FUJII, MICHIHIKO

KOJIMA, SADAYOSHI

Osaka City University Repository

Fujii, M. and Kojima, S.
Osaka J. Math.
34(1997), 541-551
FLEXIBLE BOUNDARIES IN DEFORMATIONS
OF HYPERBOLIC 3-MANIFOLDS
MICHIHIKO FUJII and SADAYOSHI KOJIMA
(Received March 11, 1996)
0. Introduction
Let M be a cusped hyperbolic 3-manifold with non-empty geodesic boundary.
A small Dehn filling deformation of M on the cusps can be performed so that the
boundary is kept to be geodesic. Then assigning to each deformation a hyperbolic
structure on the boundary, we get a map BM from the space of such deformations
to the Teichmϋller space of dM. More precise argument for this fact will be given
i n § l .
Motivated by the conjectures posed in Cooper-Long [1] and Kapovich [5],
Neumann and Reid [7] discovered many examples of M such that BM is a constant
map. Fujii [3] also obtained another concrete family of small deformations for
some M such that BM maps his family to a constant structure. These examples at
first contrasted with our naive intuition that the deformation of hyperbolic structure
affects everywhere. But the fact itself would not be too surprising once we realized
that the dimension of the source can be bigger than that of the target and in that
case BM can never be injective. A more pertinent problem to set up for the moment
would be what the map BM looks like. In fact, we have known only a little about
it so far.
Under the circumstances above, it would be worth finding examples for which
we can convert this rather difficult problem to something we can do by hand. In
this paper, we will construct infinitely many one-cusped examples of M so that BM
is a local embedding at the complete structure. The polyhedral construction will
be discussed in §2. Then by using its polyhedral structure, we will compute the
derivative of BM at the complete structure by hand in the later sections.
The authors are indebted to Professor T. Soma for the argument in §1, and we
would like to thank him.
1. The map BM
We briefly review Dehn filling deformations of cusped hyperbolic 3-manifolds.
Let TV be a noncompact, orientable, complete hyperbolic 3-manifold of finite vol-
ume, and PO : πι(N} —> PSL2(C) its holonomy representation. According to
542 M. FUJII AND S. KOJIMA
Thurston [9] (cf. [2]), p0 has a lift p0 : πι(N) -> SL2(C). Since SL2(C) is an
algebraic set, the space of representations Hom(πι(TV), SL2(C)) is also an algebraic
set. To each representation p, associated is its character χp. Culler and Shalen [2]
showed that the irreducible component of Hom(πι(TV), SL2(C)) containing p0 is
mapped by this correspondence onto the space of characters X, which is an affine
variety. The preimage of a character χp near χpo consists of representations conju-
gate to p. Thus a small neighborhood of χpo in X is bijectively identified with the
set of conjugacy classes of SL2(C)-representations near the conjugacy class of p0.
This small neighborhood is also identified with the set of conjugacy classes
of PSL2(C)-representations near the conjugacy class of p0. This is because any
path in Hom(τrι(TV),PSL2(C)) based at p0 lifts to a path in Hom(πι(TV), SL2(C))
based at p0 and hence the covering projection SL2(C) —» PSL2(C) induces a local
homeomorphism of X at χpo. Through this correspondence, we will identify the
set of equivalence classes of PSL2(C)-representations near the conjugacy class of p0
with a small neighborhood of χpo in X.
It has been known by Thurston [9] and Neumann-Zagier [8] that the complex
dimension of X is equal to the number of cusps of TV and that the character of po
is a smooth point. If we choose a set of meridional elements {rrij} for all cusps of
TV, then the traces of these elements turn out to be a local coordinate of X at the
conjugacy class of po
Thurston originally introduced another parameter which is the set of complex
lengths of the p(rrij)'s. The complex length of p(nij) is well-defined with sign by
orienting the axis of p(πij). Their squares turn out to be a local coordinate at χpo
in X. To each representation p near po, he assigned a hyperbolic manifold TVp,
see [9]. It can be interpreted as a small deformation of TV and is called a Dehn
filling deformation of TV. The topological type of the deformation may be different
from the original. Thurston's parameter can be converted to the set of pairs of real
numbers which is a generalization of Dehn surgery coefficients in the classical knot
theory.
Now, suppose that M is an orientable complete hyperbolic 3-manifold of finite
volume with both cusps and compact geodesic boundaries. Let DM be the double of
M along the boundary and ρ0 be a holonomy representation of DM. The manifold
DM admits an obvious involution r switching the sides.
DEFINITION. Fix a set of meridians πij closed under r, and choose a small
neighborhood U of χpo so that the traces of the ra/s become a local coordinate
of the space of characters at χpo. Let DM be a diagonal set in U fixed by an
involution on U which is induced by r. It is a smooth submanifold of real dimension
- #{cusps of DM}.
Lemma 1. The restriction of a representation p near p0 whose conjugacy class
FLEXIBLE BOUNDARIES 543
is in DM to πι(<90M) is fuchsίan (Le., a discrete faithful representation to SL2(R)),
where <%M is a component of the boundary dM.
Proof. Choose p as above, and denote the associated Dehn filling deformation
by DMP. There is also a topological involution rp on DMp switching sides. Then
since p is close to p0? τp induces a nearby deformation of DM and the induced
structure is the same as DMp by the local parametrization around the complete
structure p0 (see [8] [9]). Hence τp can be deformed to an isometry Tp by a tiny
isotopy. We want to show that Tp is an involution fixing a geodesic surface isotopic
to dM.
There is a dense subset Y in VM near [p0], corresponding to a set of rational
rays in the Dehn filling coefficient space, so that the deformation DMy for y G Y is
a cone manifold with cone angle more than 2π. Then by changing the metric near
the cusps as in [4], DMy can be modified to a compact negatively curved manifold
DMy with obvious involutive symmetry so that Ty is also modified near the cusps
to an isometry Ty of DMy. Then Ty is an isometry on the compact negatively
2
curved manifold which is isotopic to the identity. Thus Ty must be the identity
and Ty is an isometric involution. Since Ty and Ty are the same map away from
neighborhoods of the cusps, the fact implies that Ty itself is an involution.
Since Ty is deformed to ry by a tiny isotopy, dM admits an equivariant collar
neighborhood with respect to ry which separates DMy into two parts. We may
choose one of them A such that Ty(A) Γ\ A = φ and DMy — (Ty(A) U A) is homeo-
morphic to dM x /. This implies by Kim and Tollefson [6] that Ty fixes a surface
isotopic to dM. Since Ty is an isometry, the fixed surface must be totally geodesic.
Let py be a corresponding representation. Then since dM is realized as the
geodesic surface, the restriction of py to πι(d$M] is fuchsian. Since Y is dense in
T>M, and since the set of fuchsian representations is closed, we are done. D
Assigning the hyperbolic structure of the boundary to such a deformation DMP
where p G DM, we get a map
BM : VM -> T(dM)
where T is the Teichmϋller space of dM.
2. Examples
Consider the Whitehead link L = K\ U K2 in S
3
. Removing a thin tubular
neighborhood of K^ from the complement of L, we obtain a manifold W with one
compact toral boundary and one toral end. Choose an arc Σ connecting two points
on dW as in Figure 1.
544 M. FUJI! AND S. KOJIMA
K
Fig. 1.
To give hyperbolic orbifold structures O
n
 on W with the singular set Σ indexed
by natural numbers n > 2, we recall the fact, for instance in [9], that the regular ideal
octahedron is a fundamental domain of the hyperbolic manifold homeomorphic
to the Whitehead link complement. Replace the regular ideal octahedron by the
truncated octahedron as in Figure 2, where the dihedral angle along each edge
connecting truncated faces is π/2n and that of each edge through oo is π/2. Then
the faces topologically identified to create the Whitehead link complement are still
isometric and the identification gives a hyperbolic orbifold O
n
 whose underlying
space is W where the singular set is Σ with rotation angle 2τr/π.
Decompose the truncated octahedron into 4 truncated tetrahedra and we will
deform them to obtain deformations of O
n
 in the next section. The edge lengths of
the truncated tetrahedra determine their shapes. As we will see later, they are subject
to the relations required by the gluing consistency to give nonsingular hyperbolic
orbifold structures with geodesic boundary on W. Thus we obtain a space W of the
deformations of O
n
 parameterized by the edge lengths of the truncated tetrahedra.
Let π°rb(W) be the fundamental group of the orbifold O
n
 and po a lift of the
holonomy representation of ττ°rb(VF) in SL2(C). Take a meridional element m for
the cusp KI. To each deformation of O
n
 represented by an element w G W, we have
two data. One is a representation p near ρQ and the other is a canonical direction
of the axis of ρ(m) to which the end of truncated tetrahedra spiral. Assigning to
each w £ W the complex length of ρ(πι) with respect to this orientation, we get a
map G : >V —» C. It will be shown in Lemma 3 that G is a local diffeomorphism
at O
n
.
On the other hand, each representation corresponds to a pair of complex lengths
FLEXIBLE BOUNDARIES 545
Fig. 2.
which differ only in sign. Hence by assigning to each w G W the character χp of p,
we get a map π of W to the space of characters of the representations of π°rb(VK)
in SL2(C). The space of characters at χpo has complex dimension > 1 since W
has a boundary with negative Euler characteristic, and π doubly covers the image
Ί)o
n
 = τr(W) of complex dimension = 1 branched at χpo.
The orbifold O
n
 has a toral boundary with two cone points of rotation angle
2π/n. Let T(dO
n
) be the Teichmϋller space of the orbifold dO
n
. It is homeomor-
phic to R4. Assigning the hyperbolic structure of the boundary to each χp, we get
a map
B0n :
Our goal is
Theorem. The derivative of the map BOn at χpo has rank 2.
Now let P
n
 be an n-fold cyclic covering of W branched along Σ. The manifold
P
n
 supports a complete hyperbolic structure M
n
 which covers O
n
. Pulling back
the hyperbolic structure of the geodesic boundary associated to each w e W by
the covering : P
n
 — > W, we get an embedding of T(dO
n
) into T(ΘM
n
). Also the
covering induces an injective homomorphism : πι(P
n
) — > πJrb(VF), and hence a map
: V0n —* ΏMn on the characters by the restriction. It is a local diίϊeomorphism onto
the image at χpo because πι
n
 lies in τrι(P
n
), and actually a local diίfeomorphism
546 M. FUJII AND S. KOJIMA
Fig. 3.
by dimension count. Therefore, by Theorem,
embedding at M
n
. Varying n, we have
VM
n
T(dM
n
] is a local
Corollary. There are infinitely many hyperbolic 3 -manifolds M with both a
cusp and a boundary such that the map BM : T^M —> T(dM) is a local embedding
near the complete structure.
3. Truncated Tetrahedra and Gluing Consistency
The truncated octahedron to create O
n
 is decomposed into four congruent trun-
cated tetrahedra as in Figure 3. We will parametrize the deformations of O
n
 in
terms of the deformations of the shapes of these blocks.
First of all, label the triangular faces by A, B, and their edges by Ai9 Bi
(ί = 1,2,3) as in Figure 4. We call each of these edges an external edge, and denote
the length of Aj and Bj by a,j and bj respectively. These lengths are subject to two
equations.
One is the following. If we let / be the length of the edge shared by two
pentagonal faces, then regarding it as the bottom of the left pentagon, we obtain an
expression o f / in terms of αi and 62,
cosh a\ cosh &2
cosh/ =
Simultaneously, if we regard it as the bottom of the right pentagon, we obtain an
expression of / in terms of α2 and 61,
cosh/ = cosh α2 cosh &ι + 1
sinh 0,2 sinh bι
FLEXIBLE BOUNDARIES 547
Then since these two are the same quantity, we obtain one identity involving edge
lengths, which we call the relation of type (1).
The other is concerned with angles. By the hyperbolic cosine rule for the top
triangle, we have
cosh αi cosh a^ — cosh 0,3
COS #t0p =
where #top is the angle between AI and A2. If we look at the bottom triangle,
then the corresponding angle Bottom has an expression in terms of the 6/s. They
represent the same dihedral angle, and we obtain another relation,
(2) f\ f\C'top — ^bottom
It is not hard to verify that the set of six length variables subject to the relations
(1) and (2) parametrizes isometry classes of labelled truncated tetrahedra.
To create a nonsingular but not necessarily complete hyperbolic orbifold struc-
ture on W, it is sufficient to verify gluing consistency which consists of the iso-
metricity conditions for faces to be identified, and the cone angle conditions along
edges. We will see when these are satisfied.
If the external edges to be identified have the same length, then the isometricity
condition for face identification is satisfied. Since there are twelve such pairs, there
are twelve simple identities in {α?,&?} we must obviously require. For simplicity,
we just assign the same variable to each pair to be identified from the beginning
and reduce the number of the variables to the half.
Then the relations of type (1) and (2) for the four truncated tetrahedra become
dependent after gluing. In fact, reading off the lengths of the bottom edges of the
pentagonal faces in order, we can see that one of the four equations of type (1)
becomes a consequence of the other three.
548 M. FUJII AND S. KOJIMA
To compute the cone angle conditions along edges, we label them by E\, E^, E%
and Σ as in Figure 1. The dihedral angle of each edge is described in terms of the
lengths of external edges as the above expression of θtop. To obtain a nonsingular
orbifold structure, the total sum of dihedral angles around the first three edges
must be 2π and the last 2π/n. These constraints give four identities. The last one is
independent from the others, however one of the first three identities is a consequence
of the other two. To see this, recall that a toral section of the end always admits
a similarity structure. Then the total sum of angles of triangles appeared in the
horospherical triangulation is 4 x 2τr. It is equal to the sum of the total sum of
dihedral angles along E\ and E2 and the double of that of £3.
We thus have obtained ten relations with twelve variables from gluing consis-
tency. These relations define a map
/ . R12 _^
 R10?
such that its zero set f~l(0) consists of the points in R12 satisfying the gluing
consistency.
Let WQ G R12 be the point corresponding to the complete hyperbolic structure.
Denote by x and y the two variables indicated in Figure 2, and by zi, ZIQ the
other 10 variables. It is not hard to see that the rank of the matrix ( d f i / d z j ( w 0 ) )
is 10 and also find the unique solutions of the following two linear equations in
terms of u and v respectively:
Denote the unique solutions by UQ and VQ respectively. Then we have obtained
Lemma 2. A neighborhood W of w0 in /-1(0) is a 2-dimensional smooth
manifold and we have two paths on W C R12,
ξ(t) = WQ + xt + (higher order),
η(t) =w0 + yt + (higher order),
such that
where the y-component o f ξ ( t ) and the x-component ofη(t) are constant, and the
x-componentofξ(t) and the y-component of'η(t) have no terms of degrees n (n > 2).
FLEXIBLE BOUNDARIES 549
cone points
Fig. 5.
4. Dehn Filling Space and Computation
Lemma 3.
W0 € W.
The map G : C defined in §2 is a local dίffeomorphism at
Proof. The complex length of a meridional element can be read off from
the triangulation by horospherical sections as in [8]. By direct computation using
triangulation induced from the polyhedral decomposition of W9 we can verify that
the rank of the Jacobian of G is 2. D
Recall that π : W —» T>On is a 2-fold covering branched at π(wQ). In fact π is
a map which locally looks like a square function : z —> z2.
The lengths Li of geodesic segments 5» {i = 1,... ,4) which are illustrated by
thick lines in Figure 5 give rise to a quadruple (Li, 1/2,1/3,^4) which defines a
global coordinate ofT(dO
n
).
Let B be a map assigning to the element of W the corresponding hyperbolic
structure of the boundary. Then its induced map from T>o
n
 is BOn. Let ^  (resp.
BI) be the composition of B (resp. ^cU with I/^.
T(do
n
) R4
Now consider a quadrilateral in general. If the lengths of four sides and one
550 M. FUJII AND S. KOJIMA
of diagonals are known, then the length of the other diagonal can be expressed in
terms of them by hyperbolic trigonometry. Applying this to the quadrilateral in
Figure 5 which is made of two triangular faces, we have an expression of BI as a
function of our length parameters.
Because of the local picture of π, letting f (ί) = π o ξ(Vt) and η(i) — π o η(\/t),
we obtain smooth paths on £>o
n
 such that its tangent vectors
are nontrivial. The images of these vectors by the derivative dBi are now expressed
by
dBi(ξ(t))
dt
dt
t=o
t=0
dt
dBi(η(Vt))
dt
t=o
t=0
To carry out the actual computation of the right hand sides, we used the Taylor
expansions of ξ(t) and η(t) up to the second degree, which can be derived from the
formula,
j .fc
By performing this rather lengthy but direct computations by hand, we verified the
following:
Lemma 4.
dBι(w) =
7 ^ -where c = cos — ana s = sin —.
2n 2n
), dB2(w)
Lemma 4 shows that these tangent vectors on X>o
n
 go to a linearly independent
pair in the tangent space of T(ΘO
n
) at the original structure and we complete the
proof of Theorem.
FLEXIBLE BOUNDARIES 551
Finally, we would like to mention that our naive method works equally well to
verify the vanishing of the derivative of BM for Neumann and Reid's example. The
details were worked out by T. Hayakawa at Tokyo Institute of Technology.
References
[1] D. Cooper and D. Long: An undetected slope in a knot manifold, In : Topology '90, Pro-
ceedings of the Research Semester in Low Dimensional Topology at Ohio State University,
111-121, Berlin New York : de Gruyter 1992.
[2] M. Culler and P.B. Shalen: Varieties of group representations and splittings of 3-manifolds,
Ann. Math. 117 (1983), 109-146.
[3] M. Fujii: Deformations of a hyperbolic 3-manifold not affecting its totally geodesic bound-
ary, Kodai Math. J. 16 (1993), 441-454.
[4] M. Gromov and W.P. Thurston: Pinching constants for hyperbolic manifolds, Invent. Math.
89 (1987), 1-12.
[5] M. Kapovich: Eisenstein series and Dehn surgery. (Preprint)
[6] P.K. Kim and J.L. Tollefson: PL involutions of βbered 3-manifolds, Trans. A.M.S. 232
(1977), 221-237.
[7] W.D. Neumann and A.W. Reid: Rigidity of cusps in deformations of hyperbolic 3-orbifolds,
Math. Ann. 295 (1993), 223-237.
[8] W.D. Neumann and D. Zagier: Volumes of hyperbolic 3-manifolds, Topology 24 (1985),
307-332.
[9] W.P. Thurston: The geometry and topology of 3-manifolds, Lect. Notes, Princeton Univ.,
1978.
M. Fujii
Department of Mathematical Sciences
Yokohama City University
22-2 Seto, Kanazawa-ku, Yokohama
Kanagawa-ken 236, Japan
e-mail: mfujii@yokohama-cu.ac.jp
S. Kojima
Department of Mathematical
and Computing Sciences
Tokyo Institute of Technology
Ohokayama, Meguro, Tokyo 152, Japan
e-mail: sadayosi @ is.titech.ac.jp



Flexible boundaries in deformations of hyperbolic 3-manifolds

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