LINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF
PUNCTURED TORIYOHEI KOMORI AND YASUSHI YAMASHITA
Abstract. After ﬁxing a marking (V, W ) of a quasifuchsian punctured
torus group G, the complex length λV and the complex twist τV,W
parameters deﬁne a holomorphic embedding of the quasifuchsian space
QF of punctured tori into C2 . It is called the complex Fenchel-Nielsencoordinates of QF . For c ∈ C, let Qγ,c be the aﬃne subspace of C2
deﬁned by the linear equation λV = c. Then we can consider the linear
slice Lc of QF by QF ∩ Qγ,c which is a holomorphic slice of QF . For
any positive real value c, Lc always contains the so called Bers-Maskit
slice BM γ,c deﬁned in [4]. In this paper we show that if c is suﬃciently
small, then Lc coincides with BM γ,c whereas Lc has other components
besides BM γ,c when c is suﬃciently large. We also observe the scaling
property of Lc .

1. Introduction
The quasifuchsian space QF of once punctured tori can be embedded in
C2 = {(λ, τ )} by the complex Fenchel-Nielsen coordinates (c.f. [4, 8, 14, 15]).
By varying the complex twist τ and keeping the complex length λ being ﬁxed
as a positive real value c, we can deﬁne the linear slice Lc ⊂ C of QF. In
this paper we investigate the global properties of Lc realized in the complex
plane. To state our results, recall that Lc has a component containing theopen interval (2, +∞) which was studied in [4] and also in [6, 14]. In thispaper we call this component the standard component and the others nonstandard. We will show
Theorem 5.1. There exists some positive constant c0 such that for any csatisfying 0 < c < c0 , Lc coincides with the standard component.
Theorem 6.1. There exists some positive constant c1 such that for any csatisfying c > c1 , Lc contains non-standard components.
In section 7, we also consider the scaling property of Lc .
Corollary 7.3. Linear slice has an asymptotic scaling constant.
See Figure 1 for theorem 5.1 and Figure 2 and 3 for theorem 6.1 andcorollary 7.3. The parameters used in the ﬁgures are explained in 4.1.
Let us describe some historical background of our subject. A marked
quasifuchsian punctured torus group G is a free marked two generator discrete subgroup of P SL2 (C) such that the commutator of the generators is
1

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YOHEI KOMORI AND YASUSHI YAMASHITA

parabolic, and the regular set Ω consists of two non-empty simply connected
invariant components Ω± . Quasifuchsian space QF is the space of marked
quasifuchsian punctured torus groups modulo conjugation in P SL2 (C). The
convex core C/G has two boundary components ∂C ± /G each of which is aonce-punctured torus and admits an intrinsic hyperbolic structure making
it a pleated surface.
In their seminal paper [4] L. Keen and C. Series deﬁned the Bers-Maskit
slice BM µ,c for a ﬁxed measured lamination µ and c > 0, as the subset of
QF on which the bending lamination of ∂C − /G and µ belong to the same
projective class and the length of µ in ∂C − /G is equal to c. By using their
theory of pleating coordinates, they showed that BM µ,c is simply connected.
J. Parker and J. Parkkonen also studied these slices for the case that µ isa rational lamination (they call them the λ-slices), and considered a generalization of I. Kra’s plumbing construction and degeneration of BM µ,c to
the Maskit slice M (c.f. [14]). The ﬁrst author and J. Parkkonen further
studied BM µ,c ; they showed that the boundary of BM µ,c is a Jordan curve
which is cusped at a countable dense set of points (c.f. [6]). In this papaer
we would like to study the outside of BM µ,c in Lc and its scaling property.
This paper is organized as follows. In section 2 we will review the basic
notions of the quasifuchsian space QF of once punctured tori and its pleating
varieties following [4]. The complex Fenchel-Nielsen coordinates of QF will
be introduced in section 3, and we will deﬁne the main subject of this paper,
the linear slice Lc of QF in section 4. In sections 5 and 6 we will study
connected components of Lc and prove our main theorems. And in the lastsection we will observe the asymptotic self similarity of Lc .
We are grateful to Caroline Series for showing us the preprint [13] of
Otal, and Raquel Diaz for explaining us her idea in section 6. It was fruitfulfor us to discuss with them in Nara in January 2000; in practice this work
was almost done during their stay in Japan. We also wish to thank Hideki
Miyachi for enjoyable conversations with him on the topic in section 7, andKentaro Ito and Sara Maloni for telling their interests in our paper recently.
The ﬁrst author was partially supported by Grant-in-Aid for Scientiﬁc Research(C) (19540194), Ministry of Education, Science and Culture of Japan.
2. The quasifuchsian space QF and rational pleating varieties
2.1. Punctured torus groups and their pleating data.
2.1.1. Marking. Let S be an oriented once-punctured torus. Any pair ofsimple closed loops on S that intersect exactly once are free generators of
π1 (S). Let (α, β) be such an ordered pair of free generators, chosen so that
their commutator αβα−1 β −1 represents a positively oriented loop aroundthe puncture. The ordered pair (α, β) is called a marking.
2.1.2. QF and F. A punctured torus group is a discrete subgroup G ⊂P SL(2, C) that is the image of a faithful representation ρ of π1 (S) such that

LINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

3

the image of the loop around the puncture is parabolic. If (α, β) is a markingof S, and if A = ρ(α), B = ρ(β), then the commutator K = ABA−1 B −1 isparabolic and the ordered pair (A, B) = (ρ(α), ρ(β)) is called a marking ofG.
The group G is quasifuchsian if the regular set Ω(G) consists of two nonempty simply connected invariant components Ω± (G). The limit set Λ(G)is topologically a circle. Quasifuchsian space QF is the space of marked
quasifuchsian punctured torus groups modulo conjugation in P SL(2, C); it
has a holomorphic structure induced from the natural holomorphic structureof SL(2, C).
Let R(π1 (S)) be the set of P SL(2, C)-conjugacy classes of representations
ρ of π1 (S) such that the image of the loop around the puncture is parabolic.
Considering the compact open topology on R(π1 (S)), Minsky showed thatthe closure of QF in R(π1 (S)) is equal to D(π1 (S)), the set of punctured
torus groups modulo conjugation in P SL(2, C) (c.f. [12]).
Fuchsian space F is the subset of QF such that the components Ω± areround disks. It is canonically isomorphic to the Teichmüller space of marked
conformal structures on S.
The quotients Ω± (G)/G are punctured tori with conformal structures,and hence also with orientations inherited from Ĉ; we assume that theorientations of Ω+ (G)/G and S agree whereas those of Ω− (G)/G and S are
opposite.
A point q ∈ QF represents an equivalence class of marked groups in
P SL(2, C). We choose once and for all a triple of distinct points in Ĉ and
let G = G(q) denote the representative normalized so that the repelling andattracting ﬁxed points of A and the ﬁxed point of K are equal to the ﬁxed
triple points in Ĉ. If it is clear from the context, for readability, we suppress
the dependence on q.
2.2. Simple closed curves.
2.2.1. Enumeration. Denote by C(S), the set of free unoriented homotopy
classes of simple closed non-boundary parallel curves on S. As is well known,
this set can be naturally identiﬁed with Q̂ = Q∪∞. One way to see this is asfollows; Let L denote the integer lattice m+in, m, n ∈ Z ⊂ C. Topologically
S is the quotient of the punctured plane Ci = C − L by the natural actionof Gi = hÂ, B̂i i ≡ Z2 by the horizontal and vertical translations. A straightline of rational slope in C − L projects onto a simple closed curve on themarked punctured torus Si = Ci /Gi , and the projection of all lines of thesame rational slope with the same orientation are homotopic. We denotethe unoriented homotopy class obtained by projecting the line of slope −q/p
by [L(p/q)]. Relative to our choice of marking, [L(p/q)] is in the homology
class of α−p β q or αp β −q on Si , where α, β are projections of the horizontal
and vertical lines corresponding to Â, B̂ respectively. Setting 1/0 = ∞, we4

YOHEI KOMORI AND YASUSHI YAMASHITA

obtain that the map Q̂ → C(S) deﬁned by p/q 7→ [L(p/q)] which is welldeﬁned and bijective. The reason for the choice of convention that [L(p/q)]
corresponds to α−p β q , is the following; if we identify the Teichmüller space
Teich(S) of once punctured tori with the upper half plane H, then one caneasily compute that the boundary point p/q ∈ R̂ is the point where theextremal length of curves in the class [L(p/q)] has shrunk to zero.
2.2.2. Special word Wp/q . Suppose that ρ : π1 (S) → G ⊂ P SL2 (C) is aquasifuchsian punctured torus group, marked as usual by generators A =ρ(α), B = ρ(β). We denote the unique geodesic in the homotopy class of
ρ([L(p/q)]) in H3 /G by γp/q . In particular, for q ∈ QF, γp/q (q) represents
the corresponding geodesic in H3 /G(q).
For each p/q ∈ Q̂, we can ﬁnd an explicit word Wp/q in the marked
generators hα, βi of π1 (S) representing [L(p/q)] as follows. The words are
generated from the initial data
W0/1 = β, W1/0 = α−1
by the formulaW(p+r)/(q+s) = Wr/s Wp/q ,whenever p/q < r/s and ps − qr = −1. We denote by Wp/q (q) the corresponding special word in G(q).
2.3. Rational pleating varieties.
2.3.1. The pleating loci. We are now ready to discuss the convex hull boundary and the pleating locus. Let q ∈ QF and let G = G(q) be the corresponding marked quasifuchsian group with the regular set and the limit
set Ω(G), Λ(G) respectively. The 3-manifold H3 /G is homeomorphic to
S × (0, 1). The surfaces Ω(G)/G at inﬁnity form the boundary S × {0, 1}.
Let ∂C(G) be the boundary of the hyperbolic convex hull of Λ(G) in H3 ;it is clearly invariant under the action of G. The nearest point retractionΩ(G) → ∂C(G), deﬁned by mapping x ∈ Ω(G) to the unique point of contact
with ∂C(G) of the largest horoball in H3 centered at x with interior disjoint
from ∂C(G), can easily be modiﬁed to a G-equivariant homeomorphism.
We denote two connected components of ∂C(G) corresponding to Ω± (G)by ∂C ± (G) respectively. Thus each component ∂C ± (G)/G is topologicallya punctured torus. (In the special case in which G is Fuchsian, ∂C(G) is
a ﬂat plane whose two sides serve as a substitute for the two components∂C ± (G).)∂C ± (G)/G are pleated surfaces in H3 /G. More precisely, there are complete hyperbolic surfaces S ± , each homeomorphic to S, and maps f ± : S ± →H3 /G, such that every point in S ± is in the interior of some geodesic arc
which is mapped by f ± to a geodesic arc in H3 /G, and such that f ± induce
isomorphisms π1 (S) → G. Further, f ± are isometries onto their images
with the path metric induced from H3 (c.f. [16]). The bending or pleatingLINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

5

locus of ∂C ± (G)/G consists of those points of S ± contained in the interiorof one and only one geodesic arc which is mapped by f ± to a geodesic arc
in H3 /G. For G non-Fuchsian, the pleating loci are geodesic laminations,meaning that they are unions of pairwise disjoint simple closed geodesics
on S ± . We denote these laminations by | pl± (q)|, and usually identify sucha lamination with its image under f ± in H3 /G. A geodesic lamination iscalled rational if it consists entirely of closed leaves. We concentrate on thespecial case in which at least one of the pleating loci is rational in this sense.
Since the maximum number of pairwise disjoint simple closed curves on apunctured torus is one, such a lamination consists of a single simple closed
geodesic and is therefore of the form γp/q (q) for some p/q ∈ Q̂.
2.3.2. Rational pleating varieties and hyperbolic loci. Given p/q ∈ Q̂, we set
±
+
−
Pp/q
= {q ∈ QF : |pl± (q)| = γp/q (q)} and Pp/q = Pp/q
∪ Pp/q.

We call these sets the p/q-pleating varieties.
For any p/q ∈ Q̂, consider the trace Tr Wp/q of the special word Wp/q
associated to p/q deﬁned in 2.2.2. For q ∈ R(π1 (S)), we may consider thefunction Tp/q (q) = Tr Wp/q (q) as a rational function on R(π1 (S)). We deﬁne
the hyperbolic locus of Tp/q to be the set
Hp/q = {q ∈ R(π1 (S)) : Tp/q (q) ∈ R, |Tp/q (q)| > 2}.
Then the next result is fundamental (c.f. Proposition 22 in [4]).
Proposition 2.1. Pp/q ⊂ Hp/q . 
3. The complex Fenchel-Nielsen coordinates of QF
3.1. The complex length of a loxodromic element. The complex translation length λM ∈ C/2πiZ of M ∈ P SL(2, C) is given by the equation(1)± Tr M = 2 cosh λM /2,

where Tr M is the trace of M and we choose the sign so that <λM ≥ 0.
The complex length is invariant under conjugation by Möbius transformations and has the following geometric interpretation, provided M is notparabolic; let x be a point on the axis Ax M of M and let v̄ be a vector
normal to Ax M at x. Then <λM is the hyperbolic distance between x andM (x) and =λM is the angle mod 2π between M (v̄) and the parallel transport of v̄ to M (x), measured facing the attracting ﬁxed point M + of M . In
particular, if M is loxodromic then <λM > 0 and if M is purely hyperbolicthen in addition =λM ∈ 2πZ.
For q ∈ QF and γ ∈ C(S), we denote the element in the group G(q)representing γ by W (q). Because the trace is a conjugation invariant, thecomplex translation length λW (q) depends only on q and is independent of
the normalization of G(q). We want to deﬁne the complex length λγ (q) =λW (q) as a holomorphic function on QF with values in C, not C/2πiZ. To
do this, we choose the branch that is real valued on F. Since λγ 6= 0 on

6

YOHEI KOMORI AND YASUSHI YAMASHITAQF this choice uniquely determines a holomorphic function λγ : QF → C.
From now on, the term “complex length” will always refer to this branch.
We remark that <λγ (q) is the hyperbolic length of γ in H3 /G(q).
3.2. The complex Fenchel-Nielsen coordinates. The complex FenchelNielsen parameters were introduced in [8, 15] as a generalization to QF of theclassical Fenchel-Nielsen coordinates for Fuchsian groups. Here we brieﬂysummarize the main points as applied to the case of a punctured torus S.
Let G = hA, Bi be a marked quasifuchsian punctured torus group constructed from a pair of marked generators α, β of π1 (S) as described in 2.1.
The complex Fenchel-Nielsen coordinates (λA , τA,B ) for G = hA, Bi are obtained as follows; the parameter λA ∈ C/2πiZ is the complex translation
length of the generator A = ρ(α), or equivalently the complex length λα .
The twist parameter τA,B ∈ C/2πiZ measures the complex shear when theaxis Ax B −1 AB is identiﬁed with the axis Ax A by B. More precisely, if
the common perpendicular δ to Ax B −1 AB and Ax A meets these axes inpoints Y, X respectively, then <τA,B is the signed distance from X to B(Y )and =τA,B is the angle between δ and the parallel translate of B(δ) along
Ax A to X, measured facing towards the attracting ﬁxed point of A. The
conventions for measuring the signed distance and the angle are explainedin more detail in [3].
As shown in [14, 8, 3], given the parameters λA , τA,B , and ﬁxed a normalization, one can explicitly write down the matrix generators for a marked
two generator group G(λA , τA,B ) ⊂ P SL(2, C) in which the commutator[A, B] is parabolic as follows:()
cosh( λ2 )cosh( λ2 ) + 1
A=,cosh( λ2 ) − 1
cosh( λ2 )()
cosh( τ2 ) coth( λ4 )− sinh( τ2 )B=.
− sinh( τ2 )cosh( τ2 ) tanh( λ4 )This group may or may not be discrete. The matrix coeﬃcients of G
depend holomorphically on the parameters. The construction thus deﬁnes aholomorphic embedding of QF into a subset of C/2πiZ × C/2πiZ, in which
Fuchsian space F is identiﬁed with the image of R2 .
We want to lift this to an embedding into C2 . In 3.1 we discussed how
to lift the length function λA on QF to a holomorphic function on C. We
can similarly lift the twist parameter τA,B by specifying that it will be realvalued on F.
On F, the real valued parameters λA , τA,B reduce to the classical FenchelNielsen parameters lA , tA,B deﬁned by the above construction with λA thehyperbolic translation length lA of A and τA,B the twist parameter tA,B .
3.3. Rational quakebends and pleated surfaces. Clearly, the complex
Fenchel-Nielsen coordinates can be made relative to any marking V, W of G.
As described in detail in section 5 of [3], for ﬁxed λ ∈ R+ and τ ∈ C, theLINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

7complex Fenchel-Nielsen coordinates relative to V, W determines a pleatedsurface ψ : D → H3 . We review this process.
Write V for the set of all lifts of the simple closed curve γ corresponding to
V to D. Since γ is simple, V consists of a set of pairwise disjoint geodesics inD, namely the axis of V and all of its conjugates. These axes in V partition
D into pieces Pi . The map ψ is deﬁned in such a way that ψ is an isometryon each axis in V and on each closed piece Pi . Let x, y ∈ D − V and let β
be an oriented geodesic from x to y. Let P0 , P1 , ..., Pk be the pieces cut inorder by β, that meet along axes α1 , α2 , ..., αk ∈ V. Orient αi so that, in
D, Pi−1 lies to the left of αi and Pi to the right. Let Xi = β ∩ αi and letv̄i , w̄i be tangent vectors to ψ(Pi−1 ∩ β) and ψ(Pi ∩ β) at ψ(Xi ), oriented
in the direction inherited from β, so that v̄i points out of ψ(Pi−1 ) and w̄ipoints into ψ(Pi ). Let v̄i0 , w̄i0 be the projections of v̄i , w̄i onto the directions
orthogonal to the image of the bending axis at ψ(Xi ). Then =τ is the angle
from v̄i to w̄i measured facing along ψ(αi ). We embed D in the hyperbolicball model B3 of H3 as the equatorial plane such that the origins in D andin B3 coincide. We arrange that the axes of V and W V W −1 in G(λ, <τ ) lie
in the boundary of a piece P0 contained in D. We then choose ψ to be theidentity on P0 . We set Dγ (λ, τ ) = ψ(D) for the image of the pleated surfacein B3 . Then ψ induces the group isomorphism ψ∗ : G(λ, <τ ) → G(λ, τ )satisfying that ψ(g(z)) = ψ∗ (g)(ψ(z)) for g ∈ G(λ, <τ ) and z ∈ D.
The next proposition explains the relation between ψ and the bending
locus of ∂C − (G(q)) for q ∈ QF.
Proposition 3.1. For q ∈ QF, let (λ, τ ) be the complex Fenchel-Nielsencoordinates relative to marked generators (V, W ) of G(q), and let γ be thesimple closed curve corresponding to V . Assume that V is purely hyperbolicand let ψ : D → H3 be the pleated surface defined above. Then ψ is ahomeomorphism if and only if | pl− (q)| = γ.
Proof: First suppose that ψ is a homeomorphism. Then the boundary ofDγ (λ, τ ) is Λ(G(q)). Dγ (λ, τ ) divides H3 into two domains; one of whichis convex, hence contains C(G(q)). Moreover Dγ (λ, τ ) contains the axis of
V and all of its conjugates in G(q), and the complement of them consist of
totally geodesic pieces. Therefore it is one of the component of ∂C(G(q)) andfrom the bending construction in the above argument, it should be equal to∂C − (G(q)) (c.f. section 7.1 in [3]).
Next suppose that | pl− (q)| = γ. Then ∂C − (G(q)) consists of the axis of V
and all of its conjugates in G(q), and totally geodesic pieces. The stabilizer
subgroup of each totally geodesic piece is conjugate to the Fuchsian subgrouphV, W V W −1 i. Therefore we can construct the pleated surface satisfying thatDγ (λ, τ ) = ∂C − (G(q)), which implies that ψ is a homeomorphism. 
3.4. Rational quakebend planes. Let (λV , τV,W ) ⊂ C2 be the complex
Fenchel-Nielsen coordinates relative to marked generators (V, W ) of G, andlet γ be the simple closed curve corresponding to V . Assume that V ispurely hyperbolic and let c be the hyperbolic length of γ in H3 /G.

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YOHEI KOMORI AND YASUSHI YAMASHITA

We denote the slice {(c, τ ) ∈ C2 |τ ∈ C} by Qγ,c and call it the rationalquakebend plane.
Clearly, Qγ,c meets F along the earthquake path (c.f. [4]). The quakebend
parameter τ is a holomorphic coordinate on Qγ,c .
On Qγ,c , the quakebend parameter τ and Tr W are related byτc
Tr W = 2 coth( ) cosh( ).
22On Qγ,c , Tr W is a holomorphic function of τ , branched at τ = 2πin (n ∈ Z).
(see ﬁgure 5.1 in [14]). When Tr V is real, QF ∩ Qγ,c is contained in thestrip{τ ∈ C | − πi < =τ < πi}from the argument in 3.3. Tr W takes the right half strip{τ ∈ C | <τ > 0, −πi < =τ < πi}conformally onto the right half plane C+ minus the interval (0, 2 coth( 2c )]where the interval {τ ∈ C | <τ = 0, −1 < =τ < 1} in the imaginary axis
is folded at the origin by Tr W and its image is (0, 2 coth( 2c )]. We remarkthat QF ∩ Qγ,c is also periodic under the action of the Dehn twist (A, B) 7→(A, An B), and symmetric under the holomorphic involution τ 7→ −τ .
4. The linear slice Lc
4.1. Definition. For q ∈ R(π1 (S)), a marked group G(q) = hA, Bi modulo
conjugation in P SL(2, C) is uniquely determined by Tr A, Tr B and Tr AB.
In fact, ignoring marking, G(q) modulo conjugation in P SL(2, C) is determined only by Tr A and Tr B (more precisely, the pair (Tr A, Tr B) determines a marked group hA, Bi or hA, B −1 i modulo conjugation in P SL(2, C)).
As an application of the Jorgensen’s theory on the combinatorial structureof the Ford domain of a punctured torus group, there is an algorithm roughly
answering whether G(q) is a geometrically ﬁnite discrete group or not fromthe data (Tr A, Tr B) (c.f. [1]). Especially ﬁxing Tr A = c, then we can use
this algorithm to draw the picture of
Dc = {Tr B ∈ C+ | G(q) = hA, Bi is a geometrically ﬁnite discrete group}.
We call this set the discrete locus. Let Qγ,c be the rational quakebend planein the complex Fenchel-Nielsen coordinates relative to the corresponding
marked generators (A, B) of G where we assume that c = λA (q) is real.
The linear slice Lc in the right half plane C+ , which is the image of QF ∩Qγ,c under Tr B. Because QF is open in C2 in complex Fenchel-Nielsencoordinates and Tr B is a open map on Qγ,c , Lc is open in C+ . Then fromthe deﬁnition of Dc and Lc , Lc is a subset of the interior of Dc .
Proposition 4.1. The interior of Dc is equal to Lc .
Proof: It is enough to show that any point q0 of Dc , not contained in Lc isa boundary point of Dc . First suppose that G(q0 ) is not a free group. Thensome word, say g(q0 ) is trivial in G(q0 ). Then applying the Jorgensen’sLINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

9inequality for the subgroup H(q0 ) generated by g and K = [A, B], we cansee that if we take a small neighborhood U of q0 , for any point q of U
except q0 , H(q) is not discrete which means that G(q) is also indiscrete.
Therefore q0 is an isolated point of Dc . Next suppose that G(q0 ) is free,hence geometrically ﬁnite non-quasifuchsian punctured torus group. Thenit must be a cusp (c.f. [12]). Hence there is some word, say g(q) which isparabolic in G(q0 ). Since Tr g is a holomorphic function of Tr B, it is a open
map, hence there is a path in the Tr B-plane starting from q0 such that on
this path g is elliptic. Therefore this path is outside of Dc . This implies thatq0 is a boundary point of Dc . 
From this result, we can see Lc as the interior Dc and study them experimentally. Figures at the end of this paper drown by the second author
show computer-generated linear slices, revealing some global properties. The
black region corresponds to the discrete locus. In the ﬁrst picture, Tr A is
ﬁxed at 2 and Tr B ranges in the square of width 4 centered at Tr B = 2 sothat we see the familiar picture of the Maskit slice. By setting Tr A = 2.5,
we get the second picture. In Figure 2 and 3, the value of Tr A is ﬁxed at
8 and 100 respectively while changing the ranges of Tr B. The width of thesquares are 16, 32, 128 and 128, 2560, 12800 respectively. We can clearly
see the “rough self similarity” of the pictures between ﬁgures 3 and 5 andbetween ﬁgures 6 and 8, with which we will discuss in section 7.
4.2. Connected components of Lc .
Proposition 4.2. For any c > 0, Lc has a component containing an open
interval (2, +∞).
Proof: There exists a component in QF ∩ Qγ,c containing F ∩ Qγ,c the realline which is periodic under the action of the Dehn twist B 7→ An B andsymmetric under τ 7→ −τ (c.f. [4, 14]. see also [11]). Then its image under
Tr B is the required component. This component is called the BM-slice in [4] and also called the λ-slicein [14]. In this paper we call this component of Lc the standard component,
and call the other components the non-standard components if they exist.
Because of the existence of the standard component which contains thecritical value of Tr B, if there is a non-standard component, it is a conformal
image of a component of QF ∩ Qγ,c under the map Tr B. Therefore we canconsider that Lc describes the picture of QF ∩ Qγ,c . Next result shows that
topologically every component is a disk.
Proposition 4.3. Each component of Lc is simply connected.
Proof: This is a consequence of a result of McMullen [11] that QF is disk
convex in R(π1 (S)), that is, for any continuous map f : ∆ → R(π1 (S))
whose restriction to the unit disk ∆ is holomorphic, f (∂∆) ⊂ QF impliesf (∆) ⊂ QF. 

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YOHEI KOMORI AND YASUSHI YAMASHITA

Remark 4.4. (The Maskit slice)
If we consider the limiting case where c = 0, we can no longer consider thecomplex Fenchel-Nielsen coordinates. But by using Tr B we can realize thepart of the boundary of QF defined by the condition that A is parabolic. Thenthe standard component defined above corresponds to the so-called Maskit
slice M.
5. Non-existence of non-standard components
5.1. Otal’s result.
Theorem 5.1. There is some positive constant c0 such that for any c satisfying 0 < c < c0 , Lc coincides with the standard component.
This is an immediate consequence of the following result due to J. P. Otal [13].
Theorem 5.2. (c.f. corollaire 9.1 in [13])There exists a positive constant c0 such that for a marked quasifuchsian
punctured torus group G(q) and V ∈ G(q) representing a simple closed geodesic γ in H3 /G(q), if V is purely hyperbolic and the hyperbolic length λγ (q)of γ is less than c0 , then γ is a bending locus of ∂C(G(q)).
Following the proof of proposition 9 in [13], we will give a proof of theorem 5.2 to estimate c0 in the next subsection 5.2.
Suppose that γ is not the bending locus of ∂C(G(q)). Then the pleatedsurface ψ is not a homeomorphism by proposition 3.1. Let H be a Fuchsian
subgroup hV, W V W −1 i where (V, W ) is a marking of G(q). Denote thetotally geodesic plane whose boundary ∂D contains Λ(H) by D ⊂ H3 . Let
P be the convex hull of Λ(H) in D ⊂ H3 . Then g(P ) is the convex hull ofΛ(gHg −1 ) in g(D) ⊂ H3 . Now we have a following claim.
Proposition 5.3. If the pleated surface ψ is not a homeomorphism, thenthere exists g ∈ G(q) such that P and g(P ) intersect transversally in theaxis of V .
To show this proposition, we need two lemmas.
Lemma 5.4. For g ∈ G(q), gHg −1 ∩H is trivial or cyclic subgroup generated
by gV g −1 . 
Lemma 5.5. For g ∈ G(q), Λ(gHg −1 ) ∩ Λ(H) is empty or fixed points ofgV g −1 .
Proof: H and gHg −1 are Fuchsian subgroups of a quasifuchsian group G(q).
From a theorem of Suskind (see theorem 3.14 in [10]),Λ(gHg −1 ) ∩ Λ(H) = Λ(gHg −1 ∩ H).
Hence it concludes the proof by using lemma 5.4. 
Now we can show proposition 5.3. Since we assume that ψ is not ahomeomorphism, there exists g ∈ G such that the interior of P and theLINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

11

interior of g(P ) intersect transversally. Then from lemma 5.5, Λ(gHg −1 ) ∩Λ(H) is empty. Therefore the axis of V cuts g(P ) transversally. 
Now we assume that c0 is smaller than the Margulis constant. Then theinterior of g(P ) cut the Margulis tube T with radius r along the axis of V
transversally. Hence now we have a geodesic disk ∆ = T ∩ g(P ) on g(P ).
Lemma 5.6. The hyperbolic area of ∆ is bigger than 4π sinh2 (r/2).
Proof: If g(P ) intersects the axis of V orthogonally, then ∆ is a hyperbolic
disk of radius r, hence the hyperbolic area of it is 4π sinh2 (r/2). If g(P )intersects the axis of V not orthogonally, then ∆ contains a hyperbolic diskof radius r, hence the hyperbolic area of ∆ is bigger than 4π sinh2 (r/2). Now we can give a proof of theorem 5.2. By the Margulis lemma, ∆projects into the image of g(P ) in H3 /G(q) injectively, whereas the imageof g(P ) in H3 /G has its hyperbolic area 2π since it is the isometric image ofa punctured cylinder. Therefore if the hyperbolic length of γ in H3 /G(q) is
suﬃciently small, we can take a radius r of the Margulis tube T satisfying
4π sinh2 (r/2) ≥ 2π, which is a contradiction. This concludes the theorem.

5.2. A lower bound of c0 . Following [9], we have a formula of the radiusof a Margulis tube.
Proposition 5.7. (c.f. theorem in section 3 of [9])For q ∈ QF, assume that V ∈ G(q) representing a simple closed geodesic
γ in H3 /G(q), which√ is purely hyperbolic. If the hyperbolic length λγ of γ
satisfies cosh λγ < 2, then there is a Margulis tube with radius r satisfying√
3 − 2 cosh λγ
12sinh (r) = (− 1).
2 cosh λγ − 1
The inequality 4π sinh2 (r/2) ≥ 2π and the above formula give us a lower
bound of c0 .
√2Corollary 5.8. cosh−1 48+5
≈ 0.493 ≤ c0 . 
49

6. Existence of non-standard components
Theorem 6.1. There is some positive constant c1 such that for any c satisfying c > c1 , Lc contains non-standard components.
To prove this theorem, we use the Earle slice E of punctured tori studied
in [5, 7]. This idea is due to Raquel Diaz. We review notations of E (c.f. [7]).
The Earle slice E of QF is the set of G(q) = hA, Bi satisfying the following
symmetry; there exists an elliptic element of order 2 such that EAE = B.
Then E is a holomorphic slice of QF and considering the conformal structure of Ω+ (G(q))/G(q), it is naturally isomorphic to the Teichmüller space

12

YOHEI KOMORI AND YASUSHI YAMASHITA

of punctured tori. Any element of E can be represented by the following
matrices in SL(2, C) of the form A = Ad , B = Bd , d ∈ C − {0}, where
(
Ad =

d2 +1
d
2d2 +1dd3
2d2 +1d)(, Bd =

d2 +1
d
2− 2d d+1

3− 2dd2 +1d).

The complex parameter d gives a holomorphic embedding of E into the righthalf plane C+ and we assume that E is embedded in C+ . Then E contains
the positive real line R+ which is the Fuchsian locus E ∩ F of E. Put+0C+
d0 = {d ∈ C |< d > d }. To show theorem 6.1, we need lemmas.
Lemma 6.2. There is a positive constant d0 such that for any d0 > d0 , thehyperbolic locus H2/1 of T2/1 (d) = Tr W2/1 (d) satisfies(H2/1 − R+ ) ∩ C+
d0 6= ∅.
Proof: We remark that W2/1 = A−2 B. Then we can check our claim bydirect calculation. 
Lemma 6.3. There is a positive constant d1 such that the 2/1-pleating
variety P2/1 satisfies
P2/1 ∩ E ⊂ C+ − C+d1 .
Proof: In [7], it is shown that Pp/q ∩ E is equal to two components of
Hp/q − R+ terminating to the unique critical point of Tr Wp/q on R+ (c.f.
theorem 5.1 in [7]). Then we can check our claim by direct calculation. 
Lemma 6.4. (c.f. [5]) There is a positive constant d2 such thatC+
d2 ⊂ E.

Now we can prove theorem 6.1. There is a positive constant c1 suchthat for any c > c1 , there is d ∈ E such that the word A−2 B is purely
hyperbolic and λW2/1 (d) = c, but d is not contained in P2/1 . This concludes
the theorem. Remark: To estimate c1 , we need to know the size of the round disk
contained in E tangent to the boundary ∂E of E at the origin (see [5]).
Comparing with the results in section 5 and 6, we have the following
conjecture supported by numerical experiences by the second author.
Conjecture 6.5. There exists a unique c0 such that Lc coincides with thestandard component for any c ≤ c0 , while Lc contains infinitely many nonstandard components for any c > c0 .

LINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

13

7. Scaling property of Lc
In the ﬁnal section we will study the self-similar phenomena of Lc which
we can observe from ﬁgures of Lc in this paper. First we remark that Lc
has analytic automorphisms coming from Dehn twists.
Proposition 7.1. (A, B) ∈ Lc implies (A, An B) ∈ Lc for all n ∈ Z.
Proof: The automorphism of G deﬁned by (A, B) 7→ (A, An B) is a Dehn
twist along A which preserves QF and Qγ,c . 
Next result is easy to prove, but it induces the asymptotic self similarity
of Lc .
Proposition 7.2.
lim√
Tr An Bn→∞ Tr An−1 B= Tr A ·1+

1 − ( Tr2A )22,

which is the attractive fixed point of the map Tr A − z1 .
Proof: The following trace identity is well known:
Tr An B = Tr A · Tr An−1 B − Tr An−2 B.
n

A B
Divide both sides by (Tr A)n and put xn := Tr(Tr A)n . Then we havexn = xn−1 −1
xn−2 .
(Tr A)2nMoreover put yn := xxn−1, thenyn = 1 −1
1.
2(Tr A) yn−1

Finally put zn := Tr A · yn , thenzn = Tr A −

1
zn−1

.

Since A is purely hyperbolic, the linear fractional transformation1zis also purely hyperbolic, hence all points besides the repellingﬁxed point√
w = T rA −

1+

1−(

2

)2

Tr A
of A converge to the attracting ﬁxed point of A, Tr A ·. From2n
Tr A B
the above arguments, zn = Tr An−1 B converges to this point. 

Corollary 7.3. Linear slice has an asymptotic scaling constant Tr A·1+√
1−( Tr2A )2
.
2

14

YOHEI KOMORI AND YASUSHI YAMASHITA

Figure 1. Maskit slice (left) and Tr A = 2.5 slice (right)

Figure 2. Tr A = 8 linear slices with ranges 16(left), 32(center), 128(right)

Figure 3. Tr A = 100 linear slices with ranges 128(left),2560(center), 12800(right)
Remark 7.4. When A tends to be parabolic,
Tr An Blim
= 1,n→∞ Tr An−1 B
which relates to the fact that the Maskit slice is invariant under translations.
Remark 7.5. Even if A is loxodromic, Lc has this scaling property. Hence
we can also see that the figure 10 in [11] also has such scaling property.

LINEAR SLICES OF THE QUASIFUCHSIAN SPACE OF PUNCTURED TORI

15

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Osaka City University Advanced Mathematical Institute and Departmentof Mathematics, Osaka City University, 558-8585, Osaka, Japan
E-mail address: komori@sci.osaka-cu.ac.jp
Department of Information and Computer Sciences, Nara Women’s University, 630-8506 Nara, Japan
E-mail address: yamasita@ics.nara-wu.ac.jp