Arithmetic hyperbolic manifolds

107 Pages·2014·1.06 MB·English

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Arithmetic hyperbolic manifolds
AlanW.Reid
UniversityofTexasatAustin
CornellUniversity
June2014
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Thurston(Qn19ofthe1982BulletinoftheAMSarticle)
Findtopologicalandgeometricpropertiesofquotientspacesof
arithmeticsubgroupsofPSL(2,C). Thesemanifoldsoftenseemto
havespecialbeauty.
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Thurston(Qn19ofthe1982BulletinoftheAMSarticle)
Findtopologicalandgeometricpropertiesofquotientspacesof
arithmeticsubgroupsofPSL(2,C). Thesemanifoldsoftenseemto
havespecialbeauty.
Manyofthekeyexamplesinthedevelopmentofthetheoryof
geometricstructureson3-manifolds(e.g. thefigure-eightknot
complement,theWhiteheadlinkcomplement,thecomplementofthe
BorromeanringsandtheMagicmanifold)arearithmetic.
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Themodulargroup
Thebasicexampleofan”arithmeticgroup”is
PSL(2,Z) = SL(2,Z)/ Id.
±
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Themodulargroup
Thebasicexampleofan”arithmeticgroup”is
PSL(2,Z) = SL(2,Z)/ Id.
±
Everynon-cocompactfiniteco-areaarithmeticFuchsiangroupis
commensurablewiththemodulargroup.
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Themodulargroup
Thebasicexampleofan”arithmeticgroup”is
PSL(2,Z) = SL(2,Z)/ Id.
±
Everynon-cocompactfiniteco-areaarithmeticFuchsiangroupis
commensurablewiththemodulargroup.
SomeparticularlyinterestingsubgroupsofPSL(2,Z)offiniteindex
arethecongruencesubgroups.
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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 .
{± }
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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.
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Thestructureofcongruencesubgroups(genus,torsion,numberof
cusps)hasbeenwell-studied.
RademacherConjecture: Thereareonlyfinitelymanycongruence
subgroupsofgenus0(orfixedgenus).