SASAKI, TAKEHIKO

Osaka City University Repository

Sasaki, T.
Osaka J. Math.
33 (1996), 475-484
THE SLICE DETERMINED BY MODULI
EQUATION xy=2z IN THE DEFORMATION SPACE
OF ONCE PUNCTURED TORI
TAKEHIKO SASAKI
(Received February 9, 1995)
1. Introduction and statement of results
We recall some terminology and results in [2], [3] and [6]. Let A and B
be loxodromic elements of PSL(2, C\ that is, A and B are 2 x 2 complex matrices
with determinant 1 and their traces do not lie on the closed interval [-2,2]. By
an obvious isomorphism we identify PSL(2,C) with the Mobius transformations
group. Denote by G = <Λ,/?> the Mobius subgroup generated by A and
B. Let x, y and z be the traces of A, B and AB, respectively. The triple (x,y,z)
is called a moduli triple of G. We restrict ourselves to the case in which triple
(x,y,z) satisfies the moduli equation
Put G = < Λ > . Then0
(
x
/8,
x
/8,4) is a moduli triple of G0 satisfying (*), G0 is a Fuchsian group of the
first kind and Ω(G0)/G0 is a pair of once punctured tori, where Ω(G0) denotes
the region of discontinuity of G0. For each quasi-Fuchsian group G = (A,By such
that Ω(G)/G is a pair of once punctured tori, there is a quasiconformal mapping
/ of the extended plane such that A=fA0f~i and B=fB0f~l. Hence G is a
quasiconformal deformation of G0. The set of all such quasi-Fuchsian groups is
called a deformation space of once punctured tori and denoted by D(G0). Under
a normalization each triple satisfying (*) determines A and B uniquely
so that a group G = <Λ,β> of Z>(G0), too. We identify />(G0) with the subset
of C3 consisting of triples each of which satisfies (*) and determines a quasi-
Fuchsian group. We put
Then, by stability of quasi-Fuchsian groups, Z)(G0) is an open subset of Γ*. In [3]
the following is shown.
476 T. SASAKI
Theorem 1 ([3]). Let G = (A,By be a group generated by loxodromic elements
A and B satisfying (*). If triple (x,y,z) of G also satisfies
(1) x>2 and y>2,
then GeZ>(G0).
We put
Then Theorem 1 implies S c Z>(G0). We denote by S the closure of S in T*. Then
S\S consists of the following three sets:
|/+z2 + 4 = 2jz, y>2}
b2 = {(x,2,z)|x2 + z2 + 4 = 2xz, x> 2}
For these sets the following are noted in [6].
Theorem 2 ([6]). Let G be a group whose moduli triple lies on b1vb2 Then
G is a boundary group of D(G0). More precisely, G is a regular b-group.
Theorem 3 ([6]). Let G be a group whose moduli triple lies on b3. Then G
is a boundary group of D(G0). More precisely, G is a web group.
A regular b-group is a Kleinian group G having only one simply connected
invariant component Δ such that
2Area(Δ / G) = Area(Ω(G) / G),
where Area implies the hyperbolic area. A web group is a Kleinian group such
that each component subgroup is a quasi-Fuchsian group of the first kind. We
remark that the web group in Theorem 3 is not quasi-Fuchsian, but it has an
infinite number of components. By Theorems 2 and 3 we see that
S={(x,y,z)\x>2, 7>2}nZ>(G0),
so we shall call S the slice determined by (1).
Comparing with Theorems 1, 2 and 3, we shall investigate another slice S' of
D(G0) determined by moduli equation
(2) xy = 2z.
Explicitely,
SLICE DETERMINED BY xy = 2z 477
5" = {(xyj)\xy = 2z} n Z)(G0).
In comparison with Theorem 1 we prove the following.
Theorem 4. Let
Then E c 5".
The set E\E consists of the following three sets:
2z9 |*| = 2, \y\>2}nT*
= 2z, \x\>29 \y\ = 2}nT*
In contrast to Theorem 2 the following holds.
Theorem 5.
We also prove the following.
Theorem 6. Let G be a group whose moduli triple lies on e3. Then G is a
boundary group of D(G0).
In §2 we make a normalization of generators and then show as Theorems 7
and 8 that each group of the slice S" has a symmetric region of discontinuity and
a symmetirc fundamental domain with respect to the origin and that none of the
boundary groups of S' is a 6-group. In §3 we modify a criterion for discontinuity
of [4] and then prove Theorems 4 and 5. Lastly, we prove Theorem 6 in §4.
The author would like to thank the referee for careful reading and valuable
suggestions.
2. Normalization and symmetry
Let A and B be loxodromic elements of PSL(2,C) and put G=(A,By. We
assume that the moduli triple (x,y,z) of G satisfies (*) and (2). Firstly, we shall
normalize A and B. Conjugating by a Mόbius transformation, we normalize A
such that
and -
0 β 2 2
478 T. SASAKI
Note that αβ=l and β=\/u. We write 5 = ( 1 Then .x = α + β, y = a+d
\c d)
and z =
Proposition. Equality (2) w equivalent to a = d.
Proof. By the substitution of ;c = α + /?, >> = 0 + rf and z = &a + βd into (2),
equality (2) reduces to (a — β)(a — d) = Q. Since α^/J, we have a — d. Π
Substituting x = a. + β,y = 2a and z = (α + /?)« into (*), we have (α — β)2a2 = (α + /?)2.
We shall choose a sign of α such that α = (α + /?)/(α — /?). Conjugating by a Mobius
transformation having the same fixed points with that of A, which leaves A
invariant, we normalize B such that a = c. Then we have
™ * Λ(3) 5- , α = - - and y =
\fl a) α-/? a — β
Under this normalization we remark the following.
Theorem 7. If G lies on the slice S' determined by (2), then, under the
normalization above, Ω(G) is symmetric with respect to the origin. Furthermore,
there is a fundamental domain for G which is symmetric \vith respect to the origin.
Proof. By (3) we have B~\-z) = -B(z\ This implies that B~n(-z)= -Bn(z)
for any z and any integer n. We also have A"( — z)= —An(z). Let M be an element
of G not being the identity. Then
M=AmίBnίAm2B"2 ..-AmiBni
for some integers mp n^ (j =1,2, •••,/)• Set
By the successive use of the identities shown just above we obtain
M-(-Z)=-M(Z).
Let M be a loxodromic element of G and let z
v
 and z2 be the fixed points of
M. Then M~ is also a loxodromic element of G and equation M~( — z)= —M(z)
implies that —zl and — z2 are the fixed points of M~ . Since the limit set, Λ(G),
of G is the closure of the fixed points of loxodromic elements of G, we obtain
that Λ(G) is symmetric with respect to the origin, so is the region of discontinuity,
too. Since G is a quasi-Fuchsian group of the first kind, there is a fundamental
domain consisting of two pieces. Let D+ be one of the two components of a
SLICE DETERMINED BY xy — lz 479
fundamental domain for G lying in a component of G. We put D~ = {z\— zeD + }.
Now it is not difficult to see that D+uD~ is a funamantal domain for G. By
construction, it is symmetric with respect to the origin. Π
It is shown in [1] that each boundary group of any Bers slice has just one
simply connected invariant component. Such a group is called a 6-group. In
contrast to the Bers slices we have the following.
Theorem 8. None of the boundary groups of S' is a b-group.
Proof. Assume that G is a group of S'\S' and is a ft-group, that is, G has
just one simply connected invariant component. Let Δ be the simply connected
invariant component of G. Let D+ be the piece of a fundamental domain for G
lying in Δ. By Theorem 7 we see that the set D~ = {z\— zεD + } is a subset of
Ω(G). If D~ lies in Δ, then there is a curve Cc: Δ connecting a point peD+ to
—peD~. Then the curve symmetric to C with respect to the origin also lies in
Δ and connects p to —p. Then the closed curve CuC~ c Δ separates 0 from
oo. Since 0 and oo lie on Λ(G), this contradicts our assumption that Δ is simply
connected. Hence D~ lies in another component, say Δ'. By symmetry, Δ' is
also an invariant component of G, a contradiction. Π
3. Proof of Theorems 4 and 5
We recall a sufficient condition for G=<y4,l?> to be Kleinian.
Theorem 9 ([4]). Let G = <Λ,£> be a subgroup of PSL(2,C) generated by
loxodromic elements A=( I and B = { ], bcj^O and satisfying (*). If, for
\0 βj \c d)
each integer n, the inequality
holds, then G is quasi-Fuchsίan and represents a pair of once punctured tori.
REMARK. The moduli equation (*) is equvalent to the trace equation
tr(ABA~iB~i)=-2 (see, for example, [2] or Lemma 3 in [5]). Since (*) is
symmetric with respect to x and y or since tr(ABA ~ VB~ *) = tr(BAB~ 1A ~ *), Theorem
9 is true even if we interchange the mormalizations of A and B.
Under the moduli equation (2), Theorem 9 reduces to the following.
Theorem 10. Let G = (A,By be as in Theorem 9 and assume that G also
480 T. SASAKI
satisfies (2). If, for each positive integer n, the inequality
|α" + /Π \
Λ
-β\
holds, then G is quasi- Fuchsian.
Proof. By Proposition in §2 we see that (4) reduces to (5). We shall show
that (5) holds for each integer n whenever (5) holds for each positive integer
n. Since 0=1 /α, we have |α1 + |/?Ί=Φ~Ί + lβ~Ί and \an + βn\ = \(*-n + β-n\. Hence
(5) holds for each negative integer n whenever it holds for each positive integer n.
To show (5) holds for n = Q observe that, for n = 0 and « = 1, (5) is equivalent to
|α-/?|<|α| + |/?| and to |α-]8|<|α + jS|, respectively. Since |α + j8|<|α| + |)8|, we see
that (5) holds for « = 0 whenever it holds for n = ί . Thus we have Theorem 10
by Theorem 9. Π
Now, we shall check (5) of Theorem 10 for each positive integer «, that will
give us proofs of Theorems 4 and 5. By Remark under Theorem 9 we may assume
that (xyj)eEvel9 so \x\>2 and \y\>2.
For w = l, (5) reduces to
By (3) we see that [y| = |2(α + j8)/(α — β)\ and so condition \y\>2 implies
|α-0|<|α + j8|. Hence (5) holds for / ι=l .
In order to treat the case «>2, we shall put
2
and v =
where u. = reiB. By the inequalities |α — /?|<|α + /?|, which is shown just above, and
we have
(6) 0<ι;<l and u>4-2v.
For n = 2, in polar coordinate oc = rew, (5) reduces to
Using the equalities r4 + r~4 = u2 — 2 and cos40 = 2u2 — 1, one shows that it is
equivalent to
Inequalities (6) imply that the left hand side is not smaller than
SLICE DETERMINED BY xy = 2z 481
and that it is greater than 4, so (5) holds for n = 2.
For « = 3, in polar coordinate, (5) reduces to
r
6
-f r~
Making use of the equalities rβ + r~6 = u3 — 3w and cos60 = 4y3 — 3t;, we have
M4-2
-6t; u-2v
A calculation shows that this is equivalent to
(7) f(u,v) = (
Since
df(u,v)
du
ι;-2t;3)>3w2-10>0,
f(u,v) is an increasing function of u. Hence by (6), in order to show (7), it suffices
to show that /(4 — 2ι>, t?) > 0. A calculation shows that
/(4-2u,t;) = 4(2-f(l-t;)(10-3ι;-12t;2H-4t;3)).
We put s(t?)=10-3f?-12i72 + 4!;3. If g(v)>0, then /(4 - 2v9v) > 0. If g(υ)<09 then
we have
2 + (1 - v)g(v) > 2+g(v) = 3(1 - v2)(4 - v) + v3 > 0.
Thus, in both cases we have /(4 — 2v, v)>Q so that (7) holds. Therefore we have
shown that (5) holds for n = 3.
For «>4, in polar coodinate, (5) reduces to
r
2n
 + r~2n + 2 r2 + r~2 + 2 u + 2
r
2
 + r~2-2cos20 u-2v '
Since
to show (5) it suffices to show
u
4
-4u2 '"u-2υ
482 T. SASAKI
or, equivalently,
(8) h(u9υ) = (1 + v)u4 - 4(14- v)u2 - 2u + 4v > 0.
Making use of inequalities u>2 and ι;>0, one obtains
dh(u9υ)_ 2_
du
Hence h(u,ύ) is an increasing function of u. There are two cases to consider.
Case I: υ<\. By (6) we see that, in order to show (8), it suffices to show
h(4 — 2v9v)>0. A calculation shows that
Case II: υ=l . Since r>l, we have u>2. Hence
h(u9ΐ) = 2(u - 2)(u3 + 2u2 -1) > 0.
Thus we have shown (8). Hence (5) holds for n>4.
Therefore we have shown that (5) holds for all positive integer n. Then
Theorem 10 implies Theorems 4 and 5. Π
4. Proof of Theorem 6
We shall prove the theorem in a sequence of lemmas. Let (x9y,z)ee3.
Lemma 1. x=y =
Proof. By (3) and \x\ = \y\ = 2 we have |α - β\ = 2. Hence we have |α + β\ = |α - β\
= 2 or, in polar coordinate a. = relθ,
It follows that cos 20 = 0 and r2 + r~2 = 4. We obtain 0=±π/4 and r =
+ 1)772. Hence
and =
Therefore we have x = a + j? =
 x
3 ± / and, by (3), y =
We choose the sign such that χ=y = < 3 + L The proof for the case
jc=j? =
 v
/3 — i is similar. By (3) we see that fl = (
v
/3 — 1)/2 and b=—i. Hence we
have
SLICE DETERMINED BY xy = 2z 483
(73 + iχi+O \ / 73-1
— ί
A= I I and B= \
Q (73-1X1-/) / I Λ/3-' Λ/3-ί
Lemma 2. Lef F=(AB9BA) and C={zeC\\z-(^/3 + 3i)/2\ = ^ /2}. Then F
is a Fuchsian group of the first kind with C as the invariant circle.
Proof. Since
AB. 2 2
2-j3-i 2-Jl-i
and
BA=\
putting
Γ=
 / (^3+ 1X1-0 -73 + ϊ \
4 2(1-(2+ 73)0 / '
we calculate and obtain
f\ 1
, and
0 I/ V4 1
It is well known and is also easy to see that the group TFT~l generated by
) and I 1 is a Fuchsian group of the first kind with the extended real
P I/ \4 I/
axis as the invariant circle. A calculation shows that 1\C) is identical with the
extended real axis. Π
Lemma 3. G is not quasi-Fuchsian.
484 T. SASAKI
Proof. Since F is a subgroup of G so that C c: Λ(G), since 0 is a fixed point
of the loxodromic element A so that OeΛ(G), and since 0 does not lie on C, it
is clear that G is not quasi-Fuchsian. Π
This lemma tells us that G does not lie in D(G0). Theorem 4 tells us that
GεE c §' a D(G0) so that G lies on D(G0). These two facts imply Theorem 6.
REMARK. An argument similar to one in [6] will show that G is a web
group with two symmetric non-equvalent components; one is bounded by C and
the other is symmetric to it with respect to the origin.
References
[1] L. Bers: On boundaries of Teichmuller spaces and on Kleinian groups, I, Ann. of Math. (2)
91 (1970), 570-600.
[2] L. Keen: Teichmuller spaces of punctured tori, I, Complex Variable 2 (1983), 199-211.
[3] L. Keen and C. Series: Pleating coordinates for the Maskit embedding of the Teichmuller space
of punctured tori, Topology 32 (1993), 719-749.
[4] T. Sasaki: A fundamental domain for some quasi-Fuchsian groups, Osaka J. Math. 27 (1990),
67-80.
[5] T. Sasaki: On Keen's moduli inequality in two generator Mόbius groups, J. Math. Soc. Japan
46 (1994), 663-680.
[6] T. Sasaki: A note on the boundary of a slice in deformation space of once punctured tori, Bull.
Yamagata Univ. (Nat. Sci.) 13 (1994), 193-198.
Department of Mathematics
Faculty of Education
Yamagata University
Kojirakawa-machi 1-4-12
Yamagata 990, Japan


The slice determined by moduli equation xy=2z in the deformation space of once punctured tori

Sasaki, Takehiko

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The slice determined by moduli equation xy=2z in the deformation space of once punctured tori

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