Growth of regulators in finite abelian coverings

22 Pages·2013·0.34 MB·English

Algebraic & Geometric Topology13(2013)2383–2404
msp
Growth of regulators in
finite abelian coverings
THANG TQ LÊ
Weshowthattheregulator,whichisthedifferencebetweenthehomologytorsionand
thecombinatorialRay–Singertorsion,offiniteabeliancoveringsofafixedcomplex
hassub-exponentialgrowthrate.
54H20;57Q10,37B50,37B10
1 Introduction
1.1 Basedfreecomplexovergroupringanditsquotients
Suppose (cid:25) isafinitelypresentedgroupand ZŒ(cid:25)(cid:141) isthegroupringof (cid:25) overthering
Z ofintegers.
Let C beafinitelygeneratedbasedfree ZŒ(cid:25)(cid:141)–complex
@ @ @ @
0!Cm(cid:0)!m Cm(cid:0)1 (cid:0)m!(cid:0)1Cm(cid:0)2(cid:0)!(cid:1)(cid:1)(cid:1)(cid:0)!2 C1(cid:0)!1 C0!0:
Here “based free” means each C is a free ZŒ(cid:25)(cid:141)–module equipped with a preferred
k
base.
For a normal subgroup (cid:128) C (cid:25) let C(cid:128) WD ZŒ(cid:25)=(cid:128)(cid:141)˝ZŒ(cid:25)(cid:141)C. Assume that the index
Œ(cid:25) W (cid:128)(cid:141) is finite. Then C(cid:128) is a finitely generated based free Z–complex, where the
preferredbaseof ZŒ(cid:25)=(cid:128)(cid:141)˝ZŒ(cid:25)(cid:141)Ck isdefinedusingtheoneof Ck inanaturalway.
A prototypical case is the following. Suppose Xz ! X is a regular covering with
(cid:25) the group of deck transformations and X a finite CW–complex. Choose a lift in
z z
X of each cell of X. Then the CW–complex C of X induced from that of X is a
finitelygeneratedbasedfree ZŒ(cid:25)(cid:141)–complex. Foranormalsubgroup (cid:128) C(cid:25), C(cid:128) isthe
CW–complex ofthecovering X(cid:128),corresponding tothegroup (cid:128),and Hk.C(cid:128)/ isthe
kth homologyofthecovering X(cid:128). Usually,interestinginvariantsdonotdependonthe
choiceoftheliftsofcellsof X.
Published: 2July2013 DOI:10.2140/agt.2013.13.2383
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1.2 Twotorsions
Wecandefinetwotorsionsofthequotientcomplex C(cid:128),thehomologytorsion (cid:28)H.C(cid:128)/
andthecombinatorialRay–Singertorsion (cid:28)RS.C(cid:128)/,asfollows. Thehomologytorsion
is
(cid:18)Y(cid:3) (cid:19)(cid:0)1
(cid:28)H.C(cid:128)/WD jtorZ.Hk.C(cid:128)//j 2RC;
k
where tor .M/ isthe Z–torsionpartofthefinitelygeneratedabeliangroup M,and
Z
Q(cid:3)
a isthealternatingproduct
k k
Y(cid:3)a DYa.(cid:0)1/k:
k k
k k
TheRay–Singertorsionof C(cid:128) is
(cid:28)RS.C(cid:128)/DY(cid:3)det0.@k/2RC
k
0
where det isthegeometricdeterminantoflinearmapsbetweenbasedHermitianspaces.
0
Werecallthedefinitionof det inSection2.
1.3 Comparison: generalquestion
We want tocompare theasymptotics ofthe two torsionsas (cid:128) becomes “thinnerand
thinnerin (cid:25)”,sothat (cid:25)=(cid:128) approximates (cid:25) inthefollowingsense. Afiniteset S of
generatorsof (cid:25) definesawordlengthfunction l (andhenceametric)on (cid:25). Define
S
h(cid:128)iWDminfl .x/jx2(cid:128)nfegg:
S
Here e istheunitof (cid:25). Inallthatfollows,statementsdonotdependonthechoiceof
thegeneratorset S,sincethemetricsoftwodifferentgeneratorsetsarequasi-isometric.
We are interested in the following question: Suppose C is L2–acyclic (see eg Lück
[12]). Underwhatconditionsdoesitholdthat
(1) lim ln.(cid:28)H.C(cid:128)//(cid:0)ln.(cid:28)RS.C(cid:128)// D0?
j(cid:25) W(cid:128)j
h(cid:128)i!1;
j(cid:25)W(cid:128)j<1
Themotivationofthisquestioncomesfromthequestion[12]: canoneapproximate
L2–torsionsbyfinite-dimensionalanalogs? Insomefavorableconditions,oneexpects
thatthegrowthrateofeachof (cid:28)H and (cid:28)RS isthe L2–torsion,hencetheymustbethe
same.
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Remark 1.1 (a) If f(cid:128) ;n D 1;2;:::g is a sequence of exhausting nested normal
n
subgroupsof (cid:25),ie, (cid:128)nC1(cid:26)(cid:128)n and Tn(cid:128)nDfeg,then limn!1h(cid:128)niD1. Thelimit
in(1)ismoregeneral(stronger)thanthelimitofanexhaustingnestedsequence,aswe
donothavethe“nested”property.
(b) Thereexistsasequence (cid:128)nC(cid:25) suchthat limn!1h(cid:128)niD1 ifandonlyif (cid:25) is
residuallyfinite. Hence,thelefthandsideof(1)makessenseonlywhen (cid:25) isresidually
finite.
(c) Define tr(cid:25).x/Dıx;e for x2(cid:25). Thisfunctionaltraceisthebaseforthedefinition
ofmanycombinatorial L2–invariants. Forafixed x2(cid:25),wehave
(2) lim tr(cid:25)=(cid:128).x/Dtr(cid:25).x/:
h(cid:128)i!1
Thisisthereasonwhyoneexpectsthatas h(cid:128)i!1,many L2–invariants(undersome
technicalconditions)canbeapproximatedbythecorrespondinginvariantsof (cid:25)=(cid:128).
1.4 Mainresults
Themainresultofthepapertreatsthecase (cid:25) DZn.
Theorem1 Suppose C isan L2–acyclicfinitelygeneratedbasedfree CŒZn(cid:141)–complex.
Then(1),with (cid:25) DZn,holdstrue.
Wewillnotgivethedefinitionof L2–acyclicity. Instead,for (cid:25) DZn,wewillusean
equivalent definition (Elek [4], Lück [12]): the L.2/ homology H.2/.C/ vanishes if
k
and only if Hk.C˝ZŒZn(cid:141)F/D0. Here F is the fractional field of the commutative
domain ZŒZn(cid:141).
Remark1.2 (a) Ourresultdoesnotimplythat
lim ln.(cid:28)H.C(cid:128)// D lim ln.(cid:28)RS.C(cid:128)// ;
j(cid:25) W(cid:128)j j(cid:25) W(cid:128)j
h(cid:128)i!1; h(cid:128)i!1;
j(cid:25)W(cid:128)j<1 j(cid:25)W(cid:128)j<1
as we cannot prove the existence of each of the limits. For (cid:25) D Z, it was known
thatbothlimitsexistandareequaltothe L2–torsionof C;seeGonzález-Acuñaand
Short [6], Riley [17], and Lück [12]. Even for the case where (cid:25) D Z2 and C is a
2–termcomplex 0!C !C !0 (sothatonly H .C/ isnon-trivial),thereisstill
1 0 0
no proof of the conjecture that the L2–torsion is equal to either of the above limits.
Forresultsanddiscussionsofthisandrelatedconjectures,seeLück[12;11],Bergeron
andVenkatesh[1],Lê[10;9],FriedlandJackson[5],andSilverandWilliams[20].
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(b) Itshouldbenotedthattheexactcalculationofthetorsionpartofthehomologyof
finitecoverings,evenintheabeliancase,isverydifficult;seeHillmanandSakuma[7],
MayberryandMurasugi[13],andPorti[15]forsomepartialresults.
1.5 Refinement
Suppose (cid:25) is residually finite and the L2–homology H.2/.C/D0 for some k. For
k
anynormalsubgroup (cid:128) C(cid:25) offinite index, thehomologygroup Hk.C(cid:128)/ isa finitely
generated abelian group. Because Hk.2/.C/ D 0 one should expect that Hk.C(cid:128)/ is
negligible. Infact,atheoremofLück[11](andKazhdanforthiscase)saysthat
lim rkZHk.C(cid:128)/ D0:
j(cid:25) W(cid:128)j
h(cid:128)i!1;
j(cid:25)W(cid:128)j<1
Thismeansthefreepart Hk.C(cid:128)/free of Hk.C(cid:128)/ issmallcomparedtotheindex. There
is another measure of the free part Hk.C(cid:128)/free, denoted by Rk.C(cid:128)/ and called the
regulator, or volume; see [1] and Section 3. Another expression of the fact that
Hk.C(cid:128)/free is small compared to the index is expressed in the following statement,
whichcomplementstheresultofKazhdanandLück.
Theorem2 Suppose C isafinitelygeneratedbasedfree ZŒ(cid:25)(cid:141)–complexwith (cid:25) DZn
and H.2/.C/D0 forsomeindex k. Then
k
(3) lim lnvol.Hk.C(cid:128)/free/ D0:
j(cid:25) W(cid:128)j
h(cid:128)i!1;
j(cid:25)W(cid:128)j<1
Remark1.3 Thequestion(andsomeformoftheconjecture)aboutthegrowthrateof
regulatorswasfirstraisedin[1],where,amongotherthings,aspecialcaseofTheorem2
was established: It was proved that if (cid:25) D Zn and (cid:128) runs the set of sublattices of
the form kZn, then (3) holds. The proof there can be modified to include the case
when (cid:128) runsthesetofuniformsublattices,asdefinedin[16]. Ourresultremovesany
restrictionon (cid:128).
1.6 Ontheproofs
Fortheproofsweusetoolsincommutativealgebraandalgebraicgeometry. Inparticular,
we make essential use of the theory of torsion points in Q–algebraic set (a simple
versionoftheManin–Mumfordprinciple). Wehopethatthemethodsandresultscan
beadaptedtothecaseofelementaryamenablegroups.
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Acknowledgements
IwouldliketothankMBaker,NBergeron,HDao,WLück,AThom,UZannierand
theanonymousrefereeforhelpfuldiscussions/comments. Theauthorissupportedin
partbyanNSFgrant.
1.7 Organizationofthepaper
InSection2werecallthenotionsofgeometricdeterminantandvolume. Wediscuss
therelationbetweenhomologyandRay–SingertorsionsinSection3. Anoverviewof
thetheoryoftorsionpointsinalgebraicsetisgiveninSection4. Section5containsa
crucialgrowthestimatewhich isneededintheproofsofthemain theorems,givenin
Section6.
2 Geometric determinant, lattices and volume in based Her-
mitian spaces
Inthissectionwerecallthedefinitionofgeometricdeterminantandbasicfactsabout
volumesoflatticesinbasedHermitianspaces.
2.1 Geometricdeterminant
Foralinearmap fW V !V ,whereeach V isafinite-dimensionalHermitianspace,
1 2 i
thegeometricdeterminant det0.f/ isthe product ofallnon-zero singular values of f .
Recall that x 2R is singular value of f if x (cid:21)0 and x2 is an eigenvalue of f(cid:3)f .
Byconvention, det0.f/D1 if f isthe0map. Thuswealwayshave det0.f/>0.
Sincethemaximalsingularvalueof f isthenorm kfk,wehave
(4) det0.f/(cid:20)kfkdimV2 if f isnon-zero.
Remark2.1 Thegeometricmeaningof det0f isthefollowing. Themap f restricts
toalinearisomorphism f0 from Im.f(cid:3)/ to Im.f/;eachisaHermitianspace. Then
det0f D jdet.f0/j, where the ordinary determinant det.f0/ is calculated using or-
thonormalbasesoftheHermitianspaces.
2.2 BasedHermitianspaceandvolume
Suppose W is a finite-dimensional based Hermitian space, ie, a C–vector space
equipped with an Hermitian product .(cid:1);(cid:1)/ and a preferred orthonormal basis. The
Z–submodule (cid:127)(cid:26)W spannedbythebasisiscalledthefundamentallattice.
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Fora Z–submodule(alsocalledalattice) ƒ(cid:26)W with Z–basis v ;:::;v ,define
1 l
vol.ƒ/Dˇˇdet(cid:0).vi;vj/li;jD1(cid:1)ˇˇ1=2:
Byconvention,thevolumeofthe0spaceis1. If ƒ(cid:26)(cid:127),wesaythat ƒ isanintegral
lattice. Itisclearthat vol.ƒ/(cid:21)1 if ƒ isanintegrallattice.
Fora C–subspace V (cid:26)W ,thelattice V.Z/WDV \(cid:127) iscalledthe Z–supportof V .
Wedefine
vol.V/WDvol.V.Z//:
Alattice ƒ(cid:26)(cid:127) isprimitiveifiscutoutfrom (cid:127) bysomesubspace,ie, ƒDV.Z/ for
somesubspace V (cid:26)W . Bydefinition,anyprimitivelatticeisintegral.
As usual, we say that a subspace V (cid:26)W is definedover Q if it is defined by some
linear equationswith rational coefficients(using the coordinatesin the preferredbase).
Itiseasytoseethat V isdefinedover Q ifandonlyifitisspannedbyits Z–support.
Suppose V ;V aresubspacesof W definedover Q,and fW V !V isa C–linear
1 2 1 2
map. Wesaythat f isintegralif f.V.Z//(cid:26)V.Z/.
1 2
Wesummarizesomewell-knownpropertiesofvolumesoflattices(seeegBertrand[2]).
Proposition 2.1 Suppose V ;V are subspaces of W defined over Q of a based
1 2
Hermitianspace W and fW V !V isanintegral,non-zero C–linearmap. Then
1 2
(5) vol.V CV /(cid:20)vol.V / vol.V /;
1 2 1 2
(6) vol.kerf/volŒf.V.Z//(cid:141)Ddet0.f/ vol.V /:
1 1
Foradetaileddiscussionof(6)anditsgeneralizationstolatticesinZŒZn(cid:141),seeRaimbault
[16].
3 Regulator, homology torsion and Ray–Singer torsion
Inthissectionweexplaintherelationbetweenthehomologytorsionandthecombina-
torialRay–Singertorsion. Proposition3.1ofthissectionwillbeusedintheproofof
maintheorems.
Throughoutthissectionwefixafinitelygeneratedbasedfree Z–complex E
0!Emd(cid:0)m!(cid:0)1Em(cid:0)1d(cid:0)m!(cid:0)1Em(cid:0)2(cid:0)!(cid:1)(cid:1)(cid:1)(cid:0)d!2 E1(cid:0)d!1 E0!0:
DefineaHermitianproducton E ˝ C suchthatthepreferredbaseisanorthonormal
k Z
base. Now E ˝ C becomesabasedHermitianspace.
k Z
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Weusethenotation
Zk Dkerdk; Bk DImdkC1; Bk D.Bk˝ZC/\Ek:
Let dk(cid:3)W Ek(cid:0)1!Ek beadjointof dk and DkW Ek !Ek bedefinedby
Dk Ddk(cid:3)dkCdkC1dk(cid:3)C1:
3.1 Ray–Singertorsionandhomologytorsion
DefinetheRay–Singertorsionandthehomologytorsionof E by
(cid:28)RS.E/DY(cid:3)det0.dk/2RC;
k
(cid:18)Y(cid:3) (cid:19)(cid:0)1
(cid:28)H.E/D jtor .H .E//j :
Z k
k
Remark3.1 TheRay–Singertorsionandthehomologytorsioncanbedefinedthrough
theclassicalReidemeistertorsionasfollows.
Let hz beanorthonormalbasisof ker.D /˝ CDH .E˝ C/. Withthebases fhz g
k k Z k Z k
ofthehomologyof E˝ C,onecandefinetheReidemeistertorsion (cid:28)R.E˝ C;fhz g/,
Z Z k
defineduptosigns(seeegTuraev[21]). Itisnotdifficulttoshowthat
(cid:28)RS.E/Dˇˇ(cid:28)R.E˝ZC;fhzkg/ˇˇ:
Both B and Z areprimitivelatticesin E ,and B (cid:26)Z . Thereisacollection h
k k k k k k
ofelementsof Z (cid:26)E thatdescendtoabasisofthegroup Z =B ,thefreepartof
k k k k
H .E/. Sinceh isabasisofH .E˝ C/,theReidemeistertorsion(cid:28)R.E˝ C;fh g/
k k k Z Z k
isdefined. ItisnotdifficulttoprovethefollowinggeneralizationoftheMilnor–Turaev
formula[14;21]:
(cid:28)H.E/Dj(cid:28)R.E˝ C;fh g/j:
Z k
3.2 Regulators
By definition, H .E/DZ =B . The Z–torsion of H .E/ is B =B , and the free
k k k k k k
part H .E/ is isomorphic to Z =B . For this reason, we define the volume
k free k k
vol.H .E/ / tobe
k free
vol.Z /
R .E/WD k :
k vol.B /
k
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Herewefollowthenotationof[1],where R iscalledtheregulator. UsingIdentity
k
(6),onecanprove(see[1,Formula2.2.4])
(cid:18) (cid:19)
Y(cid:3)
(7) (cid:28)RS.E/D(cid:28)H.E/ R .E/ :
k
k
Wewillusethefollowingestimateoftheregulator.
Proposition3.1 Let Rz WDvol.kerD /. Forevery k,onehas
k k
Rz (cid:21)R (cid:21) 1 :
k k z
R
k
Proof Let W betheorthogonalcomplementof B ˝ C in Z ˝ C and pW Z ˝
k Z k Z k Z
C!W betheorthogonalprojection. Then
R Dvol.p.Z //:
k k
ByHodgetheory(forfinitelygenerated Z–complexes),
ker.D /DE \W DW.Z/:
k k
Itfollowsthat ker.D /(cid:26)p.Z /,andhence vol.p.Z //(cid:20)vol.ker.D //,or
k k k k
(8) R (cid:20)Rz :
k k
By[2,Proposition1(ii)],
vol.Z / Œ.W \Z(cid:3)/WZ (cid:141)
(9) R D k D k k ;
k vol.B / Rz
k k
where Z(cid:3) isthe Z–dualof Z in Z ˝ C undertheinnerproduct. Notethat Z(cid:3) is
k k k Z k
alsotheorthogonalprojectionof E onto Z ˝ C.
k k Z
Since the numerator of (9) is (cid:21) 1, we have R (cid:21) 1=Rz , which, together with (8),
k k
provestheproposition.
4 Abelian groups, algebraic subgroups of .C(cid:3)/n and torsion
points
WereviewsomefactsaboutrepresentationtheoryoffiniteabeliangroupsinSection4.1
andthetheoryoftorsionpointsonrationalalgebraicsets(asimpleversionofManin–
Mumfordprinciple)inSections4.2and4.3.
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4.1 Decompositionofthegroupringofafiniteabeliangroup
Suppose A is a finite abelian group. The group ring CŒA(cid:141) is an A–module (the
regularrepresentation)andisa C–vectorspaceofdimension jAj. Equip CŒA(cid:141) with
a Hermitian product so that A is an orthonormal basis. This makes CŒA(cid:141) a based
Hermitianspace,with ZŒA(cid:141) thefundamentallattice.
LetAyDHom.A;C(cid:3)/,knownasthePontryagindualofA,bethegroupofallcharacters
(cid:3)
of A. Here C is the multiplicative group of non-zero complex numbers. We have
jAyjDjAj.
Thetheoryofrepresentationsof A over C iseasy: CŒA(cid:141) decomposesasadirectsum
ofmutuallyorthogonalone-dimensional A–modules:
M
(10) CŒA(cid:141)D Ce(cid:31);
(cid:31)2Ay
where e(cid:31) istheidempotent
(11) e(cid:31) D 1 X(cid:31).a(cid:0)1/a:
jAj
a2A
The vector subspaces Ce(cid:31) are not only orthogonal with respect to the Hermitian
structure,butalsoorthogonalwithrespecttotheringstructureinthesensethat e(cid:31)e(cid:31)0D
0 if (cid:31)¤(cid:31)0. Each Ce(cid:31) isanidealofthering CŒA(cid:141).
Fromthetraceidentity(seeegSerre[19,Section2.4])wehave,forevery a2A,
X (cid:26)0 ifa¤e;
(12) (cid:31).a/D
jAj ifa¤e:
(cid:31)2Ay
Here e2A isthetrivialelement.
4.2 Algebraicsubgroupsof .C(cid:3)/n andlatticesin Zn
4.2.1 Algebraicsubgroupsof.C(cid:3)/n Analgebraicsubgroupof.C(cid:3)/n isasubgroup
thatisclosedintheZariskitopology.
Foralattice ƒ,ie,asubgroup ƒ of Zn,notnecessarilyofmaximalrank,let G.ƒ/ be
thesetofall z2Cn suchthat zkD1 forevery k2ƒ. Herefor kD.k ;:::;k /2Zn
1 n
and zD.z ;:::;z /2.C(cid:3)/n weset zkDQ zki.
1 n i i
Itiseasytoseethat G.ƒ/ isanalgebraicsubgroup. Theconverseholdstrue: Every
algebraic subgroup is equal to G.ƒ/ for some lattice ƒ; see Schmidt [18]. If ƒ is
primitive,then G.ƒ/ isconnected,andinthiscaseitiscalledatorus.
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4.2.2 Automorphismsof .C(cid:3)/n Anexampleofatorusofdimension l isthestan-
dard l–torus T D.C(cid:3)/l (cid:2)1n(cid:0)l (cid:26).C(cid:3)/n,whichis G.„n(cid:0)l/,where
„n(cid:0)l Df.k1;:::;kn/2Znjk1D(cid:1)(cid:1)(cid:1)Dkl D0g:
The following trick shows that each torus is isomorphic to the standard torus. For
details,see[18].
For matrix K 2GL .Z/ with entries .K /n , one can define an automorphism
n ij i;jD1
' of .C(cid:3)/n by
K
(cid:18) n n n (cid:19)
' .z ;z ;:::;z /D YzK1j;YzK2j;:::;YzKnj :
K 1 2 n j j j
jD1 jD1 jD1
For any lattice ƒ(cid:26)Zn, ' .G.ƒ//DG.K.ƒ//. When ƒ is a primitive lattice of
K
rank n(cid:0)l, there is K 2GLn.Z/ such that K.ƒ/D„n(cid:0)l. Then 'K.G.ƒ// is the
standard l–torus.
4.2.3 Algebraicsubgroupsandcharactergroups Fixgenerators t ;:::;t of Zn.
1 n
Wewillwrite Zn multiplicativelyandusetheidentification ZŒZn(cid:141)DZŒt˙1;:::;t˙1(cid:141).
1 n
Suppose (cid:128) (cid:26)Zn is a lattice. Every element z2G.(cid:128)/ defines a character (cid:31) of the
z
quotientgroup A(cid:128) WDZn=(cid:128) via
(cid:31) .tk1(cid:1)(cid:1)(cid:1)tkn/Dzk; where kD.k ;:::;k /:
z 1 n 1 n
Conversely,everycharacterof A(cid:128) arisesinthisway. Thusonecanidentify G.(cid:128)/ with
thePontryagindual Ay(cid:128) via z!(cid:31)z.
Wewillwrite ez fortheidempotent e(cid:31)z,andthedecomposition(10),with (cid:128) having
maximalrank,nowbecomes
M
(13) CŒA(cid:128)(cid:141)D Cez:
z2G.(cid:128)/
4.3 Torsionpointsin Q–algebraicsets
4.3.1 Torsionpoints Withrespecttotheusualmultiplication, C(cid:3)WDCnf0g isan
abeliangroup,andsoisthedirectproduct .C(cid:3)/n. Thesubgroupoftorsionelements
of C(cid:3), denoted by U,is thegroupof rootsofunity,and Un isthe torsionsubgroup
of .C(cid:3)/n.
If (cid:128) (cid:26)Zn isalatticeofmaximalrank,then G.(cid:128)/ isfinite,and G.(cid:128)/(cid:26)Un.
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