2. Schottky groups

Let $D^-$ and $D^+$ be the closure of two disjoint simply connected domains in $\hat{\mathbb{C}}$ , where $\partial D^\pm$ is a Jordan curve. A loxodromic $\gamma \in PSL_2(\mathbb{C})$ pairs $D^-$ to $D^+$ if the following two conditions hold:

1. (1) $\gamma(\partial D^-) = \partial D^+$ ;

2. (2) $\gamma(int(D^-)) = \hat{\mathbb{C}} \setminus D^+$

If $\{D_i^\pm\}_{i=1}^n$ is a collection of closures of mutually disjoint simply connected domains, and $\gamma_i$ is a loxodromic that pairs $D_i^-$ to $D_i^+$ , then $\langle \gamma_1,\ldots \gamma_n \rangle$ is a Schottky group. If each $D_i^\pm$ in the previous sentence is a geometric disk, then $\langle \gamma_1,\ldots ,\gamma_n \rangle$ is a classical Schottky group.

It is a simple matter to construct classical Schottky groups. If $D^-$ and $D^+$ are disks with centers $d^-$ and $d^+$ , and radii r and s, then

\[ {}z\mapsto \frac{rs}{z-d^-} + d^+ {}\]

is a Mobius transformation taking $\partial D^-$ to $\partial D^+$ and mapping $int(D^-)$ onto $\hat{\mathbb{C}}\setminus D^+$ . Then

\[ {}\frac{-1}{\sqrt{rs}}\begin{pmatrix} {}d^+ & rs-d^-d^+\\ {}1 & -d^-\end{pmatrix}\in PSL_2(\mathbb{C}) {}\]

will be a loxodromic pairing $D^-$ to $D^+$ when $|d^- - d^+|\gt2\sqrt{rs}$ . So if $\{D_i^\pm\}_{i=1}^n$ is any collection of pairwise disjoint disks in $\mathbb{C}$ such that the distance between the centers is much larger than their respective radii, then there exists a classical Schottky group $\langle \gamma_1,\ldots \gamma_n \rangle$ where $\gamma_i$ pairs $D_i^-$ to $D_i^+$ . In particular, for all $n\in \mathbb{N}$ , there are classical Schottky groups whose generators pair disks with radii less than $1/n$ .

Schottky groups are free, discrete, and purely loxodromic. For any loxodromic $\gamma$ , let $\gamma^+$ and $\gamma^-$ denote the attracting a repelling fixed points of $\gamma$ and let $Fix(\gamma)\;:\!=\;\{ \gamma^\pm \}$ . From the definitions it is clear that if $\gamma$ pairs $D^-$ to $D^+$ , then $\gamma\pm \in D^\pm$ . In later arguments, for words $\omega$ in a Schottky group $\langle \gamma_1,\ldots \gamma_n \rangle$ where $\gamma_i$ pairs $D_i^-$ to $D_i^+$ , we will need to specify which disk in the collection $\{D_i^\pm\}_{i=1}^n$ contains $\omega^+$ and $\omega^-$ . For this reason, we introduced the following notation.

Let $\Gamma = \langle \gamma_1,\ldots, \gamma_n \rangle$ be a Schottky group where $\gamma_i$ pairs $D_i^-$ to $D_i^+$ . We let $D^\pm (\gamma_i)\;:\!=\; D_i^\pm$ . Put $D^+(\gamma_i^{-1})\;:\!=\; D^-(\gamma_i)$ and $D^-(\gamma_i^{-1})\;:\!=\; D^+(\gamma_i)$ . The collection $\{D_i\}_{i=1}^n$ is not unique to $\Gamma$ . So $D^\pm(\gamma_i)$ just refers to any collection of disks paired by the Schottky generators.

Since the attracting/repelling fixed point of $\gamma$ is the repelling/attracting fixed point of $\gamma^{-1}$ , one can verify via induction the following basic but crucial observation:

While there is no canonical choice of disks for $\Gamma$ , all statements and arguments referring to $D^\pm(\gamma_{i_k})$ will hold for any collection of disks satisfying the Schottky definition, and will not require choices.

3. Schottky space

From now on, let $G=\langle g_1,\ldots , g_n \rangle$ denote a fixed rank n free group with $\{g_1, \ldots , g_n\}$ a fixed ordered set of generators. Schottky space, denoted S(G), is the set of all $PSL_2(\mathbb{C})$ conjugacy classes of embeddings $\sigma\;:\;G \rightarrow PSL_2(\mathbb{C})$ such that its image $\sigma(G)=\langle \sigma(g_1),\ldots ,\sigma(g_n) \rangle$ is a Schottky group. It is a classical result of Bers [ Reference Bers1 ] that Schottky space can be identified with a connected open subset of $\mathbb{C}^{3n-3}$ . We review some details of this identification.

For any point $[\sigma]$ in Schottky space, we can take a representative $\sigma$ where $\sigma(g_1)^+ = -1$ , $\sigma(g_2)^- = 0$ , and $\sigma(g_2)^+ = 1$ . Then $\sigma$ is determined by the squared traces of the generators, and the remaining fixed points of the generators which have yet to be specified. Then the following embedding identifies Schottky space with a domain in $\mathbb{C}^{3n-3}$ :

\[ {}[\sigma] \mapsto (tr^2(\sigma(g_1)), tr^2(\sigma(g_2)), \sigma(g_2)^-,\ldots , tr^2(\sigma(g_n)), \sigma(g_n)^+, \sigma(g_n)^-) {} {} {}. {}\]

From this point forward, we identify Schottky space S(G) with a domain in $\mathbb{C}^{3n-3}$ as described above. We abuse notation so that $\sigma \in S(G)$ refers to both a point in Schottky space under this identification, and an embedding $\sigma \rightarrow PSL_2(\mathbb{C})$ such that $\langle \sigma(g_1), \ldots , \sigma(g_n) \rangle$ is a Schottky group with $\sigma(g_1)^+ = -1$ , $\sigma(g_1)^- = 0$ , and $\sigma(g_2)^+ = 1$ .

We conclude this section with some remarks on holomorphic functions defined on simply connected open subsets of S(G). As a warm-up, recall that for any simply connected domain $\Omega\subset \mathbb{C}^k$ , if $f\;:\;\Omega \rightarrow \mathbb{C}$ is holomorphic and non-vanishing, then $\ln f$ is also holomorphic. For such f, $e^{\ln f/m}$ will also be holomorphic, so that f has holomorphic mth roots for all m. In particular, $\sqrt{f}$ is a holomorphic function on $\Omega$ .

Now suppose $\Omega \subset S(G)$ is a simply connected open subset of S(G) (note that Schottky space itself is not simply connected [ Reference Takashi and Yoshida7 ]). Let $g_i$ be one of the fixed generators of G. The entries of $\sigma(g_i)$ can be written as rational functions in $\sigma^\pm (g_i)$ and $\sqrt{tr^2\sigma(g_i)}$ . Notice that for all $\sigma \in S(G)$ , the quantities $(\sigma(g_i)^+-\sigma(g_i)^-)$ , $tr\sigma(g_i)$ , and $tr^2\sigma(g_i) - 4$ are never 0 because $\sigma(G)$ is purely loxodromic and discrete.

For any non-identity element $w\in G$ , we conclude $ \sigma \mapsto \sigma^+(w) - \sigma^-(w)$ defines a (necessarily non-vanishing) holomorphic function on $\Omega$ , as the entries of $\sigma(w)$ are $\mathbb{Z}$ -polynomials in the entries of the generators $\sigma(g_i)$ , and the fixed points of any

\[ {}\begin{pmatrix} {}a & b\\ {}c & d\end{pmatrix} \in PSL_2(\mathbb{C}) {}\]

are given by

\[ {}\frac{a-d \pm \sqrt{(a+d)^2-4}}{2c}. {}\]

Later, for fixed pairs of non-commuting words $w_1,w_2\in G$ , we will define functions mapping $\sigma \in \Omega$ to the cross ratio of the fixed points of $\sigma(w_1),\sigma(w_2)$ . By this discussion, such functions are holomorphic. See [ Reference Yamamoto and Taniguchi8 ] for more details on how fixed point cross ratios have been used to study Schottky space.

4. Classifying pairs of words

Given a $\sigma \in S(G)$ and two freely reduced non-commuting words $w_i,w_j\in G$ , we will need to know which $D^\pm (g_k)$ contains which fixed points of $\sigma(w_i)$ and $\sigma(w_j)$ , and whether or not some $D^\pm (g_k)$ contains multiple points in $Fix(\sigma(w_i))\cup Fix(\sigma(w_j))$ . By Lemma 2·1, this depends only on what the left-most and right-most letters of the words are. We now classify pairs of freely reduced non-commuting words in G with this goal in mind. Afterwards, we explain more explicitly the geometric intuition behind the classification. Recall that a word is cyclically reduced if its left-most and right-most letters are not inverses of each other.

We now explain the geometric intuition behind the classification. Let $\sigma \in S(G)$ . Say $w_1,w_2\in G$ is a pair of freely reduced non-commuting words. If $w_1,w_2$ is a

1. (1) type 1 pair, then each $D^\pm(\sigma(g_i))$ contains at most one point in $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ .

2. (2) type 2 pair, then some $D^\pm(\sigma(g_k))$ contains exactly two points in $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ .

3. (3) type 3 pair, then some $D^\pm(\sigma(g_k))$ contains exactly three points in $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ .

4. (4) type 4 pair, then some $D^\pm(\sigma(g_k))$ contains $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ .

While the disks $D^\pm(\sigma(g_k))$ are not unique to each $\sigma(G)$ , any collection of disks satisfying the Schottky group definition will also satisfy the containment properties listed above. Moreover, for type 2 pairs, when $D^\pm(\sigma(g_k))$ contains two points in $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ it will contain the same 2 fixed points for all $\sigma \in S(G)$ . For example, if $w_1,w_2$ is a type 2 pair and (say) for some $\sigma_0 \in S(G)$ we have $\{\sigma_0(w_1)^+,\sigma_0(w_2)^+\} = (Fix(\sigma_0(w_1))\cup Fix(\sigma_0(w_2)))\cap D^+(\sigma_0(g_1))$ , then for all $\sigma \in S(G)$ we will have $\{\sigma(w_1)^+,\sigma(w_2)^+\} = (Fix(\sigma(w_1))\cup Fix(\sigma(w_2)))\cap D^+(\sigma(g_1))$ .

This geometric picture helps one see that all pairs of non-commuting freely reduced words are indeed of one of the types listed above, since under a Schottky embedding the fixed points of any word must be in one of the Schottky disks. The four types account for all possible ways the 4 fixed points $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ can be distributed among the disks $\{D^\pm(\sigma(g_i))\}$ . See Figure 1 for some diagrams of how the translation axes of $\sigma(w_1)$ and $\sigma(w_2)$ can be positioned relative to disks paired by generators of $\sigma(G)$ , depending on what type of pair $w_1,w_2$ is.

Fig. 1.

This figure gives examples of how the endpoints of the translation axes of $\sigma(w_1)$ and $\sigma(w_2)$ can be distributed among the disks $\{D^\pm(\sigma(g_i))\}$ being paired by the generators of $\sigma(G)$ , based on what type of pair $w_1,w_2$ is. The circles represent the $D^\pm(\sigma(g_k))$ containing $\sigma^\pm(w_i)$ .

5. Proofs

Let $w_1,w_2\in G$ be a pair of freely reduced non-commuting words. We will show there exists $\sigma\in S(G)$ such that $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ is not contained in a line. Our argument will be divided into cases based on what type of pair $w_1,w_2$ is. The next lemma will help us reduce the argument to word pairs of type 1 and 2.

Proof. First suppose the pair is of type 3. Say $w_1$ is not cyclically reduced, so $w_1=gw_1'g^{-1}$ for some $g, w_1'\in G$ , where $gw_1'g^{-1}$ is freely reduced and $w_1'$ is cyclically reduced. Then $w_2$ is cyclically reduced. Without loss of generality we suppose $w_1$ and $w_2$ have the same left-most letter.

Write $g=g_{i_1}^{m_1}\cdots g_{i_r}^{m_r}$ , $m_j\in \mathbb{Z}\setminus \{0\}$ . Let $sgn(m_j)=+1$ if $m_j\gt0$ and $sgn(m_j)=-1$ if $m_j\lt0$ . We claim that after free reduction of $g^{-1}w_2g$ , the pair $w_1',g^{-1}w_2g$ is of type 1 or 2. It cannot be a type 4 pair because $w_1'$ is cyclically reduced, so it suffices to show the pair is not of type 3.

Suppose for contradiction that $w_1',g^{-1}w_2g$ is a type 3 pair. Since $w_1'$ is cyclically reduced and the pair is of type 3, then after free reduction $g^{-1}w_2g$ is not a cyclically reduced word, and it has either the same left-most letter as $w_1'$ , or the same right-most letter as $w_1'$ . We start by arguing the pair does not have the same right-most letter.

Recall that $w_2$ is cyclically reduced. Also $w_1=gw_1'g^{-1}$ and $w_2$ share the same left-most letter, which is $g_{i_1}^{sgn(m_1)}$ . This means $g_{i_1}^{-sgn(m_1)}$ cannot be the right-most letter of $w_2$ . Hence the word

\[ {}w_2g=w_2g_{i_1}^{m_1}\cdots g_{i_r}^{m_r} {}\]

is freely reduced. But the right-most letter of $w_1'$ is not $g_{i_r}^{sgn(m_r)}$ because $w_1=gw_1'g^{-1}$ is freely reduced.

Therefore, $w_1'$ and $g^{-1}w_2g$ must share the same left-most letter after free reduction of $g^{-1}w_2g$ . But the left-most letter of $w_1'$ is not $g_{i_r}^{-sgn(m_r)}$ because $w_1=gw_1'g^{-1}$ is freely reduced. This means after free reduction of $g^{-1}w_2g$ , the entire $g^{-1}$ word must have been eliminated. That is, after free reduction we have $g^{-1}w_2g=w_2'g$ for some freely reduced word $w_2'$ . As argued in the previous paragraph, $w_2'g$ is freely reduced. By assumption, $w_2'g$ is not cyclically reduced, so the left-most letter of $w_2'g$ is $g_{i_r}^{-sgn(m_r)}$ . But again, the left-most letter of $w_1'$ cannot be $g_{i_r}^{-sgn(m_r)}$ . This gives the desired contradiction.

Finally, if the pair is of type 4, we can let $g\in G$ be such that $gw_1g^{-1}$ is cyclically reduced after free reduction. This means $gw_1g^{-1},gw_2g^{-1}$ is not a type 4 pair after free reduction. If it is type 1 or 2, then we are done. If it is of type 3, then we conjugate further as in the previous case.

For $w_1,w_2$ a pair of freely reduced non-commuting words in G, put

\[R(w_1,w_2): = \{ {}\sigma \in S(G) : Fix(\sigma(w_1))\cup Fix(\sigma(w_2)) \text{ is contained in a circle} {}\}. {}\]

Our next task is to show $R(w_1,w_2)$ has empty interior. To do this, we will need to exhibit two Schottky embeddings $\sigma',\sigma''\in S(G)$ where the cross ratio of the fixed points of $\sigma'(w_1),\sigma'(w_2)$ is different from the cross ratio of the fixed points of $\sigma''(w_1),\sigma''(w_2)$ . The candidate Schottky embeddings we will consider will be a sequence of classical Schottky groups whose generators pair disks with successively smaller diameter, shrinking to zero.

To construct this sequence, fix 2n distinct points $\{d_i^\pm\}_{i=1}^{2n}$ in $\mathbb{C}$ which are pairwise distance (say) at least 100 apart. For $t\in \mathbb{N}$ , let $D_i^\pm[t]$ be a disk centered at $d_i^\pm$ with diameter less than $1/t$ . Let $\sigma_t\;:\;G \rightarrow PSL_2(\mathbb{C})$ be an embedding such that $\sigma_t(G)$ is a classical Schottky group where $\sigma_t(g_i)$ pairs $D_i^-[t]$ to $D_i^+[t]$ . Put $D_t^\pm(\sigma_t(g_i))\;:\!=\;D_i^\pm[t]$ and $D_t^\pm(\sigma_t(g_i^{-1}))\;:\!=\; D_i^\mp[t]$ . The representations $\sigma_t$ are not yet points in our model of Schottky space S(G) because the fixed points of $\sigma_t(g_1)$ and $\sigma_t(g_2)$ have not yet been normalized, but as will be clear later on, we can conjugate the representations to get the desired points in S(G). We summarize the properties about $\{\sigma_t\}_{t=1}^\infty$ that we need for future arguments in the following Lemma. The absolute value refers to the standard Euclidean distance as points in $\mathbb{C}$ .

The first property of Lemma 5·2 comes from the fact the centers are fixed during the construction of $\{\sigma_t\}_{t=1}^\infty$ .

Case 1: Say $w_1,w_2$ is a type 2 pair where (say) $w_1$ is not cyclically reduced. For any $\sigma\;:\;G \rightarrow PSL_2(\mathbb{C})$ where $\langle \sigma(g_1),\ldots , \sigma(g_n)\rangle$ is a Schottky group, put

\begin{align*} {} {}\chi_{w_1,w_2}(\sigma) &\;:\!=\; [\sigma^+(w_1), \sigma^+(w_2), \sigma^-(w_1), \sigma^-(w_2)] \\ {} {}&= \frac{(\sigma^-(w_1) - \sigma^+(w_1))}{(\sigma^-(w_1)-\sigma^+(w_2))} \frac{(\sigma^-(w_2)-\sigma^+(w_2))}{(\sigma^-(w_2)-\sigma^+(w_1))}. {}\end{align*}

That is, $\chi_{w_1,w_2}(\sigma)$ is the cross ratio of the fixed points of $\sigma(w_1),\sigma(w_2)$ . We will now show there exists $\sigma',\sigma'' \in S(G)$ such that $\chi_{w_1,w_2}(\sigma')\neq \chi_{w_1,w_2}(\sigma'')$ .

Let $\{\sigma_t\}_{t=1}^\infty$ be the sequence of Schottky embeddings as in Lemma 5·2. Since $w_1,w_2$ is a type 2 pair with $w_1$ not cyclically reduced, there is a $g_j$ in the fixed set of generators $\{g_1,\ldots g_n \}$ of G, and $s\in \{\pm 1\}$ , such that $\sigma_t^\pm (w_1)\in D_t^+(\sigma_t(g_j^{s}))$ for all t. Moreover, $\sigma^\pm(w_2) \not\in D_t^+(\sigma_t(g_j^{s}))$ . Hence, property (ii) of the sequence $\{\sigma_t\}_{t=1}^\infty$ implies

\[\lim\limits_{t\rightarrow \infty} |\sigma_t^-(w_1) - \sigma_t^+(w_1)| = 0. \]

The term $|\sigma_t^-(w_2)-\sigma_t^+(w_2)|$ either converges to 0 or remains bounded above as $t\rightarrow \infty$ , depending on whether or not $w_2$ is also cyclically reduced. Meanwhile, by property (i) of the sequence $\{\sigma_t\}_{t=1}^\infty$ , there are constants $c_1,c_2 \in (0,\infty)$ such that

\[c_1 \lt |\sigma_t^-(w_1)-\sigma_t^+(w_2)|,|\sigma_t^-(w_2)-\sigma_t^+(w_1)| \lt c_2\]

for all t. We conclude

\[ {}\lim\limits_{t\rightarrow \infty}|\chi_{w_1,w_2}(\sigma_t)|=0 {}\]

so there must be $\sigma_{t'},\sigma_{t''}$ in $\{\sigma_t\}_{t=1}^\infty$ with $\chi_{w_1,w_2}(\sigma_{t'})\neq \chi_{w_1,w_2}(\sigma_{t''})$ . There exists $\gamma\in PSL_2(\mathbb{C})$ such that $\gamma\sigma_{t'}(g_1)^+\gamma^{-1}=-1$ , $\gamma\sigma_{t'}(g_1)^-\gamma^{-1}=0$ , and $\gamma\sigma_{t'}(g_2)^+\gamma^{-1}=1$ . Let $\sigma'\in S(G)$ be such that $\sigma'(w)= \gamma\sigma_{t'}(w)\gamma^{-1}$ for all $w\in G$ . Cross ratios are invariant under Mobius transformations, and $(\gamma\sigma_{t'}(w)\gamma^{-1})^\pm = \gamma(\sigma_{t'}(w))^\pm$ , so we have

\[ {}\chi_{w_1,w_2}(\sigma')= [\gamma \sigma_{t'}^+(w_1), \gamma \sigma_{t'}^+(w_2), \gamma \sigma_{t'}^-(w_1), \gamma \sigma_{t'}^-(w_2)] = \chi_{w_1,w_2}(\sigma_{t'}). {}\]

Obtain $\sigma''$ from a $PSL_2(\mathbb{C})$ conjugation of $\sigma_{t''}$ in the same way. Then $\sigma',\sigma''$ are points in S(G) such that $\chi_{w_1,w_2}(\sigma')\neq \chi_{w_1,w_2}(\sigma'')$ , as desired.

Now for any $\sigma \in S(G)$ , since S(G) is a connected open subset of $\mathbb{C}^{3n-3}$ , there exists a smoothly embedded arc joining $\sigma,\sigma'$ and $\sigma''$ . Let $\Omega_\sigma$ be a tubular neighbourhood of such an arc. Then we see $\{\Omega_\sigma\}_{\sigma\in S(G)}$ is an open cover of S(G) by simply connected subsets, each of which contains $\sigma'$ and $\sigma''$ . Manifolds are Lindelof so we can pass to a countable sub-cover, which we denoted by $\{\Omega_m\}^\infty_{m=1}$ .

On each $\Omega_m$ , define $\chi_{w_1,w_2}|_{\Omega_m}\;:\;\Omega_m \rightarrow \mathbb{C}$ by $\chi_{w_1,w_2}|_{\Omega_m}(\sigma)\;:\!=\; \chi_{w_1,w_2}(\sigma)$ . Then $\chi_{w_1,w_2}|_{\Omega_m}$ is a non-constant holomorphic function. By the open mapping theorem for several complex variables (see [ Reference Korevaar and Wiegerinck5 ]), $\chi_{w_1,w_2}|_{\Omega_m}^{-1}(\mathbb{R})$ has empty interior. Since the cross ratio of four points is real if and only if they lie on a circle, we have that for all $\sigma \in \Omega_m \setminus \chi_{w_1,w_2}|_{\Omega_m}^{-1}(\mathbb{R})$ , $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ does not lie on a circle. That is,

\[R(w_1,w_2)= \bigcup_{m=1}^\infty \chi_{w_1,w_2}|_{\Omega_m}^{-1}(\mathbb{R}),\]

and since each $\chi_{w_1,w_2}|_{\Omega_m}^{-1}(\mathbb{R})$ is closed and has empty interior, the Baire category theorem (for locally compact Hausdorff spaces) implies $R(w_1,w_2)$ has empty interior as desired.

Case 2: Now suppose $w_1,w_2$ is a type 2 pair with both words cyclically reduced. Furthermore, replacing either word with their inverses if necessary, we assume they have the same left-most letter. Put

\begin{align*} {} {}\chi_{w_1,w_2}(\sigma) &\;:\!=\; [\sigma^+(w_1), \sigma^-(w_1), \sigma^+(w_2), \sigma^-(w_2)] \\ {} {}&= \frac{(\sigma^+(w_2) - \sigma^+(w_1))}{(\sigma^+(w_2)-\sigma^-(w_1))} \frac{(\sigma^-(w_2)-\sigma^-(w_1))}{(\sigma^-(w_2)-\sigma^+(w_1))}. {}\end{align*}

Let $\sigma_t$ denote the same sequence of representations from Lemma 5·2 as in case 1. We will show there exists $\sigma_{t'},\sigma_{t''}\in \{\sigma_t\}_{t=1}^\infty$ such that $\chi_{w_1,w_2}(\sigma_{t'})\neq \chi_{w_1,w_2}(\sigma_{t''})$ . This suffices to give the desired result because the same reasoning as in case 1, in the part after $\sigma_{t'}$ and $\sigma_{t''}$ were found, applies verbatim.

Under the current assumptions of this case, there is a $g_l$ in the fixed set of generators $\{g_1,\ldots,g_n \}$ for G and $s\in \{\pm 1 \}$ such that $\sigma_t^+(w_1),\sigma_t^+(w_2)\in D_t^+(g_l^s)$ for all t. This means

\[\lim\limits_{t\rightarrow \infty} |\sigma_t^+(w_2) - \sigma_t^+(w_1)| = 0, \]

while $|\sigma_t^-(w_2) - \sigma_t^-(w_1)|$ either remains bounded above or decays to 0 as $t\rightarrow \infty$ . Neither $\{\sigma^+(w_2),\sigma^-(w_1) \}$ nor $\{\sigma^-(w_2),\sigma^+(w_1)\}$ can be contained in a single $D_t^\pm(\sigma_t(g_i))$ for any $\sigma_t$ or generator $g_i$ because $w_1,w_2$ is a type 2 pair and $\{\sigma^+(w_2)),\sigma^+(w_1) \}$ always lies in $D_t^+(g_l^s)$ for all t all $\sigma_t$ . Hence the terms $|\sigma^+(w_2))-\sigma^-(w_1)|$ and $|\sigma^-(w_2)-\sigma^+(w_1)|$ must be bounded below strictly away from 0 for all t. From this we conclude

\[ {}\lim\limits_{t\rightarrow \infty}|\chi_{w_1,w_2}(\sigma_t)|=0, {}\]

so there must be $\sigma',\sigma''\in S(G)$ with $\chi_{w_1,w_2}(\sigma')\neq \chi_{w_1,w_2}(\sigma'')$ .

Case 3: Finally, suppose $w_1,w_2$ is a type 1 pair. Put

\begin{align*} {} {}\chi_{w_1,w_2}(\sigma) &\;:\!=\; [\sigma^+(w_1), \sigma^-(w_1), \sigma^+(w_2), \sigma^-(w_2)] \\ {} {}&= \frac{(\sigma^+(w_2) - \sigma^+(w_1))}{(\sigma^+(w_2))-\sigma^-(w_1))} \frac{(\sigma^-(w_2)-\sigma^-(w_1))}{(\sigma^-(w_2)-\sigma^+(w_1))}. {}\end{align*}

As before, it suffices to find $\sigma',\sigma''\in S(G)$ such that $\chi_{w_1,w_2}(\sigma')\neq \chi_{w_1,w_2}(\sigma'')$ . Observe that if $\sigma\in S(G)$ is such that $\sigma(G)$ is Fuchsian, then $\chi_{w_1,w_2}(\sigma)\in \mathbb{R}$ and $\chi_{w_1,w_2}(\sigma)\in (-\infty,0)$ if and only if $Fix(\sigma(w_1))$ separates $Fix(\sigma(w_2))$ i.e. if and only if the translation axes of $\sigma(w_1)$ has non-empty intersection with the translation axes of $\sigma(w_2)$ .

Since $w_1,w_2$ is a type 1 pair, for all $\sigma \in S(G)$ , each point in $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ always lies in a different Schottky disk of $\sigma(G)$ . Then it is easy to use a Ping Pong argument to obtain $\sigma',\sigma''\in S(G)$ where $\chi_{w_1,w_2}(\sigma') \in (-\infty,0)$ and $\chi_{w_1,w_2}(\sigma'')\in (0,\infty)$ .

Now we restate and prove the main Theorem.

Proof. Let

\[ {}R \;:\!=\; \bigcup_{w_1w_2 \neq w_2w_1} R(w_1,w_2), {}\]

where $R(w_1,w_2)$ is defined as in Proposition 5·3 and the union is over all pairs of non-commuting words in G. This is a countable union. It is clear each $R(w_1,w_2)$ is closed, and each $R(w_1,w_2)$ has empty interior by Proposition 5·3, so R also has empty interior by the Baire category theorem for locally compact Hausdorff spaces. For any $\sigma \in S(G)\setminus R$ , $Fix(\sigma(w_1))\cup Fix(\sigma(w_2))$ is not contained in a line for any pair of non-commuting words $w_1,w_2\in G$ , by construction of R.