ON GROUPS AND FIELDS DEFINABLE IN o-MINIMAL STRUCTURES Anand PILLAY * 0

17 Pages·2001·1.07 MB·English

lanruoJ fo eruP deilppA arbeglA 35 )8891( 552-932 932
dnalloH-htroN
ON SPUORG DNA SDLEIF ELBANIFED NI
LAMINIM-o SERUTCURTS
dnanA YALLIP *
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A.
detacinummoC yb .A ssalB
devieceR 11 rebmetpeS 6891
ehT erutcurts M = (M, ,< R,, R,, .) si o-minimal fi yreve elbanifed tes CX M si a etinif
noinu fo slavretni ,a( )b dna .stniop teL G eb a puorg elbanifed ni M .e.i( G si a elbanifed
tesbus fo M” dna eht hparg fo noitacilpitlum si osla .)elbanifed eW wohs taht G nac eb
ylbanifed deppiuqe htiw eht erutcurts fo a ’dlofinam‘ revo M ni hcihw noitacilpitlum dna
noisrevni era .suounitnoc nI eht laiceps esac M = ,w[( ,< ,+ .) ruo noitcurtsnoc sevig G eht
erutcurts fo a Nash group. esehT stluser era osla desu ot wohs taht na etinifni dleif elbanifed ni
na laminim-o erutcurts si laer desolc ro yllaciarbegla .desolc
0. noitcudortnI
eW yas the puorg G si elbanifed ni the laminim-o erutcurts M fi G dna the
graph of the puorg noitarepo are elbanifed stesbus of Mk dna k34A ,ylevitcepser
for emos 1. eW show that hcus a puorg G nac eb ylbanifed deppiuqe htiw
k 2
lacigolopot erutcurts gnikam G a lacigolopot .puorg ,revoeroM G nac eb derevoc
yb a etinif rebmun of nepo sets, each ot na nepo tesbus of M”
elbatius( .)n nI( ralucitrap fi the gniylrednu order of M si (R, ,)< neht G si a eiL
).puorg ehT emas noitcurtsnoc works for a dleif K elbanifed ni M, dna we esu siht
ot show that hcus a dleif tsum eb laer desolc or yllaciarbegla .desolc
nI noitceS 1 we xif noitaton gninrecnoc the tnatropmi noiton of .noisnemid nI
noitceS 2, we tsrif ekam emos yranimilerp snoitavresbo no spuorg G elbanifed ni
laminim-o .serutcurts etiuQ a tol of elbats-w‘ puorg ’yroeht goes .hguorht
nI ralucitrap for X a ’egral‘ tesbus of ,G yletinif ynam setalsnart of X revoc .G
eW neht show that G nac eb ylbanifed edam otni a lacigolopot .puorg yllacisaB
the lieW yroeht of puorg sknuhc seilppa ot ruo .noitautis ecnO siht si enod we nac
esu ssendetcennoc stnemugra ni noitidda ot noisnemid ,stnemugra dna ni noitceS
3 the stluser no sdleif are .deniatbo
M si tuohguorht a ledom (M, <, . . .) where < si a esned raenil order tuohtiw
,stniopdne dna M si ,laminim-o .e.i yreve elbanifed XC M si a etinif noinu of
slavretni (a, 6) (where a E M U ,}oc-{ b E M U { +m}) dna .stniop
M si deppiuqe htiw the lavretni ygolopot dna M” htiw the tcudorp .ygolopot
sselnU esiwrehto stated, these are the seigolopot referred to.
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240 A. Pillay
eW lliw yllaunitnoc refer ot suoiverp works no the ,tcejbus ylniam [3] dna [6].
tuohtiW gnillacer the noitinifed of ,sllec tel em noitnem emos laicurc .seitreporp
Fact 0.1. )i( ynA elbanifed-A XC M” (A C M) si a etinif tniojsid noinu of
elbanifed-A cells.
)ii( ynA llec XC M” si definably connected .e.i( has on proper nepolc
elbanifed .)tesbus
)iii( roF yna llec XC M” there si k I n hcus that fi rr si the noitcejorp no k
elbatius etanidrooc axes, neht r(X) si nepo ni Mk dna r si a msihpromoemoh
neewteb X dna r(X).
roF A C M, )A(lcd = )A(lca = {b E M: b si elbanifed morf A}.
)vi( oS (lcd ) SI’ t ,evitisnar dna yb [8] fi a E b(lcd U A) dna a ,)A(lcd@ neht
b E a(lcd U A).
suhT we osla evah
)v( fI X C M” si na elbanifed-A llec dna k 5 n si as ni ,)iii( neht for elbatius
19 ,i < ,i * * a(=Zynarof,n5,i<e ,,..., lia{(lcdEG,XE),a ,..., .)AU},,a
,ri( ,,i . . . , i, are the etanidrooc axes no hcihw the noitcejorp rr of )iii( .)si
roF ecneinevnoc sake we lliw emussa that M si yrev saturated. llA ruo ,stluser
,revewoh dloh for yrartibra M.
1. noisnemiD
noitinifeD 1.1. )i( Let it E M”, AC M. nehT )AlZ(mid = tsael ytilanidrac of a
elputbus a’ of a hcus that G C A(lcd U a’).
)ii( Let p(X) E S,(A). nehT mid p = )Ala(mid for emos )yna( 5 E M” gnisilaer
P.
ammeL 1.2. )i( )A/Z(mid = the cardinality of any maximal algebraically indepen-
dent over A, subtuple of a.
)ii( A C B + )Ala(mid 2 .)BlG(mid
)iii( )Al&(mid = Ala(mid .)Alb(mid
U 6) +
)vi( bla(mid U A) = ,Ala(mid iff Alb(mid U a) = .)Alb(mid
)v( If ~(4 E K(A) and A C B, then there is p E S,(B) with p C p’ and
mid p = mid p’.
Proof. )vi(-)i( wollof yb tcaF .)vi(l.O
roF )v( ti si hguone yb )ii( ot dnif p’ E S,(B) htiw mid p’ 2 mid p. esoppuS
mid p = k. Let it ezilaer p. tuohtiW ssol of ytilareneg a,, _ . . , ak are yllaciarbegla
tnednepedni revo A. yB the noitarutas of M we nac dnif ylevitcudni a;, . . . , a; E
M, yllaciarbegla tnednepedni revo B htiw ,ia(pt . . . , aLlA) = tp(a,, . . , a,lA).
sihT si ylraelc .tneiciffus 0
Groups and fields definable in o-minimal structures 241
eW lliw yas a dna b are tnednepedni revo A fi the tnelaviuqe snoitidnoc of
ammeL )vi(2.1 are .deifsitas
Definition 1.3. Let XC M” eb elbanifed-A (A C M). nehT mid X =
:)Alit(mid{xam it E }X mid{xam=( p: p E S,(A), dna p si dezilaer ni .)}X
Note. )i( yB ammeL ,)v(2.1 mid X does ton dneped no A.
)ii( Let X eb d-A e if an lb e dna a E X. eW lliw yas that a is a generic point of X
over A fi )Alit(mid = mid X.
Lemma 1.4. Let XC M” be definable. Then (for k 5 n) mid X 2 k iff some
projection of X onto Mk has interior in Mk.
Proof. esoppuS X si elbanifed-A dna mid XL k (so k I n). Let a = (a,, . . . , a,)
eb a cireneg tniop of X revo A.
tuohtiW ssol of ,ytilareneg a,, . . . , ak are yllaciarbegla tnednepedni revo A.
Let Y eb the noitcejorp of X otno the tsrif k etanidrooc axes. oS Y c Mk si
elbanifed-A dna sniatnoc the tniop (a,, . . . , ak). yB ammeL 1.4 of [6], Y has
roiretni ni Mk.
,ylesrevnoC esoppus emos noitcejorp Y of X otno Mk has .roiretni Let X dna(
so Y) eb ,elbanifed-A Now Y sniatnoc na nepo elbanifed-B xob ,.e.i( tcudorp of
)slavretni Z ni Mk (B 3 A). yB noitarutas of M ti si ysae ot dnif ylevitcudni
a,, . . . , ak hcus that (a,, . . . , ak) E Z dna a,, . . . , ak are yllaciarbegla -nepedni
tned revo B. So ,,a(mid . . . , a,lB) = k, dna fi a si a tniop ni X gnidnetxe
(a, .. . ak) we see that )Bla(mid 2 k. So mid X2 k. 0
ehT gniwollof si na etaidemmi ecneuqesnoc of the noitinifed of mid :X
Lemma 1.5. )i( Let XC M” be definable and let f : X-+ Y be a definable bijection,
where Y C Mk. Then mid X = mid Y.
)ii( Zf X,, . . . , X, are definable subsets of M”, then (mid iu )iX =
{xam mid iX : i < }r .
Proof. )i( Let A eb a set revo hcihw X dna f are .denifed Let it E X. OS
(a, f(a)) C a(lcd U A) dna (a, f(C)) C (lcd f(G) U A). ’ suhT )Ala(mid =
(mid f(a) /A).
)ii( .etaidemmI lC
Lemma 1.6. Let fi(xl * + * ,,x j) be a formula (without parameters say) and for any
6 let Xi be the subset of M” defined by 6(X, 6). Then for any k 5 n there is a
formula I/J~(y ) without parameters such that for any 6, mid XL = k iff I+!J~(~).
Proof. An etaidemmi ecneuqesnoc of ammeL 1.4. q
ammeL 1.7. Let Xc M” be definable. Then mid X = n iff X has interior in M”.
Proof. ,niagA siht swollof yb ammeL 1.4. 0
242 A. Pillay
ehT gniwollof noitaziretcarahc of noisnemid ni smret of elbanifed ecnelaviuqe
snoitaler lliw eb lufesu nehw gniyduts elbanifed :spuorg
Proposition 1.8. mid X 2 k + 1 iff there is a de$nable equivalence relation E on X
infinitely many classes of which have dimension 2 k.
Proof. fI mid X 2 k + 1, neht yb ammeL 1.4 emos noitcejorp of X otno Mk
sniatnoc na nepo xob B. enifeD na ecnelaviuqe noitaler no B yb X - p ffi X dna j
evah the emas tsrif .etanidrooc gnitfiL siht ecnelaviuqe noitaler ot X sevig su the
deriuqer E.
,ylesrevnoC emussa the dnah-thgir edis .sdloh yB tcaF )i(1.0 dna ammeL
)ii(Sl we yam emussa that X si a .llec esoppuS yb yaw of noitcidartnoc that
mid X = k. nehT ylraelc k si as ni tcaF .)iii(l.O Let E1 eb the ecnelaviuqe noitaler
no )X(n decudni yb E.
Now tel C eb a ssalc of E htiw mid C 2 k. So ylraelc mid C = k. yB ammeL 1.5,
mid r(C) = k, dna so yb ammeL 1.7, r(C) has roiretni ni Mk. ,suhT yletinifni
ynam sessalc of E1 evah roiretni ni Mk. sihT stcidartnoc [6, noitisoporP 2.11. q
eW won evig a rather erom tluciffid noitaziretcarahc of ,noisnemid ,hcihw
hguohtla ton yltcirts deriuqer for the niam enil of siht paper, si gnitseretni ni sti
nwo thgir dna ni smret of the citamoixa noiton of noisnemid nevig ni [7].
Proposition 1.9. Let Xc M” be definable. Then mid X 2 k + 1 if and only if there
is a definable Y c X such that mid Y 2 k and Y has no interior in X.
Proof. ehT tfel ot thgir noitcerid si ysae dna si devorp as ni the tfel ot thgir
noitcerid of noitisoporP 1.8.
,ylesrevnoC esoppus that Y C X, Y has on roiretni ni X dna mid Y 2 k. eW
yam emussa that mid Y = k dna Y si a .llec eW yam osla ,emussa tuohtiw ssol of
,ytilareneg that Y si cihpromoemoh ot rr(Y), the noitcejorp of Y no the tsrif k
etanidrooc axes, dna )Y(n si nepo ni Mk. Let =a (a,, . . . , ak, ak+l, . . . , a,) eb a
cireneg tniop of Y revo A (where X, Y are .)elbanifed-A oS a,, . . . , ak are
tnednepedni revo A.
Now for each j = k + 1, . . . , n, tel $j(a,, . . . , ak, xi) eb the alumrof
3X kfl . . . xj-lxj+, . . . ,x [(a,, a*, . . . , ak, x~+~, . . . , xn) E X] .
Claim. For some j, $tj(a,, . . . , ak, ),x si .etinifni
fI the mialc ,sdloh neht we nac choose bj gniyfsitas tij(a,, . . . , ak, x,) htiw
bj ,a(lcdg . . . ak U A). sihT ylraelc sevig esir ot a tniop =C (c,, . . . , )nc htiw
c,=a ,,..., ck=ak, cj = b,, hcihw si ni X dna hcus that mid (c/A) 2 k + 1. oS
mid X 2 k + 1 as .deriuqer
Groups and fields definable in o-minimal structures 243
Proof of .mialC fI ,ton ,neht as ,a(mid . . . a,lA) = k, neht there ,si yb [3] na nepo
xob Z C )Y(n gniniatnoc (a,, . . . , uk) hcus that for lla (x1, . . . , xk) E Z,
$j(xl, . . . > xk, xi) si ,etinif for lla .j
Nowfor(x,,. . . , -idroochtjeht=),x,...,,x(,gtel,nIj51+kdnaZE),x
etan of ,,x(’K . . . .,,x(ifenifeddna,),x, . . ..,lx(jg<jxtsetaerg=),x, . ),x,
hcus that ,,x(;cI . . . , ,kx .),x ,,x(ff . . . , )kx = tsael ix > ,,x(jg . . . , )kx hcus that
niagA gnisu [3] dna the fact that ,a(mid . . . a,/A) = k (so yna
$j(xI> ” ’ 3 ‘k> x,).
elbanifed-A llec ni Mk gniniatnoc (a,, . . , ak) tsum eb ,)nepo we yam emussa
f,! ,
,...,l+k=jroftaht ,II the snoitcnuf g,, :f are suounitnoc no .Z
nehT ylraelc we nac dnif na nepo xob W ni M” gniniatnoc
(a,, . , ,ku ,,+ku . . , ),a hcus that for yna .,ix . . , xk, (x1,. . . , xk,
f:+l(xP.. .,x,) ,..., f:,(x, ,..., Y=WnXylraelcnehT.2,1=irofWC))kx
dna so Y has roiretni ni X, gnitcidartnoc ruo noitpmussa dna gnivorp the ,mialc
dna the .noitisoporp 00
krameR 1.10. Let X eb a set, deppiuqe htiw a ygolopot .t esoppuS that X si
derevoc yb nepo stesbus ,,U . . . , U,. ,esoppuS ,revoerom that for each i =
1, . . . , s there si a mishpromoemoh rr, neewteb ,U dna V, where V, si a elbanifed
tesbus of M”’ htiw sti decudni ,ygolopot dna that for each ,i j the decudni
msihpromoemoh neewteb (,~7 ,U f’ ),U C V, dna ,U(jr n ),U C .y si .elbanifed
piuqE X htiw lla the elbanifed erutcurts decudni morf M yb the .s‘(v oS X si a
dnik of ’dlofinam‘ revo M, dna seifsitas the :gniwollof
)i( yrevE d e if an lb e tesbus of X si a naelooB noitanibmoc of desolc elbanifed
;stes
)ii( roF yreve elbanifed Y C X, mid Y si denifed dna seifsitas mid Y 2 k + 1 ffi
there si elbanifed Z C Y htiw on roiretni ni Y dna mid Z 2 k;
)iii( yrevE elbanifed Y C X si a etinif tniojsid noinu of ylbanifed detcennoc
elbanifed ;stes
)vi( ehT ygolopot no X si ylticilpxe .elbanifed
,)i( )iii( dna (’ vi ) are ysae dna )ii( swollof morf noitisoporP 1.9. oS X htiw sti
ygolopot dna lla sti elbanifed erutcurts si na elpmaxe of what we dellac a
yllacigolopot yllatot latnednecsnart erutcurts ni [7].
,yllaniF ni siht noitces we ecudortni large sets.
noitinifeD 1.11. Let Y C XC M” eb .elbanifed eW yas that Y si large ni X fi
X(mid - Y) < mid X.
ehT gniwollof si na ysae ecneuqesnoc of the :snoitinifed
ammeL 1.12. Let Y C X be definable. Then Y is large in X if and only if for every
A over which X and Y are defined, every generic point Z of X over A is in Y. q
244 A. Pillay
noitisoporP 1.13. Let XC M” be A-dejinable. Let 4(x,, . . . , ,,x 9) be a formula
over 0. Then (6: 4(X, b)M n X is large in X} is A-definable.
Proof. yB ammeL 1.6. 0
krameR 1.14. Note that noitisoporP 1.13 si a dnik of ytilibanifed of sepyt for
laminim-o .serutcurts roF
:b{ 4(X, 6) If X si egral ni }X
= :b{ for every cireneg tniop a of X revo ,b k 4(5, 6))
,Xmid=pmiddnap~Xhtiw)M(,S~pyreverof:b{=
C-44 b)EP).
ehT ylno ecnereffid htiw elbats-o seiroeht si that ni ruo txetnoc a elbanifed set X
yam evah ynam scireneg revo A pu ot .msihpromosi-A
Note. nA tnelaviuqe noitinifed of noisnemid ni laminim-o serutcurts was nevig yb
nav ned seirD ]ol[ who osla devresbo ynam of the facts ni siht .noitces
2. elbanifeD spuorg
G si here a puorg elbanifed ni M. ylemaN the esrevinu of G si a elbanifed
tesbus of M”, emos n, dna the puorg noitarepo si osla .elbanifed eW emussa that
G si denifed revo .0
ammeL 2.1. )i( Let b E G and let a be a generic of G over b. Then b . a is a generic
of G (over b).
)ii( For any b E G there are generics ,b , b, of G such that b = b, . b,.
Proof. )i( mid G = )bla(mid = b(mid * a/b).
)ii( Let b, eb cireneg revo b. So by’ si cireneg revo b dna yb )i( so si
b,‘*b. 0
eW edulcni the gniwollof ammel for ,tseretni dna the proof si tfel ot the reader.
ammeL 2.2. Let X be a definable subset of G. Then {a E G: a . X 17 X is large in
X} is a definable subgroup of G. II!
ammeL 2.3. Let H be a definable subgroup of G. Then mid H = mid G fi dna only
if H has finite index in G.
CTO~PS and fields deniable in o-minimal stmctures 245
yB ammeL ,)i(5.1 mid H = mid a. H rof yna a E .G oS yb noitisoporP 1.8,
Proof.
mid H = mid ,G neht H has etinif xedni ni .G ylesrevnoC fi H has etinif xedni ni
if
G neht G = a, . H H so yb ammeL ,)ii(5.1 mid G = mid H. cl
U - * * U a,.
ehT gniwollof laicurc ammel was osla devorp )yltnednepedni( yb nav ned seirD
tnereffid( .)foorp
a large de~nubZe subset of G. Then ~n~teiy zany translates of
Lemma 2.4. LA X be
X cover G.
eb a small ledom revo hcihw X is .denifed eW lliw wohs
Proof. Let MO< h4
)*( roF yna GEa ereht is ORGEN hcus that a~b.X.
tI lliw wollof yb ssentcapmoc that ereht era b,, . . . , b, E 0MG hcus that yreve
Ea G is ni rofX*jb emos i= 1,. . . , .P
So ot evorp ,)*( tsrif tel c eb a cireneg tniop fo G revo M, hcus that revoerom
,&/c(pt )a is yletinif elbaifsitas ni M,. tI swollof that c is a cireneg fo G revo
U
M, a. roF( tel Ic eb a lamixam elputbus fo c yllaciarbegla tnednepedni revo M,,
U
and a, a lamixam elputbus fo a yllaciarbegla tnednepedni revo .,ILi nehT ylisae
a, lc- is yllaciarbegla tnednepedni revo M,, os a(mid ),M/c* = dim(a/~~~) +
),l/h/c(mid and os ),Mlc(mid = ,,lrlllc(mid ).)a
U
woN as X is egral ni G ti swollof morf ammeL )i(5.1 that X. a-’ is egral ni .G
oS yb ammeL 1.13, EC X. .‘-a suhT a E .I--c .X As ,Mlc(pt is yletinif
U a)
elbaifsitas ni M,, ereht is b E ”MG hcus that a E b * X. sihT sevorp )*( and thus
osla eht .noitisoporp q
eW lliw won wohs that G nac eb ylbanifed edam otni a lacigolopot puorg hcihw
is yllacol‘ .’naedilcuE lieW ]ll[ dewohs that an ciarbegla puorg revo an -iarbegla
yllac desolc dleif nac eb derevocer ro denifed morf lanoitarib‘ data’, ,.e.i morf a
yteirav no hcihw an ciarbegla puorg erutcurts is nevig ylno gener~caZZy. ehT dnoces
trap fo his foorp stsisnoc ni gniwohs that morf a puorg knuhc ylsuoiverp(
,)deniatbo hcihw is an nepo tes fo eht yteirav htiw suoirav elbarised ,seitreporp an
gnipolevne ciarbegla puorg nac eb .denifed Van ned seirD devresbo that this
tluser nac eb desu ot wohs that a puorg elbanifed ni an yllaciarbegla desolc dleif
citsiretcarahc( )0 nac eb ylbanifed nevig cirtemoeg erutcurts making ti an -egla
ciarb .puorg yksvohsurH ]2[ evag an tnagele foorp fo this last tluser fo s’lieW ni
eht laiceps esac that eht puorg knuhc is a tes fo lamixam yelroM knar and eerged
ni an already given definable group. tI is this foorp fo yksvohsurH hcihw seilppa
tsomla drow rof drow ot eht tneserp ,txetnoc nevig eht yrenihcam depoleved os
raf and stluser ni .]3[
teL G be a group 0 de~nab~e ni M htiw mid G = a. Then there are
Proposition 2.5.
a large O-definable subset V of G and a topology t on G such taht
246 A. Pillay
)i( G with the topology t is a topological group (i.e. inversion and multiplica-
tion are continuous operations);
)ii( V 1s a ji mte d’l sjoint union of 0-definable sets U,, . . . , U, such that for each
,...,l=i r, 17, is t-open in G and there is a @definable (in M) homeomorphism
between U, (with its t-topology) and some nepo subset v of M”.
Note. yB ammeL 2.4, yletinif ynam setalsnart of V revoc .G suhT the ygolopot t
no G si ylticilpxe ,elbanifed dna osla G stif otni the pu-tes debircsed ni krameR
1.10.
Proof. tsriF we nac yb [3] etirw G as a etinif tniojsid noinu of elbanifed-O .sllec Let
U,, . . , U, eb the sllec of noisnemid n. suhT $I = U, W . . . W U, si egral ni .G
Each U, ,si yb tcaF 0.1, ylbanifed-O cihpromoemoh htiw na nepo set ni M”, dna
we lliw yfitnedi U, htiw siht nepo set. ,revoeroM we lliw yfitnedi V, yllacigolopot
htiw the tniojsid noinu of these nepo sets U,. Now for each i, j we nac yb [3] etirw
Vi as a etinif tniojsid noinu of elbanifed-d( sllec no each of hcihw rehtie noisrevni si
ton a pam otni U,, or noisrevni si a suounitnoc pam otni U,. gnitoN that
(a) roF each cireneg a of G revo 0 there are i, j htiw a E U,, a- ’ E U, yb(
ammeL 1.12), dna
)b( yrevE elbanifed-O tesbus (of )G of noisnemid n sniatnoc a cireneg of G revo
,0
ew ees that for each i there are nepo stesbus of U,, Ul, . . , l.Jl hcus that
U l-y lJi si egral ni U, dna noisrevni si a suounitnoc pam morf Ui+ U, for yreve
J
tuP V, = 0 r,i Ui htiw( decudni ,)ygolopot neht gnimmus pu
)i( VI si egral dna nepo ni V, dna noisrevni si a suounitnoc pam VI -+ V,.
ylralimiS yb [3] we nac dnif na nepo egral elbanifed-O Y,, C V, X V, hcus that
)ii( noitacilpitluM si a suounitnoc pam morf Y, + V,.
sA( evoba Y, si deniatbo as a tniojsid noinu of nepo sets jliY C U, X Uj hcus
that noitacilpitlum si suounitnoc morf ,FY j ---f U,. eW esu the fact that for yllautum
cireneg a, b of G, (a, b) si a cireneg of G x G dna a. b si a cireneg of ,G dna esu
niaga ammeL 1.12). Note also that Y, is large in G X .G
Now enifed Vi = {a E VI : for yreve cireneg b of G revo a, (b, a) E Y,, dna
(b-l, b. a) E Y,}.
yB noitisoporP 1.13 dna( krameR 1.14), Vi si .elbanifed-O eW ,mialc ni fact,
that Vi si egral ni .G roF ,siht ti si ,hguone yb ammeL 1.12 ot show that iV
sniatnoc yreve cireneg a of G revo 0. tuB fi a si a cireneg of ,G neht a E VI (as VI
si egral dna ,)elbanifed-O dna revoerom for b cireneg of G revo a, ti si ysae ot see
that htob (b, a) dna (b-l, b . a) are scireneg of G X ,G so ni Y,.
,niagA yb gninoititrap Vi otni elbanifed-0 sllecbus of the U, dna throwing away
sllec of noisnemid ssel naht n, we niatbo elbanifed-O V, C Vl hcihw si nepo ni V,
dna egral ni .G Now yb )i( V, -’ si osla nepo ni V,, dna ylraelc egral ni .G
tuP V= V, fl V,‘. nehT V si nepo ni V,, egral ni G dna V= V-‘.
Groups and fields definable in o-minimal structures 247
,suhT osla V x V si nepo ni V, x V, dna egral ni G x G.
Let Y = (V x V) II {(a, b) E Y,: a . b E }V . yB )ii( evoba dna the ssennepo of
V ni V,, Y si nepo ni V, x V, dna ,revoerom as for yllautum cireneg a, b of ,G
(a, b) E Y,, dna a. b si ,cireneg so ni V, we see that Y si egral ni G X .G
oS tuohtiw ssol of ,ytilareneg we :evah
)iii( V= U, W . . . 0 U, where the ,U are nepo elbanifed-O stesbus of M”, dna V
has the decudni .ygolopot
)vi( V si egral ni .G
)v( Y si egral ni G X G dna( .)elbanifed-O
)iv( noisrevnI si a suounitnoc pam morf V otno V.
)iiv( Y si esned nepo ni V x V dna noitacilpitlum si a suounitnoc pam morf Y
ot .v
)iiiv( roF yna a E V, fi b si a cireneg of V revo a, neht (b, a) E Y dna
(b-‘, b . u) E Y.
,)iv(( ,)iiv( ( mv )‘“ are the puorg knuhc .)smoixa tA siht tniop we ypoc yksvohsurH
dna so we are .feirb
.ammeL (a) For any a, b E G, the set 2 = {x E V: a. .x b E V} is open in V, and
the map x -+ a . x . b is a homeomorphism (in V) Z -+ a . Z * b.
)b( For any a, b E G, the set Z = {(x, y) E V x V: a. .x b. y E V} is open in
V X V and the corresponding map Z+ V is continuous.
Proof. (a) Let xO E .Z eW show that x,) si ni emos nepo set ,Z of V htiw ,,Z C ,Z
dna that revoerom the pam +-x a . x . b si suounitnoc no .,Z tsriF etirw b = 6, . ,h
htiw b,, b, E V yb( ammeL 2.1, as yreve cireneg si ni V). Let c E ,G htiw c
cireneg revo ,a{ xO, b,, b2}. So cEV dna C.UEV.
Let Z,={xEV:(c*a,x)EY,(c~~~~,b,)EY,(~~u~x~b,,b~)~Y
dna ,‘-c( c.u.x.b,.b,)E Y}.
nehT yb ,)iiv( ,Z si nepo ni V, Z, C Z dna noitacilpitlum .Y +
a. x . b(=c? . c . .a x . b, . b2) si suounitnoc morf +-,Z V. yB )iiiv( revoerom
xO E .,Z sihT si .tneiciffus )b( si devorp .esiwekil 0
Now, enifed the ygolopot t no G :yb Z C G si nepo-t ffi for lla g E ,G g. Z n V
si .nepo ylraelC siht ygolopot si ylticilpxe elbanifed ni( M).
oT hsinif the proof of noitisoporP 2.5,
mialC I. Let Z C V and a E G. Then a. Z is t-open iff Z is open in V. (So in
particular Z is t-open iff Z is open).
Proof. fI a. Z si ,nepo-t neht 1ma . (a . Z) II V= Z si nepo ni V. ,ylesrevnoC fi Z si
trapybVnineposiVnZ.)a.g(=Vn)Z.u(.g,GEgynarofneht,Vninepo
(a) of the evoba ,ammel so a. Z si .nepo-t 0
248 A. Pillay
mialC II. Inversion is a t-homeomorphism on G.
Proof. Let W eb nepo-t ni G. sA yletinif ynam setalsnart of V revoc ,G we yam
emussa yb( mialC )I that W C a * V for emos a E G. yB mialC ,I a-’ . W si nepo ni
V, dna so yb )iv( (a-‘. W)-’ = W-l. a si nepo ni V. nehT for yna g E ,G
g*W-’ nV=g.(WP1*a)a-’ n V si nepo ni V yb part (a) of the .ammel oS W- ’
si .nepo-t 0
mialC III. Multiplication is t-continuous on G.
Proof. Let W C G eb .nepo-t eW tsum show that ,x({ )y E G x :G ,x. y E W} is
nepo-t ni G X .G sA yletinif ynam setalsnart of V revoc ,G we yam emussa that
W C c * V for emos c E G dna tsuj show that for yna a, b E G, Z = {(x, y) E a. V:
x. y E W} si .nepo-t tuB Z = {a. ,w b. z) E G X G: (w, z) E V x W dna .l-c a *
x . b . y E l-c . W}. yB mialC I dna part )b( of the ,ammel Z si .nepo-t 0
smialC ,I II dna ,III together htiw )iii( evoba etelpmoc the proof of noitisoporP
2.5. 0
krameR 2.6. tI si ysae ot see that noitisoporP 2.5 si dilav for yna laminim-o M
ton( ylno detarutas M). nI ,ralucitrap fi the order epyt of M si (R, ,)< neht G
htiw the ygolopot t si a yllacol naedilcuE lacigolopot ,puorg dna suht yb
,yremogtnoM nippiZ dna s’nosaelG noitulos ot s’trebliH 5th ,melborp a eiL
.puorg
nI the laiceps case where M = (R, <, + , .), ti si nwonk that fi U si na nepo
elbanifed tesbus of M” dna f : +Jl Mk si ,elbanifed neht there si na nepo esned
elbanifed tesbus ’U of U no hcihw f si analytic. suhT ni siht case, we dluoc ni the
proof of noitisoporP 2.5 choose V dna Y hcus that noitacilpitlum +Y V, dna
noisrevni V+ V are analytic .snoitcnuf ehT proof of noitisoporP 2.5 neht -nifed
ylba stneserp G as na citylana .puorg tahT ,si a ciarbeglaimes puorg .e.i( a puorg
elbanifed ni the dleif of laer )srebmun nac eb yllaciarbeglaimes deppiuqe htiw
citylana erutcurts gnikam G a eiL .puorg
eW won esu noitisoporP 2.5 ot show that niatrec yelroM‘ degree’ elyts
stnemugra go hguorht for .G G sniamer a puorg elbanifed ni M. eW tel t eb the
ygolopot no G nevig yb noitisoporP 2.5 dna V the egral tesbus of G osla nevig yb
noitisoporP 2.5. nI the ,gniwollof yb definable we lliw naem elbanifed ni M,
hguohtla we lliw ekam tneuqesbus stnemmoc tuoba what sneppah nehw we
tcirtser sevlesruo ot sets elbanifed ylno ni the erutcurts ,G( .)v
ammeL 2.7. Any dejinable set Z C G is a finite union of t-locally closed definable
subsets of G.