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1、AlanF.BeardonTheGeometryofDiscreteGroupsWith93IllustrationsSpringerAlanF.BeardonUniversityofCambridgeDepartmentofPureMathematicsandMathematicalStatistics16MillLaneCambridgeCB21SBEnglandEditorialBoardS.AxierF.W.GehringP.R.HalmosDepartmentofDepartmentofDepartmentofMathematic
2、sMathematicsMathematicsMichiganStateUniversityUniversityofMichiganSantaClaraUniversityEastLansing,MI48824AnnArbor,MI48109SantaClara,CA95053USAUSAUSAMathematicsSubjectClassifications(1991):30-01,30CXX,20F32,30FXX,51Mb,20HXXLibraryofCongressCataloginginPublicationDataBeardon
3、,AlanF.Thegeometryofdiscretegroups.(Graduatetextsinmathematics;91)Includesbibliographicalreferencesandindex.1.Discretegroups.2.Isometries(Mathematics)3.Möbiustransformations.4.Geometry,Hyperbolic.I.Title.II.Series.QA17I.B3641983512'.282-19268©1983bySpringer-VerlagNewYorkIn
4、c.Allrightsreserved.NopartofthisbookmaybetranslatedorreproducedinanyformwithoutwrittenpermissionfromSpringer-Verlag,175FifthAvenue,NewYork,NewYork10010,U.S.A.TypesetbyCompositionHouseLtd.,Salisbury,England.PrintedandboundbyR.R.Donnelley&Sons,Harrisonburg,VA.PrintedintheUni
5、tedStatesofAmerica.98765432(Correctedsecondprinting,1995)ISBN0-387-90788-2Springer-VerlagNewYorkHeidelbergBerlinISBN3-540-90788-2Springer-VerlagBerlinHeidelbergNewYorkToToniPrefaceThistextisintendedtoserveasanintroductiontothegeometryoftheactionofdiscretegroupsofMöbiustran
6、sformations.Thesubjectmatterhasnowbeenstudiedwithchangingpointsofemphasisforoverahundredyears,themostrecentdevelopmentsbeingconnectedwiththetheoryof3-manifolds:see,forexample,thepapersofPoiricaré[77]andThurston[101].About1940,thenowwell-known(butvirtuallyunobtainable)Fench
7、el—Nielsenmanuscriptappeared.Sadly,themanuscriptneverappearedinprint,andthismoremodesttextattemptstodisplayatleastsomeofthebeautifulgeo-metricalideastobefoundinthatmanuscript,aswellassomemorerecentmaterial.Thetexthasbeenwrittenwiththeconvictionthatgeometricalexplana-tionsa
8、reessentialforafullunderstandingofthematerialandthathoweversimpleamatrixproofmightseem,ag
