Finite-volume hyperbolic 4-manifolds that

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1、DifferentialGeometryanditsApplications10(I999)205-223205North-HollandFinite-volumehyperbolic4-manifoldsthatshareafundamentalpolyhedron*DubravkoIvanSiCllr?i~ervit~ofOklrrhoma.MathemuticsDepartment,Normun,OK730/Y-03/5.USA

2、mefunctionforhyperbolicmanifoldsofdimension>3isfinite-to-one.Weshowthatthenumberofnonhomeomorphichyperbolic4-manifoldswiththesamevolumecanbemadearbitrarilylarge.Thisisdonebyconstructingasequenceoffinite-sidedfinite-volumepolyhedrawithside-pairingsthatyieldmanifolds.Infact,weshowthatarbitrarilymanyn

3、onhomeomorphichyperbolic4-manifoldsmayshareafundamentalpolyhedron.Asaby-productofourexamples,wealsoshowinaconstructivewaythatthesetofvolumesofhyperbolic4-manifoldscontainsthesetofevenintegralmultiplesof4n2/3.Thisishalfthesetofpossiblevaluesforvolumes,whichistheintegralmultiplesof4rr/3duetotheGauss-

4、BonnetformulaVol(M)=4n2/3x(M).Ke~word.v:Hyperbolic4-manifolds,volumefunction,Poincarespolyhedrontheorem,embeddedtotallyFe-odesichypersurfaces.Gauss-Bonnetformula,setofvolumes.iUSc./cl.v.siiir,ritiorr:5IMIO.5IM25.0.IntroductionandstatementofresultsTheoriginalaimofresearchthatproducedthispaperwastoco

5、nstructnoncompacthyperbolic4-manifoldsbymeansofside-pairingsofpolyhedra.Previousexamplesofhyperbolicmanifoldswithdimensionhigherthanthreewererestrictedtoconstructionsviaarithmeticgroups(see,forexample,[2,13]),orviainterbreedingarithmeticgroupstogetnonarithmeticones]9]andtherewasonlyone(compact)exam

6、pleusingside-pairings,thatofDavisin[5].Wewereabletoproduceanumberofexamplesofside-pairingsofhyperbolic4-polyhedraandget,inaddition.newinformationaboutvolumesofhyperbolic4-manifolds.Furtherresearchledtoconsiderationofembedabilityofthesemanifoldsascomplementsofsurfacesincompact4-manifolds-wedealwitht

7、hisin]I1J.Itisknown(see1171)thatforeveryconstantc;>0thereareonlyfinitelymanycompletenonhomeomorphichyperbolicn-manifoldswithvolume