Table Of ContentGeodesic ideal triangulations exist virtually
FengLuo,SaulSchleimerandStephanTillmann
7
0
0 Abstract Itisshownthateverynon-compacthyperbolicmanifoldoffinitevol-
2
ume has a finite cover admitting a geodesic ideal triangulation. Also, every hy-
n perbolicmanifoldoffinitevolumewithnon-empty,totallygeodesicboundaryhas
a
afiniteregularcoverwhichhasa geodesicpartiallytruncatedtriangulation. The
J
proofsuseanextensionofaresultduetoLongandNibloconcerningthesepara-
6
bilityofperipheralsubgroups.
1
AMSClassification 57N10,57N15;20H10,22E40,51M10
]
T
Keywords hyperbolicmanifold,idealtriangulation,partiallytruncatedtriangu-
G
lation,subgroupseparability
.
h
t Epstein and Penner [2] used a convex hull construction in Lorentzian space to show
a
m thateverynon-compact hyperbolic manifoldoffinitevolumehasacanonical subdivi-
[ sion into convex geodesic polyhedra all of whose vertices lie on the sphere at infinity
ofhyperbolicspace. Ingeneral,onecannotexpecttofurthersubdividethesepolyhedra
1
v into ideal geodesic simplices such that the result isanideal triangulation. That this is
1 possible after lifting the cell decomposition to an appropriate finite cover is the first
3
mainresult ofthis paper. Acelldecomposition ofahyperbolic n–manifold into ideal
4
1 geodesic n–simplices all of which are embedded will be referred to as an embedded
0 geodesic idealtriangulation.
7
0
/ Theorem1 Anynon-compacthyperbolicmanifoldoffinitevolumehasafiniteregu-
h
larcoverwhichadmitsanembeddedgeodesicidealtriangulation.
t
a
m
The study of geodesic ideal triangulations of hyperbolic 3–manifolds goes back to
:
v Thurston [11]. They are known to have nice properties through, for instance, work
i
X by Neumann and Zagier [8]and Choi[1]. Petronio and Porti[9]discuss the question
r ofwhether everynon-compact hyperbolic 3–manifold offinitevolume hasageodesic
a
idealtriangulation —thisquestion stillremainsunanswered.
Kojima[5]extended theconstruction byEpsteinandPennertoobtain acanonical de-
compositionintopartiallytruncatedpolyhedraofanyhyperbolicmanifoldwithtotally
geodesic boundary components. A cell decomposition of a hyperbolic n–manifold
with totally geodesic boundary into geodesic partially truncated n–simplices all of
which are embedded will be referred to as an embedded geodesic partially truncated
triangulation.
0Thisworkisinthepublicdomain.
1
Theorem2 Anyfinite-volumehyperbolicmanifoldwithnon-empty,totallygeodesic
boundaryhasafiniteregularcoverwhichadmitsanembeddedgeodesicpartiallytrun-
catedtriangulation.
An ideal polyhedron willbe viewed as aspecial instance of a partially truncated one,
which allows a unified proof of Theorems 1 and 2. They are proved by showing
that any cell decomposition lifts to some finite regular cover where it can be subdi-
vided consistently. Inparticular, onehasthefollowing application. Kojima[4]shows
that any 3–dimensional finite-volume hyperbolic manifold with non-empty, totally
geodesic boundary has a decomposition into geodesic partially truncated polyhedra
eachofwhichhasatmostoneidealvertex. Frigerio[3]conjecturesthatsuchadecom-
position exists where each polyhedron is a tetrahedron; avirtually affirmative answer
isanimmediateconsequence oftheproofofTheorem2:
Corollary 3 Any3–dimensionalfinite-volumehyperbolicmanifoldwithnon-empty,
totallygeodesicboundaryhasafiniteregularcoverwhichadmitsadecompositioninto
partiallytruncatedgeodesictetrahedraeachofwhichhasatmostoneidealvertex.
ThekeyresultusedintheproofofTheorem2isthefollowingtheoremwhichfollows
easily from work by Long and Niblo [6]. A subgroup H of a group G is separable
in G if given any element g ∈G\H, there is a finite index subgroup K ≤G which
contains H but g∈/ K. If M is a hyperbolic manifold of finite volume with (possibly
empty)totallygeodesicboundary, thenasubgroupof p (M) istermedperipheral ifit
1
iseitherconjugatetothefundamentalgroupofatotallygeodesicboundarycomponent
ortothefundamental groupofacuspor ¶ –cusp.
Theorem 4 (Long–Niblo) Let M beahyperbolicmanifoldoffinitevolumewith
(possiblyempty)totallygeodesicboundary. TheneveryperipheralsubgroupofM is
separableinp (M).
1
Acknowledgements The authors thank Chris Leininger for an enlightening conver-
sation. They also thank Daryl Cooper for bringing the work by Long and Niblo to
theirattention. ResearchofthefirstauthorissupportedinpartbytheNSF.Thesecond
author ispartly supported bytheNSF(DMS-0508971). Thethird author issupported
under the Australian Research Council’s Discovery funding scheme (project number
DP0664276).
1 Subgroup separability
Let M be a finite-volume hyperbolic n–manifold with non-empty totally geodesic
boundary. Following Kojima [5], the periphery of M is made up of three parts:
2
first, ¶ M consisting of totally geodesic closed or non-compact hyperbolic (n−1)–
manifolds;second, (internal)cuspsmodelledonclosedEuclidean (n−1)–manifolds;
and third, ¶ –cusps modelled oncompact Euclidean (n−1)–manifolds withgeodesic
boundary. Theboundary of ¶ –cusps iscontained onnon-compact geodesic boundary
components.
Fortheremainderofthispaper, M denotesafinite-volumehyperbolicmanifoldwhich
is either non-compact or has non-empty totally geodesic boundary. Without loss of
generality, itmaybeassumedthat M isorientable. Notethateither Me =IHn oritcan
be viewed as the complement of an infinite set of hyperplanes in IHn; in either case
thereisanidentification p (M)=G ≤Isom+(IHn).
1
Proposition5 (Long–Niblo) LetX beatotallygeodesiccomponentof¶ M.Choose
abasepointx∈X.Thenp (X,x) isaseparablesubgroupofp (M,x).
1 1
Proof Let D denotethemanifoldobtainedbydoubling M along X.Then D ishyper-
bolicwith(possiblyempty)totallygeodesicboundary,andhence p (D)≤Isom+(IHn)
1
is residually finite due to a result by Mal′cev [7]. The proof in §2 of Long and Niblo
[6]nowappliestothisset-up. (cid:4)
Proposition 6 Let X beahorosphericalcrosssectionofacuspor ¶ –cuspof M.
Chooseabasepointx∈X. Thenp (X,x) isaseparablesubgroupofp (M,x).
1 1
Proof Thisfollowsfromthewell-knownresultthatamaximalabeliansubgroup ofa
residually finitegroup G isseparable in G (seeRatcliffe[10]). (cid:4)
ProofofTheorem4 Firstnote that if j : G →G isanisomorphism and H ≤G
1 2 1
isseparablein G , then j (H)≤G isseparablein G . Inparticular, referencetobase
1 2 2
points can be omitted. Next, note that if H ≤ G is separable, so is g −1Hg for any
g ∈G . Thus,Theorem4followsfromtheabovepropositions fororientablemanifolds.
If M is non-orientable, denote by G a subgroup of index two of G corresponding
0
to the fundamental group of the orientable double cover. Then any subgroup H ≤G
is separable in G if and only if H∩G is separable in G . Now if H is a peripheral
0 0
subgroup of G , then H∩G isaperipheral subgroup of G . (cid:4)
0 0
2 Partially truncated polyhedra
Certain convex geodesic polyhedra in IHn are termed geodesic partially truncated
polyhedra and can be described intrinsically. However, reference to the projective
ballmodel Bn⊂IRn willbemadehere,and IHn willbeidentifiedwith Bn throughout.
3
Let Pˆ be an n–dimensional convex Euclidean polyhedron in IRn such that (1) each
vertex is either called ideal or hyperideal, (2) its ideal vertices are contained on ¶ Bn,
(3) itshyperideal vertices arecontained in IRn\Bn, and(4) each face ofcodimension
two meets Bn. Then a convex geodesic polyhedron P⊂Bn is obtained by truncating
Pˆ along hyperplanes canonically associated to its hyperideal vertices as follows. If
v∈IRn is a hyperideal vertex then the associated hyperplane H(v) is the hyperplane
parallel to the orthogonal complement of v (with respect to the standard Euclidean
innerproducton IRn)whichmeets ¶ Bn inthesetofallpoints x withthepropertythat
there is a tangent line to ¶ Bn passing through x and v. The polyhedron P is termed
a geodesic partially truncated polyhedron, and Pˆ the Euclidean fellow of P. Combi-
natorially, P is obtained from Pˆ by removing disjoint open stars of all the hyperideal
verticesaswellasalltheidealvertices.
If a codimension one face of P is contained in a face of Pˆ, then it is called lateral;
otherwiseitisatruncationface. Lateralfacesandtruncationfacesmeetatrightangles.
If P hasnotruncation faces,thenitisalsotermedageodesic idealpolyhedron.
Anysubdivision of Pˆ into n–simplices withoutintroducing newverticesuniquely de-
termines a subdivision of P into geodesic partially truncated n–simplices. Thepoly-
hedron Pˆ istermedtheEuclidean fellowof P.
3 The pulling construction
Let (C,F ) be a geodesic partially truncated cell decomposition of M, that is, C is
a disjoint union of geodesic partially truncated polyhedra, each element in F is an
isometricfacepairing,and M=C/F . Then (C,F ) pullsbacktoa G –equivariantcell
decomposition of Me ⊆Bn, andforeach P∈C onemaychoose anisometric lift P˜ to
Bn andhenceaEuclideanfellow Pˆ⊂IRn. Thehyperideal verticesof Pˆ correspond to
totally geodesic boundary components of M, theidealvertices of Pˆ tointernal cusps,
andtheintersection ofcodimension-two facesof Pˆ with ¶ Bn to ¶ –cusps.
Let Cˆ=∪{Pˆ} be the finite disjoint union of the Euclidean fellows, and view P⊂Pˆ.
The cell decomposition of M induces face pairings Fˆ such that M is obtained from
the pseudo-manifold Mˆ =Cˆ/Fˆ by deleting the ideal vertices and open stars of the
hyperideal vertices, andeachelementof Fˆ restrictstoanelementof F .
Lemma7 (Pullingconstruction) SupposethatnopolyhedroninCˆhastwodistinct
verticesidentifiedinMˆ. ThenM hasanembeddedgeodesicpartiallytruncatedtrian-
gulation.
4
Proof Itsufficestoshowthatthereisasubdivisionof (Cˆ,Fˆ) suchthat(1)eachpoly-
hedron in Cˆ is simplicially subdivided into straight Euclidean n–simplices without
introducing newvertices,and(2)theelementsof Fˆ restricttosimplicialfacepairings
withrespecttothesubdivision.
Chooseanorderingofthecuspsandtotallygeodesicboundarycomponentsof M.This
determines a well-defined, unique ordering of the 0–skeleton of Mˆ and, by assump-
tion, of the vertices of each polyhedron in Cˆ. One thus obtains the following unique
subdivision ofeachpolyhedron.
Let P∈C, andlabelitsvertices v ,v ,...,v suchthat v >v if i< j. Subdivide Pˆ by
0 1 k i j
coningto v eachelementofits i–skeleton, 0≤i≤n−1, whichdoesnotcontain v .
0 0
The result is a collection of polyhedra, P , together with well-defined face pairings
0
F such that the identification space P /F is Pˆ. One now proceeds inductively.
0 0 0
Given P and F , subdivide each polyhedron in P containing v by coning to
j j j j+1
v eachelementofits i–skeleton, 0≤i≤n−1, whichdoesnotcontain v . This
j+1 j+1
gives P , togetherwithwell-definedfacepairings F suchthattheidentification
j+1 j+1
space P /F is Pˆ.
j+1 j+1
ItneedstobeshownthatthesetP isacollectionofn–simplices. Indeed,letQ∈P ,
k k
and assume that v is its smallest vertex. Then Q is the cone to v of an (n−1)–
h h
dimensional face Fn−1 not containing v . The face Fn−1 is the cone to its smallest
h
vertex ofan (n−2)–dimensional face Fn−2 notcontaining that vertex, anditfollows
inductively that Q hasexactly n+1 vertices.
Let Pˆ,Pˆ′∈Cˆwithtop-dimensionalfaces Fˆ, Fˆ′ suchthatthereisafacepairing j ∈Fˆ
with j (Fˆ)=Fˆ′. Therespectivesubdivisionsof Fˆ and Fˆ′ intoideal (n−1)–simplices
dependuniquelyontheorderingoftheirvertices. Whence j issimplicialwithrespect
to the subdivisions, and restricts to a simplicial face pairing for each n–simplex in
the subdivision. Moreover, the resulting decomposition of Mˆ is simplicial since any
n–simplexhasnotwoverticesidentified, andhencemustbeembeddedin Mˆ. (cid:4)
4 Proof of the main results
The strategy of the proof is to create a finite regular cover N of M with the property
that Lemma 7 can be applied to the pull back of C. The notation of the previous
sections will be used. Recall that for each P∈C, there is the fixed Euclidean fellow
Pˆ ⊂IRn. The action of G on Bn extends to IRn\Bn via the action on the associated
hyperplanes. In particular, if v ∈ Pˆ is a vertex, then the subgroup StabG (v) ≤ G is
peripheral.
5
Let D(M) be the following set of pairs of points in IRn : (v,w)∈D(M) if and only
if there is some P∈C such that v and w are distinct vertices of Pˆ. Note that D(M)
is finite; its elements are termed diagonals for M. A diagonal (v,w) is said to be
returning ifthere is g ∈G suchthat g v=w. Notethatthepulling construction canbe
applied to Cˆ ifnodiagonal isreturning. Henceassumethatthisisnotthecase.
If p: N →M isafinitecover, thenthecelldecomposition (C,F ) pullsbacktoacell
decomposition of N, and there is a corresponding set of diagonals for N. If P∈ C
pulls back to P ,...,P, then (uptorelabelling) onemaychoose Pˆ=Pˆ . Inparticular,
1 k 1
it may be assumed that D(M) ⊂D(N); any other element of D(N) is of the form
g ·(v,w) = (g v,g w) for some g ∈ G and (v,w) ∈ D(M). This choice will be made
throughout. If (v,w)∈D(M) is not a returning diagonal for M, then it is also not a
returning diagonal for N.
Assumethat (v,w)∈D(M) isareturning diagonal.
Lemma 8 Thereisafinite(possiblynotregular)cover p: N → M suchthat
(v,w)
(v,w)∈D(N) isnotareturningdiagonal.
Proof Since (v,w) isareturning diagonal for M, thereis g ∈G suchthat g v=w. In
particular, g ∈/ StabG (v) because v and w are distinct. Since StabG (v) is a peripheral
subgroup, Theorem 4 yields a finite index subgroup K ≤G which contains StabG (v)
but g ∈/ K. Denote by p: N →M the finite cover corresponding to the subgroup
(v,w)
e
K, i.e.N =M/K.
(v,w)
Assume that (v,w) ∈ D(N) is a returning diagonal. Then there is d ∈ K with the
property that d v = w. Thus, g −1d ∈ StabG (v) ≤ K which implies g ∈ K. But this
contradicts thechoiceof K. Whence (v,w) isnotareturning diagonal for N. (cid:4)
Lemma9 IfN→M isaregularcoverwhichfactorsthroughN ,thennoelement
(v,w)
oftheorbitG ·(v,w) canbeareturningdiagonalforN.
Proof If N → M is a cover which factors through N , then (v,w) cannot be a
(v,w)
returning diagonal for N. If N →M is a regular cover, then no element of the orbit
G ·(v,w) is a returning diagonal for N since the action of the group of deck transfor-
mationsistransitive and p (N) corresponds toanormalsubgroup of p (M). (cid:4)
1 1
Foreachdiagonal (v,w) chooseafinitecover N →M withthepropertythat (v,w)
(v,w)
is not areturning diagonal for N . This gives afinite collection of covers, and one
(v,w)
may pass to a common finite cover N with the property that N →M is regular. In
particular, no element of the orbit of any diagonal for M can be a returning diagonal
for N. The cell decomposition (C,F ) of M lifts to a polyhedral cell decomposition
of N, towhichthepulling construction canthusbeapplied. Thiscompletes theproof
ofTheorems1and2. (cid:4)
6
References
[1] Young-Eun Choi: Positively oriented ideal triangulations on hyperbolic three-
manifolds,Topology43,no.6,1345–1371(2004).
[2] David B.A. Epstein & Robert C. Penner: Euclidean decompositions of noncompact
hyperbolicmanifolds.J.DifferentialGeom.27(1988),no.1,67–80.
[3] RobertoFrigerio:Ondeformationsofhyperbolic3–manifoldswithgeodesicboundary.
Algebr.Geom.Topol.6(2006),435–457(electronic).(math.GT/0504116)
[4] SadayoshiKojima: Polyhedraldecompositionofhyperbolic3–manifoldswith totally
geodesic boundary.Aspects of low-dimensionalmanifolds, 93–112,Adv.Stud. Pure
Math.,20,Kinokuniya,Tokyo,1992.
[5] SadayoshiKojima: PolyhedralDecompositionof HyperbolicManifoldswith Bound-
ary,Onthegeometricstructureofmanifolds,editedbyDongPyoChi,Proc.ofWork-
shopinPureMath.,10,partIII(1990),37–57.
[6] Darren D. Long, Graham A. Niblo: Subgroup separability and 3–manifold groups.
Math.Z.207(1991),no.2,209–215.
[7] Anatoli˘ıIvanovichMal′cev: Onisomorphicmatrixrepresentationsofinfinitegroups.
Rec.Math.[Mat.Sbornik]N.S.8(50),(1940).405–422.
[8] Walter D. Neumann,Don Zagier: Volumesof hyperbolicthree–manifolds,Topology,
24,307-332(1985).
[9] Carlo Petronio, Joan Porti: Negatively oriented ideal triangulations and a proof of
Thurston’s hyperbolic Dehn filling theorem. Expo. Math. 18 (2000), no. 1, 1–35.
(math.GT/9901045)
[10] JohnG.Ratcliffe: Foundationsofhyperbolicmanifolds.GraduateTextsinMathemat-
ics,149.Springer-Verlag,NewYork,1994.
[11] William P. Thurston: The geometry and topology of 3–manifolds, Princeton Univ.
Math.Dept.(1978).Availablefromhttp://msri.org/publications/books/gt3m/
DepartmentofMathematics,RutgersUniversity,NewBrunswick,NJ08854,USA
DepartmentofMathematics,RutgersUniversity,NewBrunswick,NJ08854,USA
DepartmentofMathematicsandStatistics,TheUniversityofMelbourne,VIC3010,Australia
Email: [email protected], [email protected],
[email protected]
7
