数理解析研究所講究録
第 1991 巻 2016 年 69-81

69

ADDENDUM TO “COMMENSURABILITY BETWEEN
ONCE-PUNCTURED TORUS GROUPS AND ONCE-PUNCTUREDKLEIN BOTTLE GROUPS”
MIKIO FUROKAWA

1. INTRODUCTION
The main purpose of this addendum to [4] is to present a proof to [4, Proposition
3.7] which gives a classification of elliptic generator triples of the fundamental group of
the quotient orbifold of the once-punctured Klein bottle (see Definition 2.1 and Proposition 2.2). We also prove the “converse” of [4, Theorem 5.1], namely, we give a condition-representation of the fundamental group of thefor a faithful type-preserving
once-punctured torus to be “commensurable” with that of the once-punctured Klein bot)(see Definitions 3.1, 3,2
tle by using Proposition 3.7 and Theorem 5.1 in the original
and Theorem 3.13).
The rest of this paper is organized as follows. In Section 2, we give a proof to [4, Proposition 3.7] (see Proposition 2.2). In Section 3, we prove the “converse” of [4, Theorem 5.1]
$PSI_{I}(2, \mathbb{C})$$pa\mathfrak{x}$$er$(see Theorem 3.13).

2. CLASSIFICATION OF ELLIPTIC GENERATOR TRIPLESIn this section, we give a proof to [ Proposition 3.7]. To this end, we first introduce
some notations and recall the definition of elliptic generators,
be the involution:Letbe the once-punctured Klein bottle and let
andbyillustrated in Figure!. Then we denote the quotient orbifold
identify
with
.
We
by
to
denote the covering projection from. Thenthe image of the inclusion\’induced by the projectionof index 2,
is regarded as a normal subgroup of$4_{1}$$N_{2,1}$$\iota_{N_{2,1}}$$N_{2,1}arrow N_{2,1}$$N_{2,1}/\iota_{N_{2,1}}$$N_{2,1}$$\mathcal{O}_{N_{2,1}}$$\mathcal{O}_{N_{2,1}}$$\pi_{1}(N_{2,1})$$p_{N_{2,1}}$$\pi_{1}(N_{2,1})arrow\pi_{1}(\mathcal{O}_{N_{2,1}})$$p_{N_{2,1}}$$\pi_{1}(\mathcal{O}_{N_{2,1}}\rangle$$\prime/r_{1}(N_{2,1})$$\prime;r_{1}(N_{2,1})=\langle Y_{1}, Y_{2}|-\rangle\triangleleft\pi_{1}(\mathcal{O}_{N_{2,1}})=\langle Q_{0}, Q_{1}, Q_{2}|Q_{0}^{2}=Q_{1}^{2}=Q_{2}^{2}=1\rangle,$$K_{0}=Q_{1}^{Qo}$ andsuch that $Y_{1}=Q_{0}Q_{1}$ and $Y_{2}=Q_{0}Q_{2}$ . Set
andis represented by the puncture of, where $A^{B}=BAB^{-1}$ . Thenofreflector
thecornerare represented by the reflector lines which generateandorder . By the identification, we also obtain $K_{N_{2,1}}=K_{2}K_{0}.$$K_{N_{2,1}}=(Y_{1}Y_{2}Y_{1}^{-1}Y_{2}\rangle^{-1},$$K_{2}=Q_{1}^{Q_{2}}$$N_{2,1_{\rangle}}$$K_{N_{2,1}}$$K_{2}$$K_{(j}$$\infty$

FIGURE 1. The involution
Received December 29, 2015.

$\iota_{N_{2,1}}$

of$N_{2,1}$

70
Definition 2.1. An ordered tripleof elements ofis called an elliptic
generator triple ofif its members generateandand. A member of an elliptic generator triple ofis called
elliptic
generator ofan.
$(Q_{0}, Q_{1}, Q_{2})$$\pi_{1}(\mathcal{O}_{N_{21}})$$sat\dot{i}s\mathfrak{h}\prime Q_{0}^{2}=Q_{1}^{2}=Q_{2}^{2}=$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$Q_{1^{Q_{2}}}Q_{1^{Q_{0}}}=K_{2}K_{0}$$1$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\pi_{1}(\mathcal{O}_{N_{2.1}})$

Now we introduce Proposition 3.7 in the original paper.
Proposition 2.2. The elliptic generator triples ofare characterized as follows.
(1) For any elliptic generator triple, the following hold:of(1.1) The triples in the following -infinite sequence are also elliptic generator triples$\pi_{1}(\mathcal{O}_{N_{2,1}})$$(Q_{0}, Q_{1}, Q_{2})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$

$bi$

of

$\pi_{1}(\mathcal{O}_{N_{2,1}})$. .. ,.

$(Q_{0^{K_{0}K_{2}}},Q_{1^{K_{0}K_{2}}}, Q_{2}^{KoK_{2}})\}(Q_{2^{K_{0}}}, Q_{1^{K_{0}}}\rangle Q_{0^{Ko}})$,$(Q_{0_{\rangle}}Q_{1}, Q_{2})$$(Q_{2^{K_{2}})}Q_{1^{K_{2}}}, Q_{0^{K_{2}}}))(Q_{0^{K_{2}K_{0}}}, Q_{1^{K_{2}K_{0}}}, Q_{2^{K_{2}K_{0}}})$,, .. .

To be precise, the following holds. Let
be the sequence of elements
obtained fromby applying the following rule:$\{Q_{j}\}$$of\pi_{1}(O_{N_{2.1}})$$(Q_{0}, Q_{1}, Q_{2})$$Q_{j}^{Ko}=Q_{-j-1}, Q_{j}^{K_{2}}=Q_{-j+5}.$

Then the triple $(Q_{3k)}Q_{3k+1}, Q_{3k+2})$ is also an elliptic generator triple offor any(1.2)
is also an elliptic generator triple of.
(2) Conversely, any elliptic generator triple
is obtained from a given elliptic
generator triple ofby successively applying the operations in (1).
$\pi_{1}(\mathcal{O}_{N_{2,1}})$$k\in \mathbb{Z}.$$(Q_{2}, Q_{1^{Q_{2}Q_{0}}}, Q_{0^{Q_{2}}})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$of\pi_{1}(\mathcal{O}_{N_{2,1}})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$

To prove Proposition 2.2, we need to introduce some definitions and notations. By a
, we mean a finite sequence
word in
where
Here we callthe k-th letter of the word. In particular, the first letterof the word
is called the initial letter of the word and the last letterof the word is called theterminal letter of the word. The inverse of a wordinis the word
. The word length of is denoted by $l(V)$ . A wordinis reduced if
for any $k=1$ , . . ., $t-1$ . Noteanythatelement in
is uniquely represented by a reduced word. For two words$U,$in, by $U\equiv V$ we denote the visual equality of and , meaning that$\{Q_{0}, Q_{1}, Q_{2}\}$$Q_{i_{k}}\in\{Q_{0}, Q_{1}, Q_{2}\}.$$Q_{i_{1}}Q_{i_{2}}\ldots Q_{i_{t}}$$Q_{i_{k}}$$Q_{i_{1}}$$Q_{i_{t}}$$\{Q_{0}, Q_{1}, Q_{2}\}$$V=Q_{i_{1}}Q_{i_{2}}\ldots Q_{i_{t}}$$V$$V^{-1}=Q_{i_{t}}Q_{i_{t-1}}\ldots Q_{i_{1}}$$V=Q_{i_{1}}Q_{i_{2}}\ldots Q_{i_{t}}$$\{Q_{0}, Q_{1}, Q_{2}\}$$Q_{i_{k}}\neq Q_{i_{k+1}}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$

$V$

if

$U$

$\{Q_{0}, Q_{1}, Q_{2}\}$

and

$V$

then $t=u$ andfor each $k=1$ , . . . , . For example, two words
andare not
visually equal, thoughandare equal as elements of.
$U=Q_{i_{1}}Q_{i_{2}}\ldots Q_{i\iota}$$V=Q_{j_{1}}Q_{j_{2}}\ldots Q_{j_{u}}(Q_{i_{k}},$ $Q_{j_{l}}\in\{Q_{0},$ $Q_{1},$ $Q_{2}$$Q_{i_{k}}=Q_{j_{k}}$$t$$Q_{0}Q_{1}Q_{1}Q_{2}$$Q_{0}Q_{1}Q_{1}Q_{2}$$Q_{0}Q_{2}$$Q_{0}Q_{2}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$

Proof of Proposition 2.2. The author got the idea of the proof from the proof of [2, Proposition 10.7] and [1, Lemma 2.1.7].
Since (1) can be proved by direct calculation, we give a proof of (2). For a given elliptic
generator triple, set, and let and a be theandautomorphism ofdefined by$(Q_{0}, Q_{1}, Q_{2})$$K_{0}=Q_{1}^{Q_{0}}$$K_{2}=Q_{1}^{Q_{2}}$$\tau$$\pi_{1}(\mathcal{O})$$(\tau(Q_{0}), \tau(Q_{1}), \tau(Q_{2}))=(Q_{2}^{K_{2}}, Q_{1}^{K_{2}}, Q_{0}^{K_{2}})$,$(\sigma(Q_{0}), \sigma(Q_{1}), \sigma(Q_{2}))=(Q_{2}, Q_{1\rangle}^{Q_{2}Q_{0}}Q_{0}^{Q_{2}})$.

Then and preserveand hence they map elliptic generator triples to elliptic
generator triples. Moreover, the operations in (1.1) is given by , and the operation in(1.2) is given by . Hence we have only to show the following lemma.
$\tau$$K_{N_{2,1}}$$\sigma$$\tau^{n}$$\sigma$

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Lemma 2.3. The group of automorphisms ofandpreserving$K_{N_{2,1}}$

is generated bywhich preserves$K_{N_{2,1}}$. Then for each$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\tau,$$\sigma$

To prove this lemma, we prepare two claims.

Claim2.4. Let be an automorphism of$j=0$ , 2, we have$f$$f(K_{j})=K_{j}^{K_{N_{2,1}}^{n}}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$for some$n\in \mathbb{Z}$and some$j’\epsilon\{O$

,2

$\}.$

is regarded as a subgroup of $Isom^{+}(1ffl^{3})$ .
and $K_{N_{2,1}}=K_{2}K_{0}$ is regarded as ais regarded as the stabilizer ofThenother hand, since $f(K_{2})f(K_{0})=$.
theOnparabolic transformation $K_{N_{2,1}}(z)=z+2$$K_{2}K_{0}=K_{N_{2,1}}$ , we see that

Proof of Claim 2.4. We first note that$\langle K_{0},$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\infty$$K_{2}\rangle$$f(K_{0})K_{N_{2,1}}(f(K_{0}))^{-1}=f(K_{\zeta)})f(K_{2})f(K_{0})(f(K_{0}))^{-1}=f(K_{0})f(K_{2})=K_{N_{21}}^{-1},\cdot$

This implies that $f(K_{0})K_{N_{2,1}}(f(K_{0}))^{-1}$ is parabolic and that Fix $く f(K_{0})K_{N_{2,1}}(f(K_{0})\rangle^{-1})=$. By Fix, where Fix (A) denotes the fixed point set of in$=f(K_{0})(Fix(K_{N_{2,1}}))$, we have $f(K_{0})(\infty)=\infty.$
)
and
$A$

$\{\infty\}$$(K_{N_{2,1}}\rangle=$$\partial \mathbb{H}^{3}=\mathbb{C}\cup\{\infty\}$$F\dot{r}x(f(K_{0})K_{N_{2,1}}\langle f(K_{0}))^{-1}$$\{\infty\}$$K^{n}$and some
and therefore $f(K_{0})=K_{j}^{N_{2,1}}$ for some).
$j’\in\{0$ , 2 . By a similar argument, we obtain the desired result forthat $f(K_{j})=K_{j}$ for each
Claim 2.5. Let be an automorphism of$f(Q_{s})=W_{s}Q_{s}W_{s}^{-1}$2, whereis a reduced
1,
$s=0$
,
$j=0_{\}}2$ . Suppose thatfor each
whose terminal letter is diﬀerent from . Then the following hold.
word inis also a trivial word for each $j=0$ , 2.
(1) Ifis a trivial word, thenword,
one of the following holds for each $j=0$ , 2.
then(2) Ifis a non-trivial$particular_{f}$isthe initial letter of
. In
(i)
is
. In particular, the terminal letter of(ii)
is diﬀerent from. In $particular_{f}$ the terminal letter of
(iii)
Hence

$n\in \mathbb{Z}$$f(K_{0})\in\langle K_{0},$ $K_{2}\rangle$$\square$$f\langle K_{2}$$\}$$\pi_{1}(\mathcal{O}_{N_{2,1}})\mathcal{S}uch$$f$$W_{s}$$Q_{s}$$\{Q_{0}, Q_{1}, Q_{2}\}$$W_{j}$$W_{1}$$W_{1}$$W_{1}$$Q_{j}.$$W_{1}\equiv W_{j}Q_{j}W_{j}^{-1}Q_{j}$$W_{1}$$Q_{j}.$$W_{1}Q_{j}\equiv W_{j}Q_{j}W_{j}^{-1}$$W_{1}$$W_{1}Q_{1}Q_{j}\equiv W_{I}Q_{j}W_{j}^{-1}$$Q_{j}.$

Proof of Claim 2.5. For each $j=0$ , 2, we have the following identity:$Q_{j}Q_{1}Q_{j}=K_{j}=f(K_{j})=f(Q_{j}Q_{1}Q_{j})=W_{j}Q_{j}W_{j}^{-1}\cdot W_{1}Q_{1}W_{1}^{-1}\cdot W_{j}Q_{j}W_{j}^{-1}.$. Sincecommutes withThis implies that2,
we haveorderofgroupsto the free product of three cyclic$Q_{1}$$Q_{j}\cdot W_{j}Q_{j}W_{j}^{-1}\cdot W_{1}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$is isomorphic$\langle Q_{s}\rangle$(Eql)$Q_{j}\cdot W_{j}Q_{j}W_{j}^{-1}\cdot W_{1}=Q_{1}$or 1.

is a trivial word. Then, by the identity
To show the assertion (1), we assume thatabelianization of this identity, we have
By
the
or
1.
(Eq2), we haveor 1.
commutes with , and hence. This implies that
$W_{j}=1$
the.
weobtain
So
,
we
have
from
is
diﬀerentofletterterminal
Since thedesired result.
is a trivial word, then the identity
or
Next, we show the assertion (2). If either$W_{1}=Q_{1}$is also a non-trivialHenceacontradiction.
1.
This
is
or
implies(Eql)
thatword for any $j=0$ , 2.
is also a reduced
is a reduced word. Then
Suppose first thatpossibly for
exceptword. Hence the identity (Eql) implies that the word
$W_{1}$$Q_{j}\cdot W_{j}Q_{j}W_{j}^{-1}=Q_{1}$$Q_{j}\cdot W_{j}Q_{j}W_{j}^{-1}=1$$W_{J}\prime=Q_{j}$$Q_{j}$$W_{j}$$Q_{j}$$W_{j}$$W_{0}$$W_{2}$$W_{f}$’$Q_{j}\cdot W_{j}$$Q_{j}\cdot W_{j}Q_{j}W_{j}^{-1}$$Q_{j}\cdot W_{j}Q_{j}W_{j\}}^{-1}$72
the first letterholds.

$Q_{j}$, is cancelled out by the word

$W_{1}$, and therefore one of the following$\bullet W_{1}\equiv W_{j}Q_{j}W_{j}^{-1}Q_{j}Q_{1},$and $Q_{j}=Q_{1},$$W_{1}\equiv W_{j}Q_{j}W_{j}^{-1}$$\bullet$$\bullet W_{1}\equiv W_{j}Q_{j}W_{j}^{-1}Q_{j}.$However, the first identity can not hold because the terminal letter ofis diﬀerent fromby the assumption, and second identity can not hold because $j=0$ , 2. Hence the third
identity holds. So we obtain the identity in the condition (ii).
Suppose next thatis not a reduced word, i.e.,
for some reduced
word . Then, by the identity (Eql), we have$W_{1}$$Q_{1}$$Q_{j}\cdot W_{j}$$W_{j}\equiv Q_{j}\cdot V_{j}$$V_{j}$(Eq2)$V_{j}Q_{j}W_{j}^{-1}\cdot W_{1}=Q_{1}$

Since

or 1.

is a reduced word, it must be cancelled out by
the initial letter of , and therefore one of the following hold.
$V_{j}Q_{j}W_{j}^{-1}$$W_{1}$, except possibly for$V_{j}$$\bullet W_{1}\equiv W_{j}Q_{j}V_{j}^{-1}Q_{1},$and$W_{1}\equiv W_{j}Q_{j}V_{j}^{\prime-1}$$\bullet$$V_{j}\equiv Q_{1}V_{j}’$for some reduced word

$V_{j}’.$$\bullet W_{1}\equiv W_{j}Q_{j}V_{j}^{-1}$

The first identity can not hold by the fact that the terminal letter ofis diﬀerent from. If the second identity or the third identity holds, then the condition (i) or (iii) holds
accordingly.
$W_{1}$$Q_{1}$$\square$

We now begin to prove Lemma 2.3 by using the above claims.
Let be an automorphism ofpreserving
Step 1. For each $j=0$ , 2, we show that we may assume $f(K_{j})=K_{j}$ by post composing
a power of to if necessary. By Claim 2.4, we have $f(K_{0})=K_{j},$for some
and for some $j’\in\{0$ , 2 . Since
is an inner-automorphism by, we may assume$f(K_{0})=K_{j’}$ by post composing a power ofto if necessary. By the assumption$f(K_{2})f(K_{0})=f(K_{N_{2,1}})=K_{N_{2,1}}$ , we have $f(K_{2})=K_{N_{2,1}}f(K_{0})=K_{2}K_{0}f(K_{0})$ .
Hence
$f$

$\pi_{1}(\mathcal{O}_{N_{2,1}})$$K_{N_{2,\lambda}}.$$K^{n}$$\tau$$N_{2,1}$$f$$n\in \mathbb{Z}$$\tau^{2}$$\}$$K_{N_{2,1}}$$\tau^{2}$$f$$f(K_{2})=K_{2}K_{0}K_{j’}=\{\begin{array}{ll}K_{2} if j’=0,K_{0}^{K_{2}} if j’=2.\end{array}$Since maps $(K_{0}, K_{2})$ to, we may assume $f(K_{j})=K_{j}$ for each $j=0$ , 2 bypost composing to if necessary.
Step 2. For each $s=0$ , 1, 2, we show that we may assume $f(Q_{s})=W_{s}Q_{s}W_{s}^{-1}$ by postcomposing to if necessary. Since $f(Q_{s})$ has order 2 and since
is isomorphic
to the free product of three cyclic groupsof order 2, we have $f(Q_{s})=V_{s}Q_{\theta(s)}V_{s}^{-1}$ forsome $\theta(s)\in\{0$ , 1, 2 , where is a reduced word whose terminal letter is diﬀerent from
. By the abelianization of the identity$(K_{2}^{K_{0}}, K_{0})$$\tau$$f$$\tau$$\sigma$$f$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\langle Q_{s}\rangle$$\}$$V_{s}$$Q_{\theta(s)}$$Q_{2}Q_{1}Q_{2}=K_{2}=f(K_{2})=f(Q_{2}Q_{1}Q_{2})=f(Q_{2})f(Q_{1})f(Q_{2})$ ,we have $\theta(1)=1$ . By Stepl, we have the following identities:$Q_{0}Q_{1}Q_{0}=K_{0}=f(K_{0})=f(Q_{0})f(Q_{1})f(Q_{0})$,$Q_{2}Q_{1}Q_{2}=K_{2}=f(K_{2})=f(Q_{2})f(Q_{1})f(Q_{2})$ .

73
By these identities, we have the following identity:$Q_{1}\cdot Q_{2}f(Q_{2})f(Q_{0})Q_{0}=Q_{2}f\langle Q_{2})f(Q_{0})Q_{0}\cdot Q_{1}.$This implies that
As in the proof of Claim 2.5, we see that

$Q_{2}f(Q_{2})f(Q_{0})Q_{0}=Q_{2}V_{2}Q_{\theta(2)}V_{2}^{-1}V_{0}Q_{\theta(0)}V_{0}^{-1}Q_{0}$$Q_{2}V_{2}Q_{\theta(2)}V_{2}^{-1}V_{0}Q_{\theta(0)}V_{0}^{-\lambda}Q_{0}=1$

or

commutes with

$Q_{1}.$$Q_{1}.$

Since the word length of the left hand side of the above identity is even, we have
. By the abelianization of this identity, we have$Q_{2}V_{2}Q_{\theta(2\rangle}V_{2}^{-1}V_{0}Q_{\theta(0\rangle}V_{く J}^{-1}Q_{0}=1$$Q_{2}Q_{\theta(2)}Q_{\theta(0)}Q_{0}=1.$. Hence must be a permutation, $9(2)\in\{0$ , 2 andThis implies thatand since maps
andon the set , 1, 2 such that $\theta(1)=1$ . Since preserves$\theta=id$
to if
composingpostby, we may assume
tois a reduced word
necessary. Hence $f(Q_{s})=W_{s}Q_{s}W_{s}^{-1}$ for each $s=0$ , 1, 2, wherefromiswhose terminal letter diﬀerent
is a trivial word foris a trivial word,
Step 3. We show that $f=(\sigma^{2})^{n+1}$ . If
is a non-trivial
any $j=0$ , 2 by Claim 2.5, and therefore $f=id$ . So we assume thatthe terminal
that
,
we
assumefrom
diﬀerent
isword. Since the terminal letter of. (The other case is treated by a parallel argument.) Then the conditionisletter of(2) $-(i)$ or (2) $-(ii)$ in Claim 2.5 holds for $j=0$ , and the condition (2) $-(i)$ or (2) $-(iii)$ in
is odd or evencontained
Claim 2.5 holds for $j=2$ . Note that the number of$-(i)$
in Claim 2.5 holds or not. If the number ofaccording to whether the condition (2)is odd, then the condition (2) $-(i)$ in Cla\’im 2.5 holds for each $j=0$ , 2. In
contained
and , a contradiction. Hence the number ofisparticular, the initial letter ofis even. Then the condition (2) $-(ii)$ in Cla\’im 2.5 holds for $j=0$ , and thecontained
andcondition (2) $-(iii)$ in Claim 2.5 holds for $j=2$ , namely, we have
implies.
This. Thus we seeis equal to $(l-i+1)$ -th letter ofthat i-th letter of$l=l(W_{0}Q_{0}W_{0}^{-1}Q_{0}Q_{2})$ .
for some $n\in N$ , and therefore
Hence
where
. Thus we see
and
$\{0$

$(Q_{0}, Q_{1}, Q_{2})$$\theta$$\theta(0)\neq\theta(2)$$\}$$\theta(0)$$K_{0}$$\sigma$$\}$$K_{2}$$\sigma$$(Q_{2}, Q_{1}^{Q_{2}Q_{0}}, Q_{0}^{Q_{2}})$$\sigma$$f$$W_{8}$$Q_{8}.$$W_{j}$$W_{1}$$W_{1}$$Q_{1}$$W_{1}$$W_{1}$$Q_{0}$$Q_{1}$$W_{1}$$Q_{1}$$W_{i}$$W_{1}$$Q_{1}$$Q_{2}$$Q_{0}$$W_{1}$$W_{1}\equiv W_{0}Q_{0}W_{0}^{-1}Q_{0}$$W_{0}Q_{0}W_{0}^{-1}Q_{0}Q_{2}\equiv W_{1}Q_{2}\equiv W_{2}Q_{2}W_{2}^{-1}$$W_{1}Q_{2}\equiv W_{2}Q_{2}W_{2}^{-1}$$W_{0}Q_{0}W_{0}^{-1}Q_{0}Q_{2},$$W_{0}Q_{0}W_{0}^{-1}Q_{0}Q_{2}$$W_{0}\equiv(Q_{2}Q_{0})^{n}Q_{2}$$W_{1}\equiv(Q_{2}Q_{0})^{2(n+1)}$$W_{2}\equiv(Q_{2}Q_{0})^{n+1}$$f(Q_{0}\rangle=Q_{0}^{(Q_{2}Q_{0})^{n+1}}, f\langle Q_{1})=Q_{1}^{(Q_{2}Q_{0})^{2(n+1\rangle}}$and $f(Q_{2})=Q_{2}^{(Q_{2}Q_{0})^{n+1}}$

On the other hand,. Hence we obtain the desired result.

$(\sigma^{2}(Q_{0}), \sigma^{2}(Q_{1}), \sigma^{2}(Q_{2}))=(Q_{0}^{Q_{2}Q_{0}}, Q_{1}^{(Q_{2}Q_{0})^{2}}, Q_{2}^{Q_{2}Q_{0}})$. Thus we have$\zeta$$f=(\sigma^{2})^{n+1}$]

Remark 2.6. It should be noted that the proof of Proposition 2.2 does not use the. Hence, in Definition 2.1, thecondition that $(f(Q_{0}), f(Q_{1}), f(Q_{2}))$ generates
is actually a consequence of thecondition that members of the triple generate3.6]).
[4,(cf.
Remarkconditionsother, the bi-infinite seof
Definition 2.7. For an elliptic generator tripleofgenerators
elliptic
sequencein Proposition 2.2(1.1) is called thequenceof(associated with
In preparation for the next section, we recall the definition of elliptic generators of thefundamental group of the quotient orbifold of the once punctured torus.
be the involution:
be the once punctured torus and let
Let
and
by
illustrated in Figure 2. Then we denote the quotient orbifold$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\pi_{1}(\mathcal{O}_{N_{2,1}}\rangle$$(Q_{0}, Q_{1}, Q_{2})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\{Q_{j}\}$$(Q_{0},$ $Q_{1},$$\Sigma_{1,1}$$Q_{2}$$\iota_{X_{1,1}}$$\Sigma_{1,1}arrow\Sigma_{1,1}$$\Sigma_{1,1}/\iota_{\Sigma_{1,1}}$$\mathcal{O}_{\Sigma_{1.1}}$

74
denote the covering projection fromimage of the inclusion
is regarded as a normal subgroup of$\Sigma_{1,1}$

toby. We identify
induced by the projectionof index 2,$\mathcal{O}_{\Sigma_{1,1}}$$p\Sigma_{1,1}$

with the$\pi_{1}(\Sigma_{1,1})$$\pi_{1}(\Sigma_{1,1})arrow\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$p\Sigma_{1,1}$. Then$\pi_{1}(\Sigma_{1,1})$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\pi_{1}(\Sigma_{1,1})=\langle X_{1}, X_{2}|-\rangle\triangleleft\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})=\langle P_{0_{\rangle}}P_{1}, P_{2}|P_{0}^{2}=P_{1}^{2}=P_{2}^{2}=1\rangle,$such that $X_{1}=P_{2}P_{1}$ and $X_{2}=P_{0}P_{1}$ . Set $K_{\Sigma_{1,1}}=[X_{1}, X_{2}]=X_{1}X_{2}X_{1}^{-1}X$ , $K=$
. Thenand $K$ are represented by the punctures ofandrespectively.
$(P_{0}P_{1}P_{2})^{-1}$$K_{\Sigma_{1,1}}$$\Sigma_{1,1}$

FIGURE 2. The involution$\iota_{\Sigma_{1,1}}$of$\mathcal{O}_{\Sigma_{1,1}},$$\Sigma_{1,1}$

Definition 2.8. An ordered tripleof elements ofis called an elliptic
generator triple ofif its members generateand satisfy
and $(P_{0}P_{1}P_{2})^{-1}=K$ . A member of an elliptic generator triple ofis called anelliptic generator of.
$(P_{0}, P_{1}, P_{2})$$\pi_{1}(\mathcal{O}_{\Sigma_{1.1}})$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$P_{0}^{2}=P_{1}^{2}=P_{2}^{2}=$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$1$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$

Definition 2.9. For an elliptic generator tripleof, let
be thebi-infinite sequence defined as follows (see [1, Proposition 2.1.6(1.1)] and [4, Proposition$(P_{0}, P_{1}, P_{2})$$\{P_{j}\}$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$3.3(1.1)])$ .

...,$P_{2}^{K^{-2}},$ $P_{0}^{K^{-1}},$ $P_{1}^{K^{-1}},$ $P_{2}^{K^{-1}},$$P_{0},$

We call the sequence$(P_{0},$ $P_{1},$$\{P_{j}\}$$P_{1_{\rangle}}P_{2},$$P_{0}^{K},$ $P_{1}^{K},$ $P_{2}^{K},$ $P_{0}^{K^{2}}$the sequence of elliptic generators of$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$,...

(associated with

$P_{2}$

3. COMMENSURABILITY
In this section, we prove the “converse” of [4, Theorem 5.1], namely, we give a condition
for a faithful type-preserving $PSL(2, \mathbb{C})$ -representation of
to be commensurable$7C_{1}(\Sigma_{1,1})$

with that of. We first introduce some notations and facts.
Letandbe the twice-punctured torus, the $(2, 2, 2, \infty)$ -orbifold (i.e.,the orbifold with underlying space a punctured sphere and with four cone points of coneangle ), the (2; 2, ]-orbifold (i.e., the orbifold with underlying space a disk and with acone point of cone angle and with a corner reflector of order 2 and a corner reflector oforder ) and the 2, 2, 2, -orbifold (i.e., the orbifold with underlying space a disk andwith three corner reflectors of order 2 and a corner reflector of order ), respectively.
Note thatis a quotient orbifold ofby an involution and that bothandare common quotient orbifolds of
andby involutions (see [4, Section2] for$\pi_{1}(N_{2,1})$$\Sigma_{1,2},$$\mathcal{O}_{\Sigma_{1,2}},$$\pi$$\mathcal{O}_{\alpha}$$\mathcal{O}_{\beta}$$\infty$$\pi$$\infty$$[$$\infty]$$\infty$$\mathcal{O}_{\Sigma_{1,2}}$$\Sigma_{1,2}$$\mathcal{O}_{\alpha}$$\mathcal{O}_{\beta}$$\mathcal{O}_{\Sigma_{1,1}}$$\mathcal{O}_{N_{2,1}}$

75
details). Their (orbifold fundamental groups have the following presentations:$\rangle$$\pi_{\lambda}(\Sigma_{1,2})=\langle Z_{1}, Z_{2}, Z_{3}|$$7r_{1}(\mathcal{O}_{\Sigma_{1,2}})=\langle R_{0}, R_{1}, R_{2}, R_{3}|R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=R_{3}^{2}=1\rangle,$$\pi_{1}(O_{\alpha})=\langle S_{0}, S_{1)}S_{2}|S_{0}^{2}=S_{1}^{2}=S_{2}^{2}=1,(S_{\lambda}S_{2})^{2}=1\rangle,$$\prime\kappa_{1}(\mathcal{O}_{\beta})=\langle T_{0},T_{1}, T_{2}, T_{3}|(T_{0}T_{1})^{2}=(T_{1}T_{2})^{2}=(T_{2}T_{3})^{2}=1T_{0}^{2}=T_{1}^{2}--T_{2}^{2}=T_{3}^{2}=1,\rangle\cdot$

Here the generators satisfy the following conditions:$Z_{1}=R_{0}R_{1}, Z_{2}=R_{2}R_{1}, Z_{3}=R_{1}R_{3}, K_{\Sigma_{1,2}}=K_{o_{\Sigma_{1,2}}}, K_{\Sigma_{1,2}}’=(K_{\overline{o}_{\Sigma_{1,2}}}^{1})^{R_{\theta}},$$P_{0}=S_{0}^{s_{2}}, P_{1}=S_{1}S_{2}, P_{2}=S_{0},$$Q_{0}=S_{0}^{s_{2}}, Q_{1}=S_{1}, Q_{2}=S_{0},$$P_{0}=T_{0}T_{\lambda}, P_{1}=T_{1}T_{2}\rangle P_{2}=T_{2}T_{3},$$Q_{0}=T_{1}T_{2}, Q_{1}=T_{3}^{\tau_{1}}, Q_{2}=T_{0}T_{1},$, which are represented
and
where $K_{8_{1,2}}=Z_{1}Z_{2}Z_{3},$(see Figure 3).
by the punctures ofandIn summary, we have the commutative diagram of double coverings as shown in Figure 3.
Every arrow represents a double covering (see [4, Section2] for details).
$K_{8_{1,2}}’=Z_{2}Z_{1}Z_{3}$$\Sigma_{\lambda,2}$$K_{O_{\Sigma}X,2}=R_{0}R_{1}R_{2}R_{3}$$\mathcal{O}_{\Sigma_{1,2}}$

FIGURE 3, a repreor
Definition 3.1. (1) For $F=\Sigma_{1,1},$$\pi_{1}(F)arrow
PSL(2,\mathbb{C})$it(equivalently,
ifisirreducibleittype-preserving:issentation$N_{2,1}\Sigma_{1,2},$$\rho$$\mathcal{O}_{\Sigma_{1,1}},$$\mathcal{O}_{N_{2,1}},$$\mathcal{O}_{\Sigma_{1,2}},$$\mathcal{O}_{\alpha}$$\mathcal{O}_{\beta}$

76) and sends peripheral elements to parabolic
does not have a common fixed point intransformations.
-representations and(2) Type-preserving
are equivalent if $i_{g}o\rho=p’,$determined bywhere is the inner automorphism, $i_{g}(h)=ghg^{-1}$ , of$\partial \mathbb{H}^{3}$$PSL(2, \mathbb{C})$$\rho’$$\rho$$PSL(2, \mathbb{C})$$i_{g}$$g.$In the above definition, if $F$ is an orbifold with reflector lines, an element of, wheresaid to be peripheral if it is (the image of) a peripheral element oforientation double covering of $F.$$\pi_{1}(\tilde{F})$isis the$\pi_{1}(F)$$\tilde{F}$-representations ofDefinition 3.2. Letandbe type-preserving(resp.
), respectively. The representations) and(resp.
and(resp.
) toareurable if there exist a double coveringfrom(resp.
(resp.
) such(resp.
) to) and a double covering
from
that
are equivalent, namely
for someand$g\in PSL(2, \mathbb{C})$ .
, without changing the equivalence class,withAfter replacing
the last identity can be replaced with the identity$\rho_{1}$$PSL(2, \mathbb{C})$$\rho_{2}$$\pi_{1}(N_{2,1})$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\rho_{1}$$commen\mathcal{S}$$\rho_{2}$$\Sigma_{1,2}$$p_{1}$$\mathcal{O}_{\Sigma_{1,1}}$$\Sigma_{1,2}$$p_{2}$$\rho_{1}o(p_{1})_{*}$$\pi_{1}(\Sigma_{1,1})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\Sigma_{1,1}$$\mathcal{O}_{\Sigma_{1,2}}$$N_{2,1}$$\mathcal{O}_{\Sigma_{1,2}}$$\mathcal{O}_{N_{2,1}}$$\rho_{1}o(p_{1})_{*}=i_{g}o\rho_{2}o(p_{2})_{*}$$\rho_{2}o(p_{2})_{*}$$i_{g}\circ\rho_{2}$$\rho_{2}$$\rho_{1}\circ(p_{1})_{*}=\rho_{2}\circ(p_{2})_{*}.$In this paper, we study the following problem which is

Problem 3.3. For a given type-preserving, when does there exist a type-preserving(resp.
) which is commensurable with ?

$PSL(2, \mathbb{C})$$a^{(}$

converse of [4, Problem 2.3].

-representation of-representation$\rho_{1}$$\pi_{1}(\Sigma_{1,1})$$PSL(2, \mathbb{C})$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}}))$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\rho_{2}$of(resp.

$\pi_{1}(N_{2,1})$$\rho_{1}$To answer this problem, we recall the definitions of complex probabilities of typepreserving representations of(see [1, Section 2] and [4, Section 4]
andfor details).
The following fact is well-known (cf. [5, Section 5.4] and [1, Proposition 2.2.2]).
$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$

Proposition 3.4. For, the following hold.
or(1) The restriction of any type-preserving
representation of$F=\Sigma_{1,1}$$N_{2,1}$$PSL(2, \mathbb{C})-$is type-preserving.
(2) Conversely, every type-preserving-representation ofunique type-preserving.
-representation of$\pi_{1}(\mathcal{O}_{F})$to$\pi_{1}(F)$$PSL(2, \mathbb{C})$$PSL(2, \mathbb{C})$$\pi_{1}(F)$extends to a$\pi_{1}(\mathcal{O}_{F})$By this proposition, the following are well-defined.
, the symbol
Definition 3.5. (1) Forordenotes the space of all
type-preserving.
-representations of(2) For $F=N_{2,1}$ or) denotes the space of(resp.
the symbol-representations
all type-preservingofsuch that $tr(\rho_{2}(K_{N_{2,1}}))=-2$
resp. $tr(\rho_{2}(K_{N_{2,1}}))=+2)$ .
$F=\Sigma_{1,1}$$PSL(2, \mathbb{C})$$\Omega(\Sigma_{1,1})$$\mathcal{O}_{\Sigma_{1,1}}$$\rho_{1}$$\mathcal{O}_{N_{2,1\rangle}}$$\pi_{1}(F)$$\Omega’(N_{2,1})$$\Omega(N_{2,1})$$PSL(2, \mathbb{C})$$\pi_{1}(F)$$\rho_{2}$$($, the isometries
Remark 3.6. For any
a common fixed point (see [3, Lemma 4.5(ii)]), and hence(see [3, Lemma 4.7]).
$\rho_{2}\in\Omega’(N_{2,1})$

Definition 3.7. (1) Let. Setof$\{P_{j}\}$$\rho_{1}$be an element of

and
have
is indiscrete or non-faithful$\rho_{2}(Q_{0}Q_{2})=\rho_{2}(Y_{2})$$\Omega(\Sigma_{1,1})$$\rho_{2}$$\rho_{2}(K_{N_{2,1}})$. Fix a sequence of elliptic generators$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$(x_{1}, x_{12}, x_{2})=(tr(p_{1}(X_{1})), tr(p_{1}(X_{1}X_{2})), tr(\rho_{1}(X_{2})))$,. Then we call the following
where $X_{1}=P_{2}P_{1}$ and $X_{2}=P_{0}P_{1}$ . Suppose that
triple, wherethe complex probability associated with$x_{1}x_{12}x_{2}\neq 0$$(a_{0}, a_{1}, a_{2})\in(\mathbb{C}^{*})^{3}$$\{\rho_{1}(P_{j})\}$$\mathbb{C}^{*}=$

77$\mathbb{C}-\{0\}.$$a_{0}= \frac{x_{1}}{x_{12}x_{2}}, a_{1}=\frac{x_{12}}{x_{2}x_{1}}, a_{2}=\frac{x_{2}}{x_{1}x_{12}}.$(2) Letbe an element of. Set$\rho_{2}$$\Omega(N_{2,1})$. Fix a sequence of elliptic generators$\{Q_{j}\}$of$\pi_{1}(\mathcal{O}_{N_{2,1}})$$(y_{1}, y_{12},y_{2})=(tr(\rho_{2}(Y_{1}))/i, tr(\rho_{2}(Y_{1}Y_{2}))/i, tr(p_{2}(Y_{2})))$,

where $Y_{1}=Q_{0}Q_{1}$ and $Y_{2}=Q_{0}Q_{2}$ . Set $y_{12}’=tr(p_{2}(Y_{1}Y_{2}^{-1}))/i=y_{1}y_{2}-y_{12}$ . Suppose thatthe complex probability. Then we call the following triple
associated with$(b_{0}, b_{1}, b_{2})\in(\mathbb{C}^{*})^{3}$$y_{1}y_{2}y_{12}’\neq 0$$\{p_{2}(Q_{j})\}.$$b_{0}+b_{1}+b_{2}=1$, where$b_{0}= \frac{y_{1}}{y_{2}y_{12}},$$b_{1}= \frac{4}{y_{1}y_{2}y_{12}}$Remark 3.8. (1) For any sequence of elliptic generators} with, the complex probal)ility (identity (see [1, Lemma 2.4.1 (1)] for details):$\{P_{j}\}$$a_{0}, a_{1}, a_{2})associate\langle$$\Omega(\Sigma_{1,1})$$b_{2}= \frac{y_{12}’}{y_{1}y_{2}}.$}ofand anysatisfics the following$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\{\rho_{1}(P_{j})\}$$\beta x\in$$a_{0}+a_{1}+a_{2}=1.$(2) For any sequence of elliptic generatorscomplex probability
associated with(see [4, Section 4] for details):$\{Q_{k}\}$$(b_{0}, b_{1}, b_{2})$

of

, the
and anysatisfies the following identity$\rho_{2}\in\Omega(N_{2,1})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\{p_{2}(Q_{j})\}$$b_{0}+b_{1}+b_{2}=1.$

We introduce the following proposition (cf. [1, Proposition 2.4.4] and4.8 and 4. Il$|4$, Propositions

Proposition 3.9. (1) For any triple
such that $a_{0}+a_{1}+c\iota_{2}=1$ and,is an element
there
any
sequenceofforof elliptic generators$\{p_{1}(P_{j})\}ts$
equal to.
such that the complex probability associated with
(2) For any triple
such that $b_{0}+b_{\lambda}+b_{2}=1$ and for any sequence, theresuch that the
an elementof elliptic generatorsofcomplex probability associated with.
is equal to$(a_{0},a_{\lambda}, a_{2})\in(\mathbb{C}^{*}\rangle^{3}$$\{P_{j}\}$$p_{1}\in\Omega(\Sigma_{1,1})$$\prime\kappa_{1}(\mathcal{O}_{\Sigma_{X,1}})$$(a_{0}, (x_{1}, a_{2})$$(b_{0}, b_{1}, b_{q})\in(\mathbb{C}^{*})^{3}$$\{Q_{j}\}$$p_{2}\in\Omega(N_{2,1})$$j_{S}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$(b_{J}, b_{1}, b_{2})$$\{\rho_{2}(Q_{j})\}$and let
Notation 3.10. (1) Let be an element of. Let be the automorphism ofgenerators of(cf. [1, Proposition 2.1.6] and [4, Proposition 3.3]):$\Omega(\Sigma_{1,1})$$\rho_{1}$$7r_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\{P_{j}\}$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\xi$

be a sequence of elliptic
given by the following$(\xi(P_{0}\rangle, \xi(P_{1}), \xi(P_{2}\rangle)=(P_{2}^{P_{1}}, P_{1}, P_{0}^{K}\rangle.$

If the complex probability associated with.
by
(2) Letbe an element ofand let. Let be the automorphism of$\{\rho_{1}(\xi^{k}(P_{j}))\}$

is well-defined, then we denote it$(a_{0}^{(k)}, a_{1}^{(k)}, a_{2}^{(k\rangle})$$\Omega(N_{2,1})$$\rho_{2}$$?r_{1}(\mathcal{O}_{N_{2,1}})$$\sigma$$\{Q_{j}\}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$be a sequence of elliptic generators ofgiven by Proposition 2.2(1.2), na1nely,$(\sigma(Q_{0}), a(Q_{1}), \sigma(Q_{2})\rangle=(Q_{2)}Q_{1}^{Q_{2}Q_{0}}, Q_{0}^{Q_{2}})$

If the complex probability associated withit by.

$\{\rho_{2}(\sigma^{k}(Q_{j}))\}$.

is well-defined, then we denote$(b_{0}^{(k)}, b_{1}^{(k)}, b_{2}^{(k)})$

The following lemma can be verified by simple calculation (cf. [1, Lemma 2.4.1] and[4, Lemmas 4.10 and 4.13

78
and letbe a sequence of elliptic. Suppose that the complex probability
associated. Then we have the following identities $(cf$is well-defined for any

Lemma 3.11. (1) Let
generators ofwith
Figure 4):$\rho_{1}$be an element of$\{P_{j}\}$$\Omega(\Sigma_{1,1})$$(a_{0}^{(k)}, a_{1}^{(k)}, a_{2}^{(k)})$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\{\rho_{1}(\xi^{k}(P_{j}))\}$$k\in \mathbb{Z}$$(k+1)$( )$a_{0} =1-a_{2} , a_{1}$$a_{2}^{(k+1)}=\underline{a_{2}^{(k)}a_{0}^{(k)}}$$k$$(k+1)= \frac{a_{1}^{(k)}a_{2}^{(k)}}{1-a_{2}^{(k)}},$’$1-a_{2}^{(k)}$$a_{0}^{(k-1)}= \frac{a_{2}^{(k)}a_{0}^{(k)}}{1-a_{0}^{(k)}},\dot{a}_{1}^{(k-1)}=\frac{a_{0}^{(k)}a_{1}^{(k)}}{1-a_{0}^{(k\rangle}}, a_{2}^{(k-1)}=1-a_{0}^{(k)}.$(2) Lettors of$\rho_{2}$be a sequence of elliptic generaand let
be an element of. Suppose that the complex probability
associated with. Then we have the following identities (cf.
is well-defined for any$\{Q_{j}\}$$\Omega(N_{2,1})$$(b_{0}^{(k)},b_{1}^{(k)},b_{2}^{(k)})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\{\rho_{2}(\sigma^{k}(Q_{j}))\}$

Figure 5):$k\in \mathbb{Z}$$b_{0}^{(k+1)}=1-b_{2}^{(k)}, b_{1}^{(k+1)}= \frac{b_{1}^{(k)}b_{2}^{(k)}}{1-b_{2}^{(k)}}, b_{2}^{(k+1)}=\frac{b_{2}^{(k)}b_{0}^{(k)}}{1-b_{2}^{(k)}},$$b_{0}^{(k-1)}= \frac{b_{2}^{(k)}b_{0}^{(k)}}{1-b_{0}^{(k)}}, b_{1}^{(k-1)}=\frac{b_{0}^{(k)}b_{1}^{(k)}}{1-b_{0}^{(k)}}, b_{2}^{(k-1)}=1-b_{0}^{(k)}.$FIGURE 4. Adjacent complex probabilities of$\rho_{1}\in\Omega(\Sigma_{1,1})$

FIGURE 5. Adjacent complex probabilities of$\rho_{2}\in\Omega(N_{2,1})$

Throughout this paper, we employ the following convention.

Convention 3.12. (1) For any element, we always assume thatsome element ofidentity is satisfied.

, after taking conjugate ofbyis normalized so that the following$\rho_{1}\in\Omega(\Sigma_{1,1})$$PSL(2, \mathbb{C})$$\rho_{1}$$\rho_{1}(K)=(\begin{array}{ll}l 10 1\end{array}).$$p_{1}$

79(2) For any element, we always assume thatsatisfied.

$p_{2}\in\Omega(N_{2,1})$$PSL(2, \mathbb{C})$, after taking conjugate ofby some element of
is normalized so that the following identities are$\rho_{2}$$\rho_{2}$$p_{2}(K_{0})=(\begin{array}{l}0i0-i\end{array}), \rho_{2}(K_{0})=(\begin{array}{ll}i -2i0 -i\end{array}).$Now we give a partial answer to Problem 3.3. By [4, Lemma 4.15], we may only consider the problem for the quotient orbifolds. Our partial answer to the commensurability
problem for representations of the fundamental groups of the orbifolds
andisgiven as follows.
$\mathcal{O}_{\Sigma_{1,1}}$$\mathcal{O}_{N_{2,1}}$

Theorem 3.13. Under Convention 3.12, the following hold:(1) Let. Suppose thatbe an element ofis faithful. Then the following
conditions are equivalent.
(i) There exists a faithful representation
which is commensurable with$\Omega(\Sigma_{1,1})$$p_{1}$$p_{1}$$p_{2}\in\Omega(N_{2,1}\rangle$$\rho_{1}.$(ii) There exist a sequence of elliptic generators$\{P_{j}\}$of$x_{\lambda}(\mathcal{O}_{\Sigma_{1.1}})$such that the complex probability
associated withfollowing identity under Notation 3.10(1) (cf. Figure 6):$(n_{0}, a_{1}, a_{2})$$(a_{0}^{く k_{0})}, a_{1}^{(k_{0})}, a_{2}^{(ko)})=(a_{2}, a_{1}, a_{0})$(iii) Therea sequence of elliptic generators$pkx$ probability
associatedidentities:$exi_{\mathcal{S}}ts$$\{P_{j}\}$$(\alpha)\langle(x_{0}^{(0)}, a_{1}^{(0)}, a_{2}^{(0)})=\langle a_{2},a_{1}, a_{0})$$k_{0}$

satisfies the.

ofsuch that the comone of the following$\gamma r_{1}(\mathcal{O}_{\Sigma_{1,1}})$$with \{\rho_{2}(P_{j})\}satisfie\mathcal{S}$$(a_{0}, a_{1}, a_{2}\rangle$

and an integer$\{p_{1}(P_{j})\}$,$(\beta)(a_{0}^{(1)}, a_{1\}}^{(1)}a_{2}^{(1)})=(a_{2}, (\iota_{1}, a_{0}\rangle.$(2) If the conditions in (1) hold, the representation
unique up to precomposition
by an automorphism ofpreserving(3) Moreover, the foltoutng hold:-representation ofextends to a type-preserving
if and only if$(iii)-(\alpha)$. Moreover, if these conditions are satisfied, thesatisfies the condition
extension is unique.
extends to a type-preserving-representation of
if and only if$(iii)-(\beta)$. Moreover, if these conditions are satisfied, thesatisfies the condition
extension is unique.
$p_{2}\dot{u}$$7r_{1}(\mathcal{O}_{N_{9,1}})$$(\alpha)\rho_{1}$$K_{N_{2.1}}.$$PSL(2, \mathbb{C})$$\pi_{1}(\mathcal{O}_{\alpha})$$PSL(2, \mathbb{C})$$7r_{1}(\mathcal{O}_{\beta})$$p_{1}$$(\beta)p_{1}$$\rho_{1}$FIGURE 6.

$(a_{0}^{(k_{0})}, a_{1}^{(k_{0})}, a_{2}^{(k_{0})})=(a_{2\}}a_{1}, a_{0})$80and the assertion (2) because theother assertions can be proved by an argument similar to [4, Theorem 5.1].
We first prove the implication (1) $-(i)\Rightarrow(1)-(ii)$ . Suppose that there exists a faithfulrepresentation
which is commensurable with . Then, by [4, Theorem$5.1(1)-(ii)]$ , there exist a sequence of elliptic generators
and an integerofsatisfies the
such that the complex probability
associated withfollowing identity under Notation 3.10(2):Proof We only show the implication (1)$-(i)\Rightarrow(1)-(ii)$$\rho_{2}\in\Omega(N_{2,1})$$\rho_{1}$$\{Q_{j}\}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$(b_{0}, b_{1}, b_{2})$$k_{0}$$\{\rho_{2}(Q_{j})\}$$(b_{0}^{(k_{0})}, b_{1}^{(k_{0})}, b_{2}^{(k_{0})})=(b_{2}, b_{1}, b_{0})$.

By Proposition 3.9(1), there is an element
such that the complex probability
is equal tofor some sequence of elliptic
associated with
. Moreover, we can prove that and are commensurable
generatorsof(see proof of the implication (1) $-(ii)\Rightarrow(1)-(i)$ in [4, Theorem 5.1] for details). Hence,
preserving $K$ such thatby [4, Theorem 5.1(2)], there is an automorphism of. Set $\{P_{j}\}=\{f(P_{j}’)\}$ . Thenis also a sequence of elliptic generators. Hence the complex probabilityofand
associated
3.11,
complexproba.
By
the
equal
with
to
Lemma
is
is equal to the complex probability
bilityassociated with
. Hence we haveassociated with$\rho_{1}’\in\Omega(\Sigma_{1,1})$$(a_{0}’, a_{1}’, a_{2}’)$$(b_{0}, b_{1}, b_{2})$$\{\rho_{1}’(P_{j}’)\}$$\{P_{j}’\}$$p_{1}’$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$f$$\rho_{2}$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$\{P_{j}\}$$\rho_{1}\circ f=\rho_{1}’$$(a_{0}, a_{1}, a_{2})$$\rho_{1}(P_{j})=p_{1}’(P_{j}’)$$\pi_{1}(\mathcal{O}_{\Sigma_{1,1}})$$(a_{0}’, a_{1}’, a_{2}’)=(b_{0}, b_{1}, b_{2})$$\{\rho_{1}(P_{j})\}$$(a_{0}^{(k_{0})}, a_{1}^{(k_{0})}, a_{2}^{(k_{0})})$$\{\rho_{1}(\xi^{k_{0}}(P_{j}))\}$$(b_{0}^{(k_{0})}, b_{1}^{(k_{0})}, b_{2}^{(k_{0})})$$\{\rho_{2}(\sigma^{k}(Q_{j}))\}$$(a_{0}^{(k_{0})}, a_{1}^{(k_{0})}, a_{2}^{(k_{0})})=(b_{0}^{(k_{0})}, b_{1}^{(k_{0})}, b_{2}^{(k_{0})})=(b_{2}, b_{1}, b_{0})=(a_{2}, a_{1}, a_{0})$.

andbe elements of
Next we prove the assertion (2). Let
such that they
are commensurable with . Then there exist double coverings :and:such that $\rho_{1}o(p_{1})_{*}=\rho_{2}o(p_{2})_{*}$ and. Note that there is a uniqueof
Pick an elliptic generator triple
up to equivalence which corresponds to the epimorphism
covering fromtodefined by the following formula (see [4, Section 2] for details):$\Omega(N_{2,1})$$\rho_{2}’$$\rho_{2}$$p_{1}$$p_{1}$$p_{2},p_{2}’$$\mathcal{O}_{\Sigma_{1,2}}arrow \mathcal{O}_{\Sigma_{1,1}}$$\rho_{1}o(p_{1})_{*}=\rho_{2}’\circ(p_{2}’)_{*}.$$\mathcal{O}_{\Sigma_{1,2}}arrow \mathcal{O}_{N_{2,1}}$$(Q_{0}, Q_{1}, Q_{2})$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$\mathcal{O}_{N_{2,1}}$$\mathcal{O}_{\Sigma_{1,2}}$$\phi_{2}:\pi_{1}(\mathcal{O}_{N_{2,1}})arrow \mathbb{Z}/2\mathbb{Z}$$\phi_{2}(Q_{j})=\{\begin{array}{l}0 if j=0 or 2,1 if j=1.\end{array}$Hence there is a self-homeomorphism
Set
Claim 3.14.

$(Q_{0}’, Q_{1}, Q_{2}’)$

Proof. Note that(1)
(2)

$Q_{0}’$

andof$Q_{2}$

Since
Hence we haveandsuch that $p_{2}’=gop_{2}$ , and$\mathcal{O}_{L_{1,2}^{\urcorner}}$and$Q_{2}’=(p_{2}’)_{*}\circ(p_{2})_{*}^{-1}(Q_{2})$is also an elliptic generator triple of$Q_{2}’$$(p_{2})_{*}(\pi_{1}(\mathcal{O}_{\Sigma_{1,2}}))arrow(p_{2}’)_{*}(\pi_{1}(\mathcal{O}_{\Sigma_{1,2}}))$$\rho_{1}o(p_{1})_{*}=\rho_{2}o(p_{2})_{*}$$\pi_{1}(\mathcal{O}_{N_{2,1}})$$Q_{2}\in$.
.

$\rho_{1}o(p_{1})_{*}=\rho_{2}’\circ(p_{2}’)_{*\rangle}$

is an isomorphism andwe have$\rho_{2}o(p_{2})_{*}=\rho_{2}’\circ(p_{2}’)_{*}.$$\rho_{2}(Q_{0})=\rho_{2}\circ(p_{2})_{*}((p_{2})_{*}^{-1}(Q_{0}))$by$=\rho_{2}’\circ(p_{2}’)_{*}((p_{2})_{*}^{-1}(Q_{0}))$$=\rho_{2}’(Q\’{o})$$Q_{0},$

have order 2, because:
have order 2.
and$(p_{2}’)_{*}\circ(p_{2})_{*}^{-1}$$Q_{0}$$g$$Q_{0}’=(p_{2}’)_{*}\circ(p_{2})_{*}^{-1}(Q_{0})$$(p_{2})_{*}(\pi_{1}(\mathcal{O}_{\Sigma_{1,2}}$by$\rho_{2}\circ(p_{2})_{*}=\rho_{2}’\circ(p_{2}’)_{*}$$Q_{0}’=(p_{2}’)_{*}o(p_{2})_{*}^{-1}(Q_{0})$.

81
Similarly, we have$\rho_{2}(Q_{2})=\rho_{2}’(Q_{2}’)$. Hence we haveby$\rho_{2}’(Q_{1}^{Q_{2}’}Q_{1}^{Q_{0}’})=p_{2}(Q_{1}^{Q_{2}}Q_{1}^{Qo})$

for $j=0$ , 2$p_{2}(Q_{j})=\rho_{2}’(Q_{j}’)$$=p_{2}(K_{N_{2,1}})$by$=p_{2}’(K_{N_{2,1}})$by Convention 3.12,$Q_{\lambda}^{Q_{2}}Q_{1}^{Q_{0}}=K_{N_{2,1}}$

Since is faithful, we have
. Thus, by Remark 2.6, the triple
is an elliptic generator triple of.
By this claim, there are elliptic generator triples
andsatisfying the following identity:$Q_{1}^{Q_{2}’}Q_{1}^{Q_{0}’}=K_{N_{2,1}}$$p_{2}’$$(Q_{0}’, Q_{1}, Q_{2}’)$$\square$$\eta r_{1}(\mathcal{O}_{N_{2,1}})$$(Q_{0}, Q_{1}, Q_{2})$$(Q_{0)}’Q_{1},$ $Q_{2}’\rangle$of$\gamma_{1}(\mathcal{O}_{N_{2,1}})$$(\rho_{2}(Q_{0}), \rho_{2}(Q_{1}), p_{2}(Q_{2}))=(p_{2}’(Q_{0}’), p_{2}’(Q_{1}), p_{2}’(Q_{2}’))$

By Proposition 2.2(2), there is an automorphism ofthat $fmai$) $s(Q_{0}, Q_{1}, Q_{2})$ to. Hence we have$f$$(Q_{0}’, Q_{1}, Q_{2}’)$$\pi_{1}(\mathcal{O}_{N_{2,1}})$.

preserving$\rho_{2}=\rho_{2}’\circ f.$$K_{N_{2,1}}$such$\square$

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[4] M. Fvrokawa, Commensurability between once-punctured torus groups and once-punctured Klein bottte groups, (in preparation).
[5] W. P. Thurston, The geometry and topology of three-manifolas. Electronic version 1.1-March 2002,
http: $//$ msri. $org/publications/books/gt3m/.$$\rangle$

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