Bull. Kyushu Inst. Tech.
(Math. Natur. Sci.) No. 34, 1987, pp. 14
VOLUME OF CERTAIN REPRESENTATIONS IN PSL,(C)
By Teruhiko SoMA
(Received November 5, 1986)
Introduction
In [1], Goldman defined the volume vol (p) of a representation p: zi(N).PSL2 (C)=
SL2(C)/{Å}I}, where N is a closed, connected, orientable 3-manifold. This volume is well-defined only when N is closed. When the manifold N is hyperbolic, we denote by po: ni(N).PSL2(C) (=Isom'(H3)) the holonomy for its hyperbolic structure. Then Goldman [1] proved that, for every representation p: ni(N).PSL2 (C), vol (p)Svol (po) and the equality holds iff p is conjugate to po in PSL2 (C).
Let M be also a closed, connected, orientable 3-manifold and K a simplicial trian- gulation of M. We denote by Z the underlying polyhedron of the 1-skeleton of K and set ll==ni(M-Z). In [4], we defined the volume volK(ip) of an a-realizable, nondegenerate representation ip: II.PSL2 (C) with respect to K as follows:
In [4], we proved that, for a representation ip as above, there exists a bent manifold Bdi on K with holonomy ip, and such a bent manifold is determined uniquely up to iso- morphism. Therefore we can define the volume volK (ip) by vol (Bip), see [4] for details.
The aim of this note is to give a relation between Goldman's volume and that of ours for certain representations in PSL2 (C).
1. Goldman"s volume and definitions.
First we give the definition of Goldman's volume [1]. Let N and p: zi(N)-ÅrPSL2 (C) be as above. For each element ct ezi(N), we denote by T(ct) the covering transformation determined by ct on the universal covering space N of N. The action of ni(N) on NÅ~H3 is defined by ct•(x, y)=(T(ct)x, p(ct)y), where xEN and yEH3. Since the projection P: NÅ~H3.N to the first factor is zi(N)-equivariant, there exists the induced projection p: E,=NÅ~ H31zi(N).N,= H3 lzi(N). ' Then p: E,.N is a fiber-bundle projection with fiber H3 and structure group PSL2(C). Let q:NxH3.H3 be the projection to the second factor and co the volume form on H3. Since q*co is a ni(N)-invariant 3-form on NxH3, the 3-form co, on Ep can be induced from q*to. Since the fiber H3 is con- tractible, there exists a piecewise smooth section s: N.E,. We define the volume of p by
vol (p) = s*tu, .
N,s
2 Teruhiko SoMA
Since N is closed, the volume does not depend on the choice of the section s.
With the notation as in Introduction, we set C=M-int ./if(V), where Vis the set of all vertices of Z and vfr"(V) is a small regular neighborhood of Vin M. Let (dC, dZ) be the double of (C, C n Z) obtained by identifying the boundary (OC, O(C n Z)) of (C, C n E)
with that of its copy (C', C'nZ'), i.e. dC=C v C'. Set I7d=ni(dC-dE). We can oc=ec'
regard ll as a subgroup of lld. The representation dip:Ild.PSL2(C) is called the double of Åë if ddilll =di and if there exists an orientation-reversing jsometry ct of H3 such that ctq5(/3)ct-i=dip(T(J(3)) for all /3Efl, where T: fl.7t,(C'-C'nZ')cll" is the isomorphism induced from the natural identification (C, C n Z)--År(C', C' n Z').
LEMMA1. An a-reatizable, non-degenerate representation to:fl-ÅrPSL2(C) has the double dip : lld.PSL2 (C).
PRooF. According to [4, Theorem], there exists the bent manifold Bip on K with holonomy ip. Since OBip is totally geodesic and E meets OBip orthogonally, we can con- struct the double bent manifold dBip along OBip. Then the holonomy of dBip is the double of ip. D
2. Theoreml.
First we prove the following lemma.
LEMMA 2. dC-di admits a complete hyperbolic structure offini,te volume.
PRooF. The complement R =dC-int ut/"(dZ) is a Haken manifold with toral boundary and F =R n 0C is incompressible in R, Let T be an arbitrary incompressible torus in R.
Since n is a free group, TnF#ip. By modifying Tby an isotopy in R, we can assume that Tmeets F transversely and each component of Tn OF is essential in T. We can also assume that the closure A of each component of T- TnOF is an annulus such that two components of OA are contained in distinct components of F. By the elementary cut and paste argument, one can show that (A, OA) is isotopic into (aR, ORnF) in (R, F). This implies that T is parallel to some component of OR, so R is (geometrically) atoroidal. By Thurston's Uniformization Theorem (see [3]), either jntRfudC-dZ admits a complete hyperbolic structere of finite volume or R is Seifert-fibered. If R were Seifert-fibered, there would exist a fibered structure .:`Jh on R such that each fiber of ,E`RZ' meets F transversely.
Since each fiber of ,9e:' 1 C n R is a closed interval, R n C is homeomorphic to Fo Å~ I, where Fo is a components ofF and I= [O, 1]. Therefore dC-dZ iS homeonorphic to GÅ~S', where G is a component of OC--O(CnZ), which is absurd. Therefore dC-dZ is hyper- bolic. D
Volume of Certain Representations 3
THEoREM 1. Let O be a 3-orbifold with combinatorial type (dC, dZ) and such that the order in each edge ei is niÅr2. Then O is a hyperbolic 3-orbifold.
We refer to [5] for the fundamental notation on orbifolds.
PRooF oF HEoREM 1. First we show that O does not contain any closed, incom- pressible, euclidean 2-suborbifold. If there were such a 2-suborbifold O' in O, then by the proof of Lemma 2 we would have O' nZ;ip. Since all niÅr2, the underlying space of O' is a 2-sphere S and S meets dZ transversely in three points. We can assume that S meets eC transversely and each component of S n OC is non-contractible in both S-S n dZ and OC-eCnZ. At least one of the innermost loops ofSnaC, say l, boundsadick D in S such that DndZ is a single point. By an isotopy in (dC, dZ), we can move (S, SndZ) to (S', S'ndZ) such that S'naCcSn0C-l. Therefore (S, SndZ) can be moved into either C or C', say ScC. Since S meets dE in three points, we may assume that S meets only two 3-simplices Ai, A2 ofK. Therefore Ai meets A2 in three faces. This contradicts that K is a simplicial triangulation.
By Thurston's Orbifold Geometrization Theorem (see [2]), O is either hyperbolic or Seifert-fibered. Since niÅr2 for all i, if O were Seifert-fibered, then O could have the fibration ge which contains dE as a set of fibers. Hence ge 1dC-dZ defines the Seifert- fibered structure on dC-dZ, contrary to Lemma 2. This completes the proof. O
3. Theorem2.
Let O be the hyperbolic 3-orbifold given in Theorem 1. There exists the natural epimorphism p: lld.n2'b(O) such that the kernel ig. the normal closure of {ptr•i} in lld.
By Selberg's Lemma, ng'b(O) has a torsion-free subgroup T of finite index n, So there exists the n-fold orbifold covering q: N-ÅrO associated to r. Then N is a closed, hyperbolic 3-manjfold.
Let ip: ll.PSL2(C) be an arbitrary representation satisfying the following con- dition.
Condition ("). ip is a-realizable, non-degenerate and, for the meridean element pai agsociated to each edge ei of Z, ip(pai) has a finite order niÅr2.
By Lemma 1, ip has the double dip: lld.PSL2(C). By Condition ("), there exists the representation dip: nl'b(O).PSL2(C) such that ddi=ddiop. Since N is closed, we can apply the Goldman's definition of volume to dip 1 zi(N).
Now we have the following commutative diagram.
4 Teruhiko SoMA nÅë
n
nd dip
IP PSL,(C)
zrrb (O) dip n
ni(N) 7il'Tiplzi(N)
THEoREM 2. PVith the notation as above, we have the equality:
vol. (ip)= -21 h-- vol (dMip l z,(N)) .
PRooF. Let q- : N--ÅrdC is the projection induced from the orbifold covering q: N.O.
By this projection, the double bent manifold structure dgip on N is induced from the structure dBdi on dC. We note that vol. (ip) == -21 - vol (dBip) =-21 i-- vol (dgip). Let g: N. fi7 Å~
H3 be the section defined by sN (x)==(x, D(x)) for xEN, where N is the universal covering space ofN and D: N-•H3 is the developing map for dB'" ip. We set p=ZITip1ni(N). Since
s" i' s ni(N)-equivariant, there exists the section s:N-ÅrE, covered by sN. Let A be any
3-TH-simplex of dgip. According to the construction of s, we have:
vol(zl)=j, s*(CO,).
Therefore we have vol(dB"V ip)=vol(p). This completes the proof. O
References
[1] W.Goldman, DiscontinuousgroupsandtheEulerclass, Thesis,U.C.Berkeley,l980.
[2] S. Kojima, K. Ohshika and T. Soma, Towards a proof of Thurston's geometrization theorem for orbifolds, Informal note, Tokyo Metro. Univ., 1985.
[3] J. Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, "The Smith Conjecture" (ed. by J. Morgan and H. Bass), pp. 37-125, Academic Press, 1984.
[4] T.Soma, Constructionofhyperbolic3-orbifoldsfromcertaintepresentations, toappear.
[5] W. Thurston, The geometry and topology of3-manifolds, Mimeographed notes, Princeton Univ.
1977178.
Department of Mathematics Kyushu Institute of Technology
