The Hyperbolic Volume of 3-manifolds
Y. Liu, S. Wang and P. Derbez
October 29, 2013
Volume of representations
Let G be a real semi-simple Lie group and let K denote its
maximal compact subgroup.
The homogeneous space X = G/K , with base point x0 = {K }, is
contractible and we fix a G-invariant Riemannian metric on it.
Let ωX be the G-invariant volume form.
Let M be a closed, connected, oriented n-manifold such that
dimM = dimX = n.
e →X
For each ρ : π1 M → G, there is a developing map Dρ : M
∗
which is equivariant. Then one can identify Dρ (ωX ) with a n-form
on M and
Z
∗
volG (M, ρ) = Dρ (ωX )
M
Constructing developing maps
Fix a triangulation T = {∆1 , ..., ∆s } of M and a Dirichlet domain
e Denote by {x1 , ..., xN } the vertices of T
e lying in intΩ and
Ω in M.
let {y1 , ..., yN } be N arbitrary points in X .
We first set Dρ (xi ) = yi for i = 1, ..., N.
e and next
We then extend Dρ equivariantly to all the vertices of T
to all the edges, faces etc... straightening their image w.r.t. the
homogeneous metric on X .
e 1 , ..., ∆
e s a lift of ∆1 , ..., ∆s . The volume of ρ can be
Denote by ∆
computed as follows
Z
volG (M, ρ) =
M
Dρ∗ (ωX )
=
s
X
e i ))
εi volX (Dρ (∆
i=1
e i is orientation
where εi = ±1 depends on whether Dρ |∆
preserving or not.
Volume and Chern-Simons functional
The Chern Simons invariants are based upon the observation if
ω is a connection on a principal G-bundle P over M then f (Ω ∧ Ω)
is an exact form on H 4 (P), where f = Tr : g ⊗ g → C is an
invariant polynomial. A primitive is given explicitly in their work by
1
cs(ω) = f (dω ∧ ω) + f (ω ∧ [ω, ω])
3
This is the Chern-Simons class of ω. If the bundle P → M is
trivial, the Chern-Simons functional is then defined by
Z
csM (ω, δ) =
δ ∗ cs(ω) ∈ C
M
Suppose from now on G = PSL(2; C). For each representation
ρ : π1 M → G admitting a lift into SL(2; C), we have the (trivial)
principal bundle M ×ρ G and the associated bundle M ×ρ H3
e × G and M
e × H3 by the diagonal action of π1 M.
obtained from M
Denote by A the flat connection over M corresponding to ρ.
This connection is defined by A = q ∗ (ωM.C. ) where ωM.C. = dLg −1
denotes the Maurer Cartan form of G which is the unique left
invariant g-valued 1-form such that ωM.C. (X ) = X for any X ∈ g.
GO
p
/ H3
O
q
q
e ×G
M
−
P = M ×ρ G
p
p
O
δ
M
s
e × H3
/M
−
/ E = M ×ρ H3
6
1 0
0 0
0 1
The matrices X =
,Y =
,Z =
form a
0 −1
1 0
0 0
basis of the Lie algebra g = sl(2; C) with commutators relations
[X , Y ] = −2Y , [X , Z ] = 2Z , [Y , Z ] = −X
Denote by ϕX , ϕY , ϕZ the dual basis of sl∗ (2; C). The Maurer
Cartan form of G is
ωM.C. = ϕX ⊗ X + ϕY ⊗ Y + ϕZ ⊗ Z
one can consider the 3-form split by a formula of Yoshida
cs(ωM.C. ) =
1
1
ϕX ∧ ϕY ∧ ϕZ = 2 p∗ ωH3 + icsL.C. (H3 ) + dγ
π2
π
inducing a closed 3-form on P
cs(A) =
1 ∗
q (ϕX ∧ ϕY ∧ ϕZ )
π2
The Chern-Simons functionnal is then defined by
Z
csM (A, δ) =
δ ∗ cs(A) ∈ C
M
Plugging this formula in the above diagram we get
csM (A, δ) = cs(Mρ ; δ) −
i
vol (M, ρ)
π2 G
Note that this formula was proved by Kirk and Klassen when M
is closed hyperbolic and ρ is the discrete and faithful
representation.
The Chern-Simons invariant csM (A, δ) is well defined modulo Z
when the section changes. Then define cs∗M (A) to be the class of
csM (A, δ) in C/Z which can be seen as
cs∗M (A) = e2iπcsM (A,δ)
Abelian representation
Any abelian representation of a closed oriented 3-manifold M
into G has a zero volume.
Suppose the image of ρ : π1 M → PSL(2; C) is torsion free. Then
there is a path of representations
ρt : π1 M → PSL(2; C)
such that ρ1 = ρ and ρ0 is the trivial representation.
Consider the associated path of flat connections At . This path
defines a connection A on the product M × [0, 1] that is no
longer flat but whose curvature satisfies the equation Ω ∧ Ω = 0
(this latter point follows from the fact that ΩAt = 0 for any t).
Since dcs(A) = Tr(Ω ∧ Ω) = 0 it follows from the Stokes formula
that cs∗M (A0 ) = cs∗M (A1 ) = 1. Therefore vol(M, ρ) = 0.
From local to global representations
Suppose M = V ∪T W is a closed oriented 3-manifold which is a
union two manifolds with incompressible toral boundary such
that ∂V = ∂W = T and fix a regular neighborhood T × [−1, 1]
with T × [−1, 0] ⊂ V and T × [0, 1] ⊂ W . We suppose each
component T of T is endowed with a meridian-parallel system
(mT , lT ) and we denote by xT = mT ∩ lT .
Let ρ : π1 V → PSL(2; C) and ϕ : π1 W → PSL(2; C) denote two
representations which lift into SL(2; C). They both induce
representation still denoted by ρ and ϕ of π1 (T × [−1, 1], xT ) into
PSL(2; C) and we will consider the case of non-parabolic
representations.
T'x[-1,1]
V
l_T
m_T
Tx[-1,1]
W
After conjugation in PSL(2; C), we may assume that there exist
αT , βT ∈ C such that
2iπα
2iπβ
T
T
e
0
e
0
ρ(mT ) ∼
ρ(lT ) ∼
0
e−2iπαT
0
e−2iπβT
and αT0 , βT0 ∈ C
2iπα0
T
e
ϕ(mT ) ∼
0
0
0
e−2iπαT
ϕ(lT ) ∼
0
e2iπβT
0
0
0
e−2iπβT
Denote by A and B two flat connections over V and W
corresponding to ρ and ϕ. Then A|T × [−1, 1] and B|T × [−1, 1]
can be turned into normal form after gauge-transformation.
Namely, there exist gT ∈ G and gT0 ∈ G such that
iαT dx + iβT dy
0
∗
gT A|T × [−1, 1] =
0
−iαT dx − iβT dy
where
gT : T × [−1, 1] → SL(2; C)
and
hT∗ A|T
0
iαT dx + iβT0 dy
× [−1, 1] =
0
0
0
−iαT dx − iβT0 dy
where
hT : T × [−1, 1] → SL(2; C)
By obstruction theory one can extend
a
a
gT ,
hT : T × [−1, 1] → SL(2; C)
T ∈T
T ∈T
to gauge transformations g : V → SL(2; C), h : W → SL(2; C).
Eventually if we manage to find representations ρ and ϕ with the
same eigenvalues when restricted to each torus T then the
connections g ∗ A and g ∗ B match on T × [−1, 1] and their union
define a flat and smooth connection C over M and an associated
representation ψ of π1 M into PSL(2; C).
Additivity principle
Denote by [M] the orientation class of M and by [V , ∂V ] and
[W , ∂W ] the induced orientations so that
[M] = [(V , ∂V )] + [(W , ∂W )]
(the orientations induced on ∂V and ∂W are opposite on T ).
By linearity of the integration
cs∗M (C) = cs∗V (CV )cs∗W (CW )
Along each component T of T we perform a Dehn filling
identifying each mT with the meridian of a solid torus and denote
b = V ∪ V − and W
c = W ∪ V + the resulting closed manifolds.
by V
Suppose both CV and CW smoothly extend to flat connections
cV , resp. C
d
b , resp. W
c , and denoted by C
over V
W.
These extensions are possible provided [mT ] lies in ker ρ and
ker ϕ. Again by the linearity we have
cV )cs∗ (C
d
cs∗Vb (C
W) =
c
W
+
cV |V − )cs∗ (CW )cs∗ + (C
d
= cs∗V (CV )cs∗V − (C
W |V )
W
V
Since the extensions from ∂V , resp. ∂W , to V − , resp. V + based
on the normal form on [−1, 1] × T , are the same on the T
direction but opposite on the [−1, 1] direction
+
cV |V − )cs∗ + (C
d
cs∗V − (C
W |V ) = 1
V
Eventually we get
b , ρb) ± volG (W
c , ϕ)
volG (M, ψ) = volG (V
b
b , resp. of ϕ, to π1 W
c.
where ρb, resp. ϕ,
b is the extension of ρ to π1 V
Maximal volumes
Given (G, K , X ) and M the Chern Simons rigidity theorem claims
that the forms Dρ∗ (ωX ) represent at most finitely many classes in
H 3 (M). Hence it makes sense to define VG (M) as the maximum
of volG (M, ρ) when ρ : π1 M → G runs over the representations.
The invariant VG is a lower volume, namely for any map
f : M → N then
VG M ≥ |degf |VG N
According to the terminology of Reznikov, a non-negative
manifolds invariant V is a volume if for any finite covering map
f : M → N then
V (M) = |degf |V (N)
Example: The Euler characteristic of surfaces and more
generally the Gromov simplicial volume.
Three-dimensional examples of VG
The universal covering
+ 2
^
SL
2 (R) → PSL(2; R) = Iso H
is endowed with the (pull-backed) hyperbolic metric.
^
^
One gets (SL
2 (R), 1, SL2 (R)).
g2 ), SO(2), SL
g2 ), where Isome (SL
g2 )
More generally (Isome (SL
denotes the identity component of the whole isometry group.
When G = PSL(2; C) we get the hyperbolic geometry
(PSL2 (C), SO(3), H3 ) and the associated hyperbolic volume
HV = VPSL(2;C) .
The other 3-dimensional geometries give always a 0 maximal
volume.
Hyperbolic manifolds
Theorem (Reznikov, Gromov)
Let M be a closed hyperbolic manifold. Then
HV (M) = volH3 (M) = µ3 kMk
and the maximum is reached only by the discrete and faithful
representation.
One can check basically that for any closed oriented 3-manifolds
M (not necessarily hyperbolic)
HV (M) ≤ µ3 ∆3 (M)
Reznikov gave a sharper bound proving that
HV (M) ≤ µ3 kMk
Structure of 3-manifolds
Let M be a closed oriented 3-manifold. By the Kneser Milnor
decomposition theorem, M splits into
M = M1 ]....]Mr ](S1 × S2 )]....](S1 × S2 )
where the Mi ’s are irreducible 3-manifolds.
By the works including those of Jaco, Shalen, Johannson,
Thurston, Hamilton, Perelman, each irreducible closed oriented
3-manifold M is either geometric or can be cut out along a family
of tori and Klein bottles TM such that
M \ TM = S(M) ∪ H(M)
where the components of S(M) are H2 × R and those of H(M)
are hyperbolic. When both TM and H(M) are non-empty M is
termed a mixed manifold.
Gromov proved that k.k is additive w.r.t connected sum and toral
decomposition. Moreover when M is irreducible
kMk =
X
Q∈H(M)
kHk =
X
Q∈H(M)
volH Q
µ3
In particular if HV (M) is positive then M is necessarily hyperbolic
or a mixed 3-manifold.
Questions for mixed 3-manifolds
1) Is HV (virtually) positive for mixed 3-manifolds ?
2) Is HV a volume?
3) What kind of information is detected by the hyperbolic
volume?
If the answer to the first question is yes then HV detects the
hyperbolic pieces of M as the simplicial volume does but the
latter only depends on the geometric pieces of M and not on the
way they are glued together. One can therefore wonder whether
HV measures the complexity of the sewing involution between
the geometric pieces.
l
s
FxS^1
m
h
Bi-reducible hyperbolic
manifold
Solid Torus
Manifolds with HV = 0 and kMk > 0
Let M1 = F × S1 where F is a surface with positive genus and
connected boundary. Let (s, h) denote the associated
section-fibre basis of H1 (∂M1 ; Z).
Let M2 be a one-cusped, complete, finite volume hyperbolic
manifolds endowed with a basis (m, l) of H1 (∂M2 ) such that both
M2 (l) and M2 (m) are connected sums of lens spaces.
Such manifolds do exist by a construction of Hoffman and
Matignon.
Denote by ϕ : ∂M1 → ∂M2 the homeomorphism defined by
ϕ(s) = m and ϕ(h) = l −1 . Then M = M1 ∪ϕ M2 satisfies kMk > 0
and HV (M) = 0.
Let ρ : π1 Mϕ → PSL(2; C) be any representation and denote by
A the resulting connection over Mϕ . Notice that either ρ(s) or
ρ(h) is trivial.
Indeed if ρ(h) 6= 1, its centralizer Z (ρ(h)) in PSL(2; C) must be
abelian. Since h is central in π1 M1 , this means that ρ(π1 M1 ) is
abelian. Since s is homologically zero in M1 , then ρ(s) = 1.
Let ζ be either s or h so that ρ(ζ) = 1. After putting A in normal
form with respect to T , denote by A1 and A2 the flat connections
over M1 and M2 respectively. Since ρ(ζ) = 1 then both A1 and A2
b 1 and A
b2
do extend over M1 (ζ) and M2 (ζ) to flat connections A
such that
b
b 1 ) × cs∗
cs∗Mϕ (A) = cs∗M1 (ζ) (A
M2 (ζ) (A2 )
Eventually taking the imaginary part we get
vol(Mϕ , ρ) = vol(M1 (ζ), ρb1 ) ± vol(M2 (ζ), ρb2 )
where ρbi denotes the extension of ρ|π1 Mi to π1 Mi (ζ).
Since both vol(M1 (ζ), ρb1 ) and vol(M2 (ζ), ρb2 ) do vanish by the
construction, this proves that HV (M) = 0.
Answer to Questions 1 and 2
Theorem (L-W-D)
Any closed mixed 3-manifold has a virtually positive hyperbolic
volume.
Remark
Since there are mixed 3-manifolds with zero hyperbolic volume
then HV is only a lower volume and not a volume whenever M is
not hyperbolic which provides a negative answer to Question 2.
However HV can be rescaled to define a honest volume
(
)
e
HV (M)
HV (M) = sup
≤ µ3 kMk
e → M)
deg(M
and HV (M) > 0 iff kMk > 0.
Proof of the theorem
Some examples
Let f : M → N be a map between oriented closed 3-manifolds. If
ρ : π1 N → G is a representation then it induces a representation
ρ ◦ f∗ : π1 M → G and
volG (M, ρ ◦ f∗ ) = |deg(f )|volG (N, ρ)
Let M = Q ∪T (F × S1 ) where Q has a hyperbolic interior and
T = ∂Q = ∂F × S1 is connected. The torus T has a natural
meridian-parallel basis (m, l) given by ∂F and the fiber S1 .
There is a degree one ”pinching” map π : F × S1 → D 2 × S1
b
inducing a degree one map M → Q(m)
implying that
b
HV (M) ≥ HV (Q(m))
\tilde{Q}
\tilde{N}
T
Q
N
Consider M = Q ∪T N where Q has a hyperbolic interior and
T = ∂Q = ∂N is connected. Then
i]
Rank H1 (∂N; R) → H1 (N; R) = 1
We may therefore choose a meridian-parallel basis (m, l) of
H1 (∂N; Z) such that i] (l) has infinite order whereas i] (m) is a
torsion element in H1 (N; Z).
Lemma
There is a prime number p0 such that for any slope m ⊂ ∂Q and
any prime number q ≥ p0 one can find
e → Q inducing the q × q-characteristic
- a finite covering p : Q
covering over ∂Q, and
e −1 (m)) → PSL(2; C) of positive
- a representation ρ : π1 Q(p
volume.
Choose q big enough so that Q(q.m) is a hyperbolic orbifold.
There is therefore a representationρ : π1 Q → PSL(2;
C) with
eiπ/q
0
positive volume such that ρ(m) ∼
and
0
e−iπ/q
x
0
ρ(l) ∼
for some x ∈ C∗ .
0 x −1
By a result of Hempel, if q is big enough there exists a finite
e → Q inducing the q × q-characteristic covering
covering p : Q
on the boundary. Meanwhile p induces an orbifold covering
e −1 (m)) → Q(q.m). Accordingly the composition ρ ◦ p∗ is
b : Q(p
p
a representation factoring through the fundamental group of the
e −1 (m)) whose volume is
(non-singular) hyperbolic manifold Q(p
deg(p)volH3 Q(q.m) > 0
e → M be a finite covering such that each component of
Let p : M
e Each component Q
e of p−1 (Q),
p−1 (Q) is homeomorphic to Q.
has a representation
e m)
e → PSL(2; C)
ρ : π1 Q(
q
x
0
e
of positive volume where ρ(l) ∼
.
0 x −q
Since l ∈ H1 (N; Z)/TorH1 (N; Z) there is a homomorphism
x
0
H1 (N; Z) → PSL(2; C) sending m trivially and l to
. For
0 x −1
e of p−1 (N), consider the representation η
each component N
p]
e → H1 (N;
e Z) →
π1 N
H1 (N; Z)→PSL(2; C)
q
x
0
e
e
with η(m) = Id and η(l) =
and such that
0 x −q
e m),
e η) = 0.
volG (N(
By the additivity principle we conclude
e m))
e ≥ dvol 3 (Q(
e
HV (M)
H
where d is the number of components of p−1 (Q).
Przytycki-Wise surfaces
Let M be a closed oriented mixed 3-manifold without essential
Klein bottles. By a PW -surface of M we mean a π1 -injective map
j : S → M of a closed orientable surface in minimal general
position w.r.t. the geometric decomposition such that:
1. for each maximal graph chunk N ⊂ M the components of
j −1 (N) are virtually embedded in N
2. for each hyperbolic piece Q the components of j −1 (Q) are
geometrically finite in Q
Remark
Any component of TM is a trivial example of a PW-surface.
A partial PW-surface is a connected subsurface R ⊂ S such that
j(∂R) is immersed in a single torus T0 of TM and for every
component T of TM the components of j −1 (T ) covers the same
slope of T .
Theorem (Haglund-Wise)
Every PW-surface subgroup of π1 M is separable and every
double coset of two PW-surface subgroups of π1 M with
non-trivial intersection is separable.
Passing to some finite covering we may assume that
1. M containg no Klein bottles,
2. each JSJ-torus is shared by two distinct geometric pieces,
3. each Seifert piece is homeomorphic to a product F × S1 ,
4. each maximal graph-manifold chunk has the antenna
property.
The antenna property of Przytycki-Wise means that if N is a
graph manifold with ∂N 6= ∅ then for any vertex v0 of ΓN there
exists an edge-path (v0 , ..., vn ) which is a full subgraph of ΓN
where vn is a boundary-vertex.
S_1
T_0
Q_1
Q_2
R
Antenna
Q_0
S_2
Proposition (Virtual boundary)
Let Q0 be a hyperbolic piece of M0 , T0 be a component of ∂Q0
and let ξ0 be a slope of T0 . There exists a partial PW -surface R
which virtually bounds ξ0 outside Q0 .
This means that j : S → M induces a proper immersion
j| : (R, ∂R) → (M \ intQ0 , T0 ) such that ∂R covers ξ0 under j.
We will denote by X (R) the carrier chunk of R which is the
minimal chunk containing R.
We are going to explain how to construct a PW-surface
j : S → M and the desired partial PW-surface R ⊂ S.
By Przytycki-Wise there exists a geometrically finite properly
immersed incompressible surface R0 in Q0 whose ∂R0 6= ∅ and
covers ξ0 .
Let Q be the geometric piece adjacent to Q0 along T0 (Q 6= Q0 ).
If Q is hyperbolic too, by the same argument as above we
choose (R, ∂R) → (Q, T0 ) where ∂R covers ξ0 .
We use the merging trick: choose a finite (possibly
f0 → R0 and R
e → R such that R
f0 and R
e
disconnected) covering R
have the same number of boundary component and all of these
components cover ξ0 with the same (unsigned) degree. This is
possible because for any hyperbolic surface S there exists an
integer K > 0 such that if we assign a positive degree ni to each
e→S
component Ci of ∂S then one can find a finite covering S
whose degree on each component over Ci is Kni .
Then eventually S is obtained taking a component of
[
f0
e
R
R
f0 =∂ R
e
∂R
If Q is Seifert denote by Qmax the maximal graph chunk ⊃ Q.
If ∂R0 is parallel to the fibre of Q then taking two copies of R0
and connecting each pair of boundary components by a properly
immersed ∂-essential vertical annulus in (Q, ∂Q) we get S.
Suppose ∂R0 is not parallel to the fibre of Q. We denote by
A = Q0 ∪ Q1 ∪ ... ∪ Qn the antenna of Qmax with Q1 = Q. There
exists a properly embedded horizontal surface E in Q1 ∪ ... ∪ Qn
meeting T0 along parallel copies of ξ0 .
This is based on the following result
Lemma (Wang-Yu)
Let S = F × S1 with boundary T1 , ..., Tl and assume there are
families of disjoint identically oriented circles
C1 ⊂ T1 , ..., Cl−1 ⊂ Tl−1 such that
[C1 ] ∩ [fibre] = ... = [Cl−1 ] ∩ [fibre] 6= 0. Then there exists a family
of disjoint identically oriented circles Cl ⊂ Tl such that ∪Ci is the
boundary of a horizontal surface.
Denote by E a surface of the antenna obtained by matching up
horizontal surfaces in Q1 ∪ ... ∪ Qn such that the components of
∂E ∩ T0 are parallel to ξ0 .
If a component of ∂E is not in ∂Qmax then one can make sure
that it lies in some JSJ-torus inside Qmax parallel to the adjacent
fibre. Then these boundaries can be closed up by an immersed
vertical annuli.
If a component ∂E in ∂Qmax then the adjacent pieces are
hyperbolic and the surface can be closed up by merging some
geometrically finite surfaces.
Proposition
e → M in which every elevation of
There exists a finite covering M
(S, R) is an embedding and intersects any elevation of T0 in at
most one slope.
The separability of PW-surface groups in π1 M implies that for
e of M such that any
any j : S → M there exists a finite covering M
e
e
e
elevation j : S → M is an embedding.
Suppose S is an embedded PW-surface in M crossing a
component T of TM . The double coset separability of any two
PW-surface groups in π1 M implies that there exists a finite
e of M such that for any elevation T
e of T and for any
covering M
e
e
e
elevation S of S the components of S ∩ T cover the same
component of S ∩ T .
Suppose that S ∩ T contains two components c1 , c2 and fix a
base points x1 ∈ c1 and x2 ∈ c2 . Let γS an arc in S and γT an
arc in T from x1 to x2 and denote by γ the closed curve γS ∪ γT .
By construction [γ] 6∈ π1 S.π1 T . There is therefore an
epimorphism ϕ from π1 M into a finite group such that ϕ([γ])
does not belong to the image of π1 S.π1 T .
e → M the elevations of
In the corresponding regular covering M
c1 and c2 never belong to the same component over T .
b of S
e
b of M
e such that any elevation S
There a finite covering M
e
meets each JSJ-torus of M in at most one slope.
e ∩T
e be a JSJ-torus of M
e and suppose S
e contains two
Let T
components c1 , c2 . Since they cover the same component of
S ∩ T then they are directly parallel with the direction induced
from S. The curve γ = γS ∪ γT meets S once. Then the abelian
covering corresponding to
π1 M → Z → Z/2Z
where the first arrow is the Poincaré Duality ∩[S] and the second
is obtained after dividing by the image of π1 T satisfied the
conclusion.
For each boundary component Ti of Q0 , 1 ≤ i ≤ n, we fix a
meridian parallel system ξi , li such that Q0 (ξ1 , ..., ξn ) is hyperbolic
and we denote by ρ : π1 Q → PSL(2; C) the representation
factoring through the discrete and faithful representation
π1 Q(ξ1 , ..., ξn ) → PSL(2; C).
Denote by xi±1 the eigenvalues of ρ(li ).
For each i we fix a partial PW -surface Ri which virtually bounds
e → M (regular and
ξi and a finite covering p : M
q × q-characteristic) such that any elevation of Ri is embedded
and meets any elevation of Ti in at most one slope. Define for
ei of X (Ri ) containing an elevation R
e i of Ri a
each elevation X
homomorphism
e i ] : π1 X
ei → H1 (X
ei ; Z) → Z
ϕ
ei = ∩[R
e i ] = 1 so that
By construction any elevation eli of li satisfies [eli ] ∩ [R
e
e
ϕ
ei (li ) = 1 ∈ Z and ϕ
ei (ξi ) = 0
Denote by αi : Z → PSL(2; C) the morphism defined by
q
xi
0
1 7→
0 xi−q
We next define on each component outside of
p−1 (Q0 ) ∪i p−1 (X (Ri )) the trivial representations. For each
f0 of Q0 in M
f0 )∗ the induced
e denote by ρe = ρ ◦ (p|Q
elevation Q
ei of Ti the elements ρe(eli ) are
representation. For any elevation T
all conjugated in PSL(2; C) to αi ◦ ϕ
ei .
Eventually the representations ρe, αi ◦ ϕ
ei , 1 match up to
conjugation on the common boundary so that one can define a
global flat connection and therefore a global representation
e → PSL(2; C) such that
ψ : π1 M
X
e ξ),
e ψ) =
e ρe) > volG Q(ξ1 , ..., ξn )
volG (M,
volG (Q(
where ξe is made of one elevation of each ξi on each component
e
of ∂ Q.
Using the same construction and adapting the gauge theory of
^
PSL(2; C) to the group of Isome (SL
2 (R)) of the Seifert geometry
we prove
Theorem (L-W-D)
Any closed mixed 3-manifold containing at least one Seifert
piece has a virtually positive Seifert volume.
Remark
We proved the same result for non-trivial graph manifold and
meanwhile we found some non-trivial graph manifolds with zero
Seifert volume so that SV is neither a volume but a lower volume
whenever M is not Seifert.
These results lead to another
Question
Is SV virtually positive for hyperbolic and mixed 3-manifolds that
contain only hyperbolic pieces in their geometric decomposition?