_i)}$ . Then we say that $x_0 y_1 x_1 \cdots y_{i-1} x_{i-1} y_i x_{i} y_{i+1} \cdots x_{\ell}$ is reduced if it satisfies the following:

1. (1) If ${\ell} = 0$ , then $x_0 \ne 1$ ;

2. (2) If ${\ell} \ge 1$ , then $x_i \not\in G_{y_i}^{t(y_i)}$ for each i such that $y_{i+1} = \overline{y_i}$ .

If $y_{i+1} = \overline{y_i}$ and $x_i \in G_{y_i}^{t(y_i)}$ , then we say that the graphical word $|c, g|$ has an incomplete backtrack at $x_i$ .

If $g = g_1 \cdots g_m \in G(K)$ is a minimal expression, then the corresponding graphical word $|c, g|$ is reduced.

In the proofs of our main results, we will use the following theorem due to Serre [ Reference Serre17 , I·5. corollary 3].

Theorem 5·3 ([ Reference Serre17 ]). Suppose that a graphical word $x_0 y_1 x_1 \cdots y_{i-1} x_{i-1} y_i x_{i} y_{i+1} \cdots x_{\ell}$ is reduced, then $x_0 x_1 \cdots x_{i-1} x_{i} \cdots x_{\ell}$ is non-trivial.

7·1. Proof of Theorem 1·3

For a given non-trivial knot K and $g\in G(K)$ , define

We will prove that if K is a prime, non-cabled knot, then $\mathcal{S}_K(g)$ is finite.

Suppose that g has a minimal expression $g_1 \cdots g_m$ , for which $g_{i_1}, \dots, g_{i_k}$ belong to $\pi_1(X)$ . For each $g_{i_s} \in \pi_1(X)$ $(1 \le s \le k)$ , following Proposition 4·2, we have a finite subset $\mathcal{S}_X(g_{i_s}) \subset \mathbb{Q}$ such that if $r \not\in \mathcal{S}_X(g_{i_s})$ , then r satisfies conditions (i) and (ii) (for $g_{i_s} \in \pi_1(X)$ ) in Definition 6·1. Let us write $\mathcal{B}(g) = \cup_{s=1}^k\mathcal{S}_X(g_{i_s})$ , which is also finite. Now we show that

\[\mathcal{S}_K(g) \subset \mathcal{B}(g).\]

Take any $r \in \mathbb{Q}$ not belonging to $\mathcal{B}(g)$ . Then, by definition, it enjoys Condition $(g|X)$ . Hence, by Lemma 6·2 we see that $p_r(g)$ is non-trivial in $\pi_1(K(r))$ . This means that $r \not\in \mathcal{S}_K(g)$ , and so $\mathcal{S}_K(g) \subset \mathcal{B}(g)$ . This completes the implication $(ii) \Rightarrow (i)$ .

Let us prove the implication $(i) \Rightarrow (ii)$ . To do this we find a non-trivial element in G(K) which can be trivialised by infinitely many Dehn fillings in the case where K is a torus knot, a cable of a non-trivial knot or a non-prime knot.

1. (1) K is a(p, q)–torus knot.

For (p, q)–torus knot $T_{p, q}$ , the slope $m/n$ with $|pqn - m| = 1$ are cyclic surgery slopes, i.e. $\pi_1(T_{p, q}(m/n))$ is cyclic. Then for such slopes $m/n$ , the epimorphism $G(K) \to \pi_1(K(m/n))$ factors through $G(K) \to G(K)/[G(K), G(K)]$ . Hence, every element in the commutator subgroup [G(K), G(K)] becomes trivial in $\pi_1(K(m/n))$ . This means that for each $g \in [G(K), G(K)]$ , $\mathcal{S}_K(g)$ is infinite.

1. (2) K is a(p, q)–cable of a non-trivial knot $(q \ge 2)$ .

Let X be the outermost decomposing piece of E(K), which is a cable space, a Seifert fiber space over the annulus with an exceptional fiber of index q. Then for any slope $m/n$ with $|pqn-m| = 1$ , $X(m/n)$ is a solid torus. Thus an epimorphism $\pi_1(X) \to \pi_1(X(m/n))$ factors through $\pi_1(X) \to \pi_1(X)/[\pi_1(X), \pi_1(X)]$ . This shows that any element $g \in [\pi_1(X), \pi_1(X)]$ becomes trivial in $\pi_1(X(m/n))$ , in particular, it becomes trivial in $\pi_1(K(m/n))$ . Hence, for each $g \in [\pi_1(X), \pi_1(X)] \subset [G(K), G(K)]$ , $\mathcal{S}_K(g)$ is infinite.

1. (3) K is a non-prime knot.

As in Subsection 4·2 we decompose E(K) as $E(K) = X \cup E_1 \cup E_2$ $(i = 1, 2)$ , where X is a 2–fold composing space, i.e. $X = [\textrm{disk with two holes}] \times S^1$ and $E_i$ is a non-trivial knot exterior. Note that the meridian of K is a regular fiber in X. For any integral slope m, $X(m) = S^1 \times S^1 \times [0, 1]$ . Since $\pi_1(X(m)) = \mathbb{Z} \oplus \mathbb{Z}$ , an epimorphism $\pi_1(X) \to \pi_1(X(m))$ factors through $\pi_1(X) \to \pi_1(X)/[\pi_1(X), \pi_1(X)]$ , and any non-trivial element $g \in [\pi_1(X), \pi_1(X)] \subset \pi_1(X) \subset G(K)$ becomes trivial in $\pi_1(X(m))$ , and hence in $\pi_1(K(m))$ . Hence, $\mathcal{S}_K(g)$ contains infinitely many slopes for $g \in [\pi_1(X), \pi_1(X)] \subset [G(K), G(K)]$ .

This completes a proof of Theorem 1·3.

The implication $(ii) \Rightarrow(i)$ in Theorem 1·3 may be obtained from [ Reference Groves and Manning5, Reference Osin15 ], together with [ Reference Aschenbrenner, Friedl and Wilton1 , theorem 7·2·2] and [ Reference Dahmani4 ], using the hyperbolicity of 3-manifold group relative to the set of parabolic subgroups which come from non-hyperbolic JSJ pieces.

7·2. Proof of Theorem 1.4

Suppose that K is a prime, non-cabled knot. Then the result follows from Theorem 1·3 ( $(ii) \Rightarrow (i)$ ).

Let us assume that K is a non-trivial knot other than above knots, i.e. K is a non-prime knot or a cabled knot. In the latter case, we separate into sub-cases depending upon its companion knot is trivial or not.

Assume first that K is a non-prime knot or a cable of some non-trivial knot. Then for each $g_{i_s} \in \pi_1(X)$ , following Lemmas 4·4, 4·5, we have a constant $N_{g_{i_s}}$ such that if $n \ge N_{g_{i_s}}$ , then the slope $1/n$ satisfies conditions (i) and (ii) (for $g_{i_s} \in \pi_1(X)$ ) in Definition 6·1. Put $N_g = \mathrm{max}\{ N_{g_{i_1}}, \dots, N_{g_{i_k}}\}$ . Then for any $n \ge N_g$ , the slope $1/n$ enjoys Condition $(g|X)$ . Hence, by Lemma 6·2 we see that $p_r(g)$ is non-trivial in $\pi_1(K(r))$ .

Finally assume that K is a torus knot. Then Lemma 4·6 gives us the desired result.

Putting $N_F = \mathrm{max}\{ N_g \mid g \in F \}$ , we finish the proof of Theorem 1·4.

7·3. Remark on hyperbolicity

Following the proof of Theorem 1·4, the knot group G(K) is fully residually perfect hyperbolic, toroidal, or Seifert 3–manifold group according to K is a hyperbolic knot, a satellite knot, or a torus knot, respectively. If K is a cable of a hyperbolic knot, then we have an epimorphism from G(K) to a closed hyperbolic 3–manifold group. However, G(K) cannot be residually closed hyperbolic 3–manifold group. Indeed, we have:

Now suppose that K is a satellite knot with non-hyperbolic JSJ-decomposing piece Y. Assume for a contradiction that G(K) is residually closed hyperbolic 3–manifold group. Then following Proposition 2·3, we may assume G(K) is fully residually closed hyperbolic 3–manifold group. Since Y is non-hyperbolic, it is Seifert fibered. Let t be a regular fiber in Y; as above we use the same symbol t to denote the element of $\pi_1(Y)$ represented by the oriented regular fiber. Write $H = \pi_1(Y)$ and take a non-trivial element $g \in [H, H]$ . Note that $t \not\in [H, H]$ . Set $F = \{ t, g \} \in \pi_1(Y) \subset G(K)$ . By our assumption we have a closed hyperbolic 3–manifold M and an epimorphism $\varphi \colon G(K) \to \pi_1(M)$ such that $\varphi(t) \ne 1$ nor $\varphi(g) \ne 1$ . Let $\rho$ be a holonomy representation $\pi_1(M) \to \mathrm{PSL}_2(\mathbb{C})$ . Consider $\rho \circ \varphi \colon G(K) \to \mathrm{PSL}_2(\mathbb{C})$ .

Now we show that $\varphi(H)$ is abelian. Take any element $h \in H$ . Assume that $\varphi(h) \ne 1$ , hence $\rho \circ \varphi(h) \ne 1$ . By the assumption $\varphi(t) \ne 1$ , and $\rho \circ \varphi(t) \ne 1$ . Since t is central in H, $[t, h] = 1$ and hence $[\rho \circ \varphi(t), \rho \circ \varphi(h)] = 1$ in $\mathrm{PSL}_2(\mathbb{C})$ . Note that $\pi_1(M)$ is torsion free and $(\rho \circ \varphi(t))^2 \ne 1$ . Then $[\rho \circ \varphi(t), \rho \circ \varphi(h)] = 1$ if and only if $\mathrm{Fix}(\rho \circ \varphi(t)) = \mathrm{Fix}(\rho \circ \varphi(h))$ ; see [ Reference Beardon2 , 4·3]. This means that for any non-trivial elements $\varphi(h)$ and $\varphi(h')$ in $\varphi(H)$ , $\mathrm{Fix}(\rho \circ \varphi(h)) = \mathrm{Fix}(\rho \circ \varphi(h'))$ . This then implies that $\varphi(h)$ and $\varphi(h')$ in $\varphi(H)$ commute, i.e. $\varphi(H)$ is abelian in $\pi_1(M)$ . Hence, $\varphi|_{H}$ factors $H \to H/[H,H]\to \pi_1(M)$ . However, this contradicts $\varphi(g)\ne 1$ .