Discrete groups of affine isometries PDF

Discrete groups of affine isometries

isometries of an affine Euclidean space E is discrete (if and) only if there is a. ?–invariant affine subspace F of E such that the restriction homomorphism

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Toute symétrie orthogonale de X (et en particulier toute réflexion) est une isométrie. Démonstration : Soit f une symétrie orthogonale d'axe Y . f est alors

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An affine isometry is a bijection. Let us now consider affine isometries f:E ? E. If. ?? f is a rotation we call

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JournalofLieTheory

Volume9(1999)321{349

1999HeldermannVerlag

Discretegroupsofaneisometries

HerbertAbels

CommunicatedbyE.B.Vinberg

testingprocedureisalgorithmic. groupasimage.

ISSN0949{5932/$2.50C

HeldermannVerlag

322Abels

wegoalong.

IthankE.B.Vinbergforhelpfuldiscussions.

1.TheBieberbachtheorems

recalltheBieberbachtheorems. (v;x)7!x+v,andametricd(x;x+v)=hv;vi1

2.LetG=Iso(E)bethe

Oof(V;h;i)wedeneanisometry

g=A(x0;t;U)(1.2by) g(x0+v)=x0+t+Uv:(1.3) formula andthedependenceonthebasepointchosen: (1.5)A(x0;t0;U0)=A(x1;t1;U1) haveahomomorphism (1.6):G!O denedby (A(x0;t;U))=U:

VwiththegroupoftranslationsofE.

Abels323

duetoBieberbacharebasic. oflinearpartsofisnite. respecttothisbasis( )hasintegerentriesforevery 2. image.

Fhenceisdiscrete.

2.Aminimalinvariantsubspace

sucestoanswerourquestionsforoneofthem. twofacts: subspace.

324Abels

V 1inV.

2.1LemmaandDenition

axisofg. empty. g.

V=V1V6=1,then(g)=t1.

andv2V6=1.Then gx=x+(g)+((g)1)v v

22V6=1.Then

gx=x+t1+((g)1)v2: nxx n andhencev2=0.Alltheclaimsarenowproved. have(g)2TFandEgintersectsF. greatergeneralitythanpresentlyneeded.

Abels325

a)Q1+Q2+Q3=1where1istheidentityofV. c)TheimageofQiiscontainedinUifori=1;2.

VontoViareink[T].

P f thepropertyg(S)=0. andQ2=P2Q4.

326Abels

TE1;2=V1\V2.

coecients. b b E

2toE1.Wehaveperp(E2;E1)=b1b22W.

Weformalizetheiterationasfollows.

York1965]LA4.1)WehaveMX=`1

n=1XnwhereXnisinductivelydened byX1=XandXn=`1 p+q=nXpXqforn1andthemultiplication M supp(m;n)=supp(m)[supp(n).

2XletE

betheaxisof ,andform;ninMSletE(m;n)bethesetEm;n followingproperties: a)E(m;n)Em.

Abels327

b)TEm=Vhsupp(m)i. andperp(Em;En)2TFform;ninM. ofE. subspaceofEcontainsaspaceFofthisform. translatesofeachotherbyavectorinV(). followingtwosetsofvectors a)fgxx;g2Sg. b)f(g);g2Sg[fperp(Em;E(g;m));g2Sg: only,by2.9b). (hgh1)=(h)(g)(2.13) by2.1a),and E hgh1=hEg(2.14) bythedenitionoftheaxes.Hence E m=

Em(2.15)

328Abels

form2Mand

2,wherem7!

mistheuniquemorphismof magmassuchthat7!

1for2.So

(2.16)perp(E m;E n)=( )perp(Em;En): theproofof2.10,itremainstoshowthat x02x0+Wfor

2andx02Em.

perpendicularfromEntoEmforeveryn2M,since by2.9b).Hencefort=perp(Em;E )wehavex+t2E andthus (2.17) x=x+(g)((g)1)t by2.1e),so x2x+Wfor 2. V bythedenitionofWin2.10.Forx02Emwehave x02x0+Wbfor 2S by2.17. is {invariantforevery

2Sandhenceforany

2.Thisimpliesthat

x inaminimal{invariantanesubspaceofE. subgroupsofGsuchthateveryelement ofhasaxedpoint,butthereis nocommonxedpoint.So( )=0forevery

2butW6=0.Anexample

Abels329

2 generatesW,bytheBieberbachtheorems. t D 0=f xx; pair( ;b);

2S;b2Bj,if(

)b2R.IfnotdeneBj+1=Bj[f bg.

3.Whenisagroupcrystallographic?

crystallographic.Hereisourtest. xx; 2gis discrete. f xx; chaptertwo,inparticular2.9. indexinD(x)andTF=D(x)R=R. F

330Abels

isomorphismr1()!r2(). ker(j)=\VandletxbeapointofE. a)D(x)isaZ[()]{modulecontaining. b)D(x)spansTFoverRforeveryx2F. c)Thereisapointx2EwithD(x)1 f. {invariantanesubspaceofE. e)SupposeD(x)Qthen fy;D(y)Qg=x+Q+VF: x=1fP ix, where achosenpointx02E.Foranotherset

0iofrepresentativesofmodulowe

have 0i= iiwithi2andthusthepoint y=1fX

0ix=1f(X(

i)(ixx)+X ix)2x+1f isinthe1 f{orbitofx.Inparticular,for

2thepoint

x=1fP ixisin the 1 f{orbitofx. xandRisanR[()]{module.Itisminimalsincef xx; 2gfor everyx2E. f

1{invariant.Then

1(xx0)2D+D0+VF1:

(x+d)x= xx)+( )dford2Dand

21,andsimilarlyforD0.Foranytwopoints

xandx0inEwehavefor 2 x0x0= xx+(( )1)(x0x): xx2D and x0x02D0for F 1in1X h2F1hwf1w2D+D0 whichimpliesthelemmainviewofP h2F1hw2VF1.

Abels331

Wearenowreadytoprovetheorem3.1.

isniteandisdiscrete.Hence1 fisadiscretesubgroupofVcontaining xxforevery D(x)1 sincecontainsofniteindex.

Assumption.()isnite.

withD(x)discreteisanalogousto2.3: x+DEintersectstheaxisE forevery 2. formx+QofE,e.g.forxasin3.3c) ofgeneratedbytheelement ,letx+Dbe{invariant,x02E and D 0=Z(

Dby{invarianceofx+D,thepower

fof

2iscontainedinand

f isthetranslationbyf( ).Soxx02D+V( )bylemma3.4.Nowuse TE =V(

TE1;2isinQ[F]andperp(E1;E2)isinD.

332Abels

Weobtainasin2.9:

is{invariant,by3.3c).Foreverypointy2x+1 kwehave yy=( xx)+(( )1)(yx)21 kf; henceD(y)1 ofVgeneratedbyf yy;

2g.Thisholdsinparticularforapointy2Em

of3.2followsfrom3.3d). hold (i) 2S. )jW=r( )is 2S. (iii)ThesubgroupofO(W)generatedby( )jW=r(

2S,isnite.

Z{submoduleofWgeneratedbyBandf

xx;

2Sg.ThenD0isalattice

inW,by(i).Thegroup( )D0iscommensurablewithD0by(ii)forevery

2Sandhenceforevery

2.ThenD1=P

2( )D0iscommensurable D

1generatedasaZ[()]{modulebyf

xx;

2Sgisdiscrete.

Abels333

foragivensubgroupofGtobediscrete. a)thetranslationalpart( )forevery 2 b)perp(Em;En)form;ninM c) xxfor

2andx2Em,wherem2MandT(Em\F)=(TF)().

M l((m;n))=l(m)+l(n). thefollowingelements: a)( )forevery 2 b)perp(Em;En)form;n2Mandl(m)N,l(n)N c) xxfor

2andx2Em,wherem2M,T(Em\F)=(TF)()and

l(m)N.

Ageometricconsequenceisthefollowing

2gis locallynite.

ThisholdsinparticularforthesetFr(

)=E \F,

2,andthederived

subgroupofGthatthesetfE

2gislocallynite.Foranexamplesee

5.3. V

H,HasubgroupofthenitegroupF=().

334Abels

2wehave

f)=f(g)and fisatranslation,i.e. f2,so( )21 f.

Thenforx02Em0theset

xx,

2,iscontainedinthelatticeD(x0),and

D(x0),by3.2.Letw=perp(Em0;E

).Then (x0+w)= x0+( )w and (x0+w)=x0+w+( sincex0+w2E ,hence (3.16) x0x0=( )1)w: so(( )1)wiscontainedinthelattice0=1 f+D(x0).

Forevery(

)2Fthereisapolynomialh12Q[t]suchthat h 1(( )1)w=w i=0ti,andwrite )forevery 2.

Wemayassumethat0isaZ[F]{module.

1; 2in andx0+wi2E i,wi=perp(Em0;E i),wehaveperp(E 1;E

2)=Q3(w2w1)

forQ3asaboveforthetwosubgroupsHi=h( i)i,seetheproofof3.7.Sothe lattice 1

M0containsperp(E

1;E

2)foreverypair

2ofelementsof.

3(w2w1)21

M(1+2),x+w1Q1(w1w2)2E(m1;m2)andx+w2+

Q asabove,rstly,x0+1 1

Ml(m1)+l(m2)0.Finally,ifx2Em,w=perp(Em;E

),x+w2E ,wehave (x+w)= x+( )w =x+w+( sincex+w2E andhence xx=( )1)w21

Ml(m)+10,which

thefootofaperpendiculartoE ,i.e.Em=E(m; )by2.9a)andb).

Abels335

4.Whenisagroupdiscrete?

Bofniteindexifisdiscrete.

weshalluseinthealgorithmicprocedure. consistsoftranslationsofFonly.

Proof.Let1bethenormalsubgroupofthose

2forwhichr(

1)isthetranslation

)r( 1)for xesTF.ItfollowsthatthenormalsubgroupB=f 2;r( )2TF;( )2L0g ofisabelianandofniteindexin. componentof(()).

Corollary4.3.IfisdiscretethenL0:=

(())0isatorusandxesTF. SoB=f 2;r( )2TF;( )2L0gisanabeliannormalsubgroupofof niteindex. ifrhasnitekernel,asfollows.

336Abels

expid:tTF!TTF.Then rankA=rankker(r)+rankr(B)+dimT: rankr(B)+dimT: Q. a)r( )isatranslationofFforevery 2S1. b)ThesubgroupofTFgeneratedbyfr(

2S1gisalatticeinTF

c)h( )iisconnectedforevery 2S1.

Test(i)ThetoriC0((

2S1,commute.

If(i)holds,letTbethesubtorusQ

2S1C0((

))ofO(V)generated bytheC0(( exponentialmap.Pute=expid:tTF!TTF.

LetbethesubgroupoftTFgeneratedbyfe1(

2S1g.

Test(ii)rankQ=dimF+dimT.

onTFby t=( )t= t

1,henceontheproductoTF.Sothefollowing

conditionmakessense.

Test(iii)Every

2mapsQtoitself.

2e(Q)for

every

2Ssuchthat

(x0)= x0

Hereisourlasttest:

Abels337

Test(iv)Thegrouph

2Siisnite.

invariantanesubspaceofE. in4.5passestheTests(i){(iv). relevantobjects(S1;C0(U); and,ofcourse,aproofof4.7.

2{dimensionalU{invariantsubspace.

Zandt

Qbethe

D(X)=f`2t

Qj`(X)2Qg.

4.9.Then

b)FortheLiealgebraLC(U)=LC0(U)wehave

LC(U)=\

`2D(X)ker` c)IfwewriteX=nP i=1 ieiwithrespecttoabasise1;:::;enof,then spannedby1,1;:::;n. thenumbers`(X),`2D(X)\t

Z.Henceif`1;:::;`disabasisofthelattice

D(X)\t

ofthenumbers`1(X);:::;`d(X). Z.

338Abels

a0+a11++ann.For(a1;:::;an)2QnwehavePn i=1aie i2D(X)i (a0=Pn i=1a1i;a1;:::;an)2Qn1\kerFwheree

1;:::;e

nisadualbasisof hencethetoriC(( B index,hence2;Q=Qandthusevery