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1 Arithmetic hyperbolic manifolds AlanW.Reid UniversityofTexasatAustin CornellUniversity June2014 2 Thurston(Qn19ofthe1982BulletinoftheAMSarticle) Findtopologicalandgeometricpropertiesofquotientspacesof arithmeticsubgroupsofPSL(2,C). Thesemanifoldsoftenseemto havespecialbeauty. 3 Thurston(Qn19ofthe1982BulletinoftheAMSarticle) Findtopologicalandgeometricpropertiesofquotientspacesof arithmeticsubgroupsofPSL(2,C). Thesemanifoldsoftenseemto havespecialbeauty. Manyofthekeyexamplesinthedevelopmentofthetheoryof geometricstructureson3-manifolds(e.g. thefigure-eightknot complement,theWhiteheadlinkcomplement,thecomplementofthe BorromeanringsandtheMagicmanifold)arearithmetic. 4 5 Themodulargroup Thebasicexampleofan”arithmeticgroup”is PSL(2,Z) = SL(2,Z)/ Id. ± 6 Themodulargroup Thebasicexampleofan”arithmeticgroup”is PSL(2,Z) = SL(2,Z)/ Id. ± Everynon-cocompactfiniteco-areaarithmeticFuchsiangroupis commensurablewiththemodulargroup. 7 Themodulargroup Thebasicexampleofan”arithmeticgroup”is PSL(2,Z) = SL(2,Z)/ Id. ± Everynon-cocompactfiniteco-areaarithmeticFuchsiangroupis commensurablewiththemodulargroup. SomeparticularlyinterestingsubgroupsofPSL(2,Z)offiniteindex arethecongruencesubgroups. 8 AsubgroupΓ < PSL(2,Z)iscalledacongruencesubgroupifthere existsann ZsothatΓcontainstheprincipalcongruencegroup: ∈ Γ(n) = ker PSL(2,Z) PSL(2,Z/nZ) , { → } wherePSL(2,Z/nZ) = SL(2,Z/nZ))/ Id . {± } 9 AsubgroupΓ < PSL(2,Z)iscalledacongruencesubgroupifthere existsann ZsothatΓcontainstheprincipalcongruencegroup: ∈ Γ(n) = ker PSL(2,Z) PSL(2,Z/nZ) , { → } wherePSL(2,Z/nZ) = SL(2,Z/nZ))/ Id . {± } niscalledthelevel. 10 Thestructureofcongruencesubgroups(genus,torsion,numberof cusps)hasbeenwell-studied. RademacherConjecture: Thereareonlyfinitelymanycongruence subgroupsofgenus0(orfixedgenus).

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These manifolds often seem to geometric structures on 3-manifolds (e.g. the figure-eight knot .. Generators for the fundamental group are (from.

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These manifolds often seem to geometric structures on 3-manifolds (e.g. the figure-eight knot .. Generators for the fundamental group are (from.

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Author:	Unknown
Publication Year:	2014
Pages:	107
Language:	English
File Size:	1.06
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