Publ. Ma . 48 (2004), 127–137
COUNTING FIXED POINTS OF A FINITELY
GENERATED SUBGROUP OF A [C]
F. Lo ay, M. an de Pu and F. Reche
Abs ac
Gi en a ini ely gene a ed subg oup Go he g oup o a ine ans-
o ma ions ac ing on he complex line C, we a e in e es ed in he
quo ien Fix(G)/G. The pu pose o his no e is o es ablish when
his quo ien is ini e and in his case i s ca dinali y. We gi e an
applica ion o he quali a i e s udy o polynomial plana ec o
ields a a neighbo hood o a nilpo en singula poin .
In oduc ion
Conside he g oup o a ine ans o ma ions ac ing on he complex
line C
A [C] = {aX +b;a, b ∈C, a 6= 0}.
Fo a ini ely gene a ed subg oup G<A [C], we deno e by Fix(G)⊂C
he se o all he poin s which a e ixed by a non- i ial elemen o G
(i.e. whose iso opy g oup G{c}={g∈G;g(c) = c}is no educed o
he iden i y {X})
Fix(G) = {c∈C;∃g∈G, g(c) = cand g6=X}.
The g oup Gac s on his se and we deno e by Fix(G)/G he se o G-o -
bi s. The pu pose o his no e is o answe o he ollowing wo ques ions:
•When is Fix(G)/G ini e?
•Suppose ha Fix(G)/G is ini e, wha is i s ca dinali y?
Mo i a ions and applica ion
Mo e gene ally, hese ques ions a ise na u ally o a ini ely gene a ed
subg oup Go he g oup Di (N) o di eomo phisms o a smoo h mani-
old Nwhen one wan s o s udy he opology o he lea es o a olia ed
mani old (M, F) gi en by suspension o a ep esen a ion
ρ:π1(B, b)−→ Di (N)
2000 Ma hema ics Subjec Classi ica ion. P ima y: 34C; Seconda y: 11A.
Key wo ds. Limi cycles, singula i ies o ec o ields, Ricca i equa ion.
128 F. Lo ay, M. an de Pu , F. Reche
whe e π1(B, b) deno es he undamen al g oup o ano he mani old B
(see also [G, Chap e I, 2.8]). Le us b ie ly ecall he cons uc ion o
he suspension (M, F). Conside he canonical ep esen a ion o he
undamen al g oup σ:π1(B, b)→Di ( e
B) in he deck ans o ma ions
o he uni e sal co e ing e
B. The image o he ep esen a ion
eρ:π1(B, b)−→ Di ( e
B×N)
γ7−→ (σ(γ), ρ(γ))
ac s p ope ly and discon inuously on he p oduc e
B×N. The quo-
ien mani old M=e
B×N/eρis he o al space o a locally i ial
bundle Π: M→Bwhose ibe s a e isomo phic o N. The ho izon-
al olia ion on e
B×Nwhose lea es a e e
B× {p},p∈N, is in a ian
unde he ac ion o eρand hus induces a egula olia ion Fon M. The
lea es o Fa e ans e sal o he ibe s and ha e he same dimension
as he basis B. The p ojec ion Π induces by es ic ion a co e ing o
each lea ΠL:L→Bon o he basis. Mo e p ecisely, i Lpdeno es he
lea passing h ough p∈Π−1(b)≃N, any loop γ∈π1(B, b) li s-up
a pas a pa h ˜γ: [0,1] →Lpjoining ˜γ(0) = p o ˜γ(1) = ρ(γ)(p). The
undamen al g oup o Lpis gi en by he exac sequence
1−→ ke (ρ)−→ π1(Lp, p)−→ G{p}−→ 1
whe e G < Di (N) deno es he image o ρand G{p}={g∈G;g(p) =
p}< G, he iso opy subg oup associa ed o p. In o he wo ds, he
undamen al g oup π1(Lp, p) o Lpis isomo phic o ρ−1(G{p}) = {γ∈
π1(B, b); ρ(γ)(p) = p}. The e o e, he lea es Lppassing h ough gene ic
poin s p(ha ing i ial iso opy g oup) a e pai wise di eomo phic (uni-
e sal p ope y o co e ings) wi h same undamen al g oup π1(Lp, p)≃
ke (ρ). Finally, he wo ques ions abo e a e ela ed o coun ing he
non-gene ic lea es, i.e. hose lea es ha ing mo e opology (o ha ing
non- i ial holonomy). The numbe o hem is hen gi en by Fix(G)/G
and hei undamen al g oup is inc eased by G{p}.
Such olia ions na u ally appea o ins ance in he phase po ai
o a di e en ial equa ion o Ricca i ype y0=a(x)y2+b(x)y+c(x)
whe e a,band cdeno e a ional unc ions o x. Deno e by Ω ⊂C
he complemen o he pola se o a,band c. The egula olia ion
induced in a iables (x, y) on Ω ×Cex ends as a egula olia ion F
on Ω ×P1(C) ans e sal o he e ical ib a ion and is ac ually he
suspension o he monod omy ep esen a ion ρ:π1(Ω, x0)→P GL(2,C)
(see [H]). The case N=Cand Di (N) = A [C] conside ed in he
p esen no e occu s when he Ricca i equa ion has a a ional solu ion.
Coun ing Fixed Poin s 129
The monod omy g oup Gis he e o e a ine in a con enien p ojec i e
coo dina e. I would be in e es ing o answe o he same ques ions in
he mo e gene al case o P GL(2,C) ac ing on he p ojec i e sphe e. This
no e unde lines he su p ising complexi y o his simple p oblem e en in
he a ine case and sugges s ha he p ojec i e case will be di icul .
Al eady, he a ine case na u ally a ises when we s udy he opology
o complex ajec o ies o a singula analy ic ec o ield in he plane
ha ing a Liou illian i s in eg al. Fo ins ance, he ajec o ies o he
nilpo en Hamil onian ec o ield V0:= y∂x+nx2n−1∂y,n≥2, a e
comple ely unde s ood by means o he i s in eg al 0(x, y) = y2−x2n:
all ajec o ies apa om he singula ibe {y2−x2n= 0}ha e he
same opology a a neighbo hood o (x, y) = (0,0) ∈C2. Now, conside
a pe u ba ion Vo V0ha ing he o m:
V= (y∂x+nx2n−1∂y) + (α+x)xn−1(x∂x+ny∂y)
o a complex numbe α∈C. As a di ec applica ion o he esul s o
his no e, we ob ain he:
Co olla y 1. When α∈C−Qand n≥3, he polynomial ec o ield V
abo e has in ini ely many complex ajec o ies ha ing non-pe iodic ho-
lonomy in any neighbo hood o he singula poin (x, y) = (0,0). When
α∈Q, hen Vhas ini ely many ajec o ies ha ing non- i ial holo-
nomy, he numbe o which depends on he a i hme ic o nand α. Fo
ins ance, when α= 0 and n=pis an odd p ime numbe , he e a e
1 + p−1
2+2p−1−1
psuch ajec o ies.
In pa icula , his co olla y p o ides explici examples o polynomial
plana ec o ields ha ing in ini ely many complex limi cycles a he
neighbo hood o a nilpo en singula poin .
P oo : The phase po ai o he ec o ield Vis de ined by he equa-
ion ω= 0 whe e ωis he holomo phic 1- o m
ω= (y dy −nx2n−1dx) + (α+x)xn−1(x dy −ny dx).
In coo dina es =y/xnand z= 1/x, we ob ain he Fuchsian Ricca i
equa ion
dz
d =( +α)z+ 1
n( 2−1) =(1 + α)z+ 1
2n( −1) +(1 −α)z−1
2n( + 1) .
The monod omy g oup o his equa ion is a ine, gene a ed by he mon-
od omy maps a ound 1 and −1
g1(z) = e2iπ (1+α)
2nz+c1and g−1(z) = e2iπ (1−α)
2nz+c−1
o some cons an s c1, c−1∈C.
130 F. Lo ay, M. an de Pu , F. Reche
We claim ha g1and g−1commu e i , and only i , (1+α)
2no (1−α)
2n
belongs o Z− {0}. Indeed, when α6=−1, he Ricca i equa ion has
wo singula poin s o e he singula poin = 1, namely z=−1
1+αand
z=∞. The e o e, he co esponding monod omy map is conjuga ed o
i s linea pa (which is he iden i y) i , and only i , (1+α)
2n∈Z. I α=−1,
he Ricca i equa ion has a double singula poin a z=∞and g1is a
ansla ion which ob iously does no commu e wi h g−1. Finally, when
g1and g−1bo h ha e a non- i ial linea pa , hen hey commu e,
i , and only i , hey sha e a common ixed poin in Cco esponding
o a a ional solu ion z( ) o he Ricca i equa ion. One can see om
he phase po ai o his equa ion in ( , z)∈P1×P1 ha he g aph
o z( ) canno in e sec he in a ian line z=∞, bu mus in e sec he
lines = 1,−1,∞ espec i ely a he singula poin s z=−1
1+α,1
1−α,0.
In o he wo ds, he a ional unc ion z( ) has no pole on he Riemann
sphe e and sa is ies z(1) 6=z(−1). This con adic s Liou ille Theo em
and p o es he claim.
When αis i a ional, g1and g−1ha e non-pe iodic linea pa s and
do no commu e. Lemma 3 p o ides in ini ely many ajec o ies ha ing
non- i ial holonomy. Mo eo e , he holonomy g oups o hose special
ajec o ies a e in ini e, and con ain con ac ions as soon as α6∈ R.
When αis a ional, g1and g−1ha e pe iodic linea pa s and he Ric-
ca i equa ion has ini ely many ajec o ies ha ing non- i ial holonomy
by Theo em 2. Fo ins ance, when α= 0, we ha e a e conjugacy
g1=ζ2nzand g−1=ζ2nz+ 1.
The e o e, he monod omy g oup Gis also gene a ed by
g1=ζ2nXand g−1◦(g1)−1=X+ 1
and he numbe o excep ional ajec o ies is gi en by Theo em 4 wi h
m= 2nand = 1.
Finally, assume ha Gis no abelian, and i s linea pa is no eal.
Fo ins ance, his is he case when n≥3 and αis ze o o i a ional. Then
we claim ha any complex ajec o y o he Ricca i equa ion ha ing a
leas one non- i ial holonomy map ac ually con ains loops a bi a y
close o z=∞wi h bounded p o iding his holonomy. The e o e, his
holonomy will occu in any neighbo hood o (x, y) = (0,0) as holonomy
o a ajec o y o he ec o ield V. In o de o p o e he claim, i su ices
o show ha gi en a poin z∈C ixed by a non i ial elemen g∈G,
and gi en a bi a y la ge cons an T0, one can ind a conjuga e
g0= ˜g−1◦g◦˜g, ˜g∈G, wi h a wo d decomposi ion
g0= (gε1)k1◦(gε2)k2◦ · · · ◦ (gεM)kM∈G,
Coun ing Fixed Poin s 131
whe e εm=±1 and km∈Z o m= 1,2,...,M, ha ing he ollowing
p ope y: he new ixed poin z0= ˜g−1(z) and all in e media e i e a es
(gεm)l◦(gεm+1 )km+1 ◦ · · · ◦ (gεM)kM(z0),m= 0,...,M,
l= 0,...,km
emain T- a om 0. Ob iously, he co esponding loop is in he same
lea (˜g∈G) and may be hough , in he undamen al g oup o he lea ,
as a p oduc o he ini ial one wi h ano he loop ha ing i ial holonomy;
hey a e e en no cohomologous in gene al. The new wo d g0may be
ob ained as ollows.
S a wi h a wo d decomposi ion o glike abo e. Deno e by h, h0∈G
wo ansla ions ha a e R-independan in G( he linea pa o Gis no
eal) oge he wi h a wo d decomposi ion. Conside also ˜
h=g◦h◦g−1.
Since z=∞is ixed by he gene a o s g1and g−1, he e is a sequence
··· > Tn>··· > T1> T0=Tsuch ha any poin z0ou side he
Tn+1-ball emains Tn- a om 0 a e one i e a ion o g1,g−1,g−1
1o g−1
−1.
Choose T0:= Tn o a nbounding he leng h o he wo ds g,h,h0and ˜
h.
The e o e, i z0is T0- a om 0, hen i s o bi unde i e a ion o g,h,h0
and ˜
h(o one o he in e ses) does no in e sec he T-ball.
Now, we choose a la ge ansla ion hNsuch ha z0=hN(z) and i s
image z00 =g(z0) a e T0- a om 0. No ice ha he new ans o ma ion
g0=h−N◦g◦hN ixes z0and admi s he wo d decomposi ion g0=
h−N◦˜
hN◦g. By cons uc ion, he i e a ion o wo d gon z0 emains
T- a om 0. Also, i he sequence
z00,˜
h(z00),...,˜
hN(z00), h−1◦˜
hN(z00),...,h−N◦˜
hN(z00) = z0
does no in e sec he T0-ball, hen he ull o bi o z00 unde h−N◦˜
hN
iewed as a wo d in g1and g−1will s ay T- a om 0.
I no , hen we can use he commu a i i y o h,h0o ˜
h o e-a ange
he wo d h−N◦˜
hN. Fo ins ance, when hand ˜
ha e R-independan , his
ollows om he ac ha he T0-ball canno disconnec he “la ice”
L= (z00 +R·h+Z·˜
h)∪(z00 +Z·h+R·˜
h)∈C.
The wo d h−N◦˜
hNco esponds o a pa h in Ljoining z00 o i s image
z0=h−N◦˜
hN(z00). I his pa h c osses he T0-ball, hen i can be
eplaced by ano he pa h in La oiding his ball. When hand ˜
ha e
R-dependan , hen we in oduce h0and use he la ice gene a ed by h
and h0. This ends he p oo o he las claim, as well as he co olla y.
132 F. Lo ay, M. an de Pu , F. Reche
Acknowledgemen . We would like o hank he e e ee who ca e ully
ead ou pape and mo i a ed us o p o ide Co olla y 1 as a conc e e
applica ion (wi h a ull p oo ).
The answe o he i s ques ion
Conside a subg oup G < A (C) gi en by gene a o s
G=haiX+bi;i= 1,...,si.
Le λ: A (C)→C∗;aX +b7→ abe he g oup homomo phism gi ing
he linea pa . Deno e by Λ := λ(G) he linea pa o Gand by
T:= ke (λ:G→Λ) i s ansla ion pa . We ha e
Λ = hai;i= 1,...,si ⊂ C∗.
One can iden i y T o a subg oup o C(s ill deno ed by T):
T={b;X+b∈G} ⊂ C.
F om he ac ion o Gby conjugacy on i s no mal subg oup T, we see
ha Tis s able unde mul iplica ion by elemen s o Λ and he e o e
inhe i s a s uc u e o module o e he ing Z[Λ]. In he sequel we will
suppose ha bo h Λ and Ta e non- i ial, o he wise he wo ques ions
we a e conce ned wi h become i ial.
We ake ca e ha , in gene al, Tis no a ini ely gene a ed subg oup
o G, bu we no e ha i is ini ely gene a ed as a no mal subg oup o G.
In o he wo ds, we claim ha Tis a ini ely gene a ed module o e Z[Λ]
(ha ing ank ≤s(s+1)
2). Indeed, he commu a o subg oup G0= [G, G]
o Gis he subg oup o Tgene a ed by all conjuga es o he elemen a y
commu a o s [aiX+bi, ajX+bj], i, j = 1,...,s,i6=jin G. The
g oup G0is hus gene a ed as no mal subg oup o G(o , equi alen ly, as
a module o e Z[Λ]) by hose s(s−1)
2elemen s. In o de o gene a e T, i
su ices o add a se o gene a o s o he quo ien T/G0. Since G/G0is
a commu a i e g oup o ank ≤s, i s subg oup T/G0has also ank ≤s.
We obse e ha he map c∈C→G{c}={g∈G;g(c) = c}induces
a bijec ion be ween Fix(G) and he se o he maximal commu a i e
subg oups Ho Gwi h H6=T. Since G{g(c)}=gG{c}g−1, one ob ain a
bijec i e co espondance
Fix(G)/G ←→ {H < G maximal commu a i e subg oup, H6=T}/G
wi h he G-conjugacy classes o such subg oups H. The image λ(H)⊂Λ
depends only on he G-conjugacy class o H.
Coun ing Fixed Poin s 133
Theo em 2. Wi h no a ions abo e, he se Fix(G)/G is ini e i , and
only i , he linea pa Λo Gis a ini e subg oup o C∗.
P oo : Suppose ha Λ is ini e. Then Λ = hζniwhe e ζndeno es a
p imi i e n h oo o uni y and Z[Λ] is equal o he ing o in ege s Z[ζn]
in C. Gi en a gene a o g∈G o Λ, g=ζnX+ , one may linea ize i by
a ansla ion and assume wi hou loss o gene ali y ha Λ is con ained as
a linea subg oup in G. The e o e, any elemen o Gw i es g=ζi
nX+ ,
∈Tand we ha e
Fix(G) =
n−1
[
i=1
1
1−ζi
n
T.
The se o T-o bi s o 1
1−ζi
nTis equal o he ini ely gene a ed mod-
ule T/(1 −ζi
n)To e he ini e ing Z[ζn]/(1 −ζi
n). Thus Fix(G)/G
is ini e. The o he implica ion o Theo em 2 ollows om he nex
lemma.
Lemma 3. Suppose ha Λis in ini e. Then he e a e in ini ely many
subg oups o Λo he o m λ(H), whe e H6=Tis a maximal commu a-
i e subg oup o G.
P oo : Le a∈Λ be an elemen o in ini e o de . A e conjuga ion o G
by an elemen o A (C), we may suppose ha aX lies in G. Choose an
in ege m > 1. In ac , we p o e ha , o any in ege m > 1, he e exis s
a maximal commu a i e subg oup H6=To Gsuch ha λ(H) con ains
some powe o abu ai6∈ λ(H) o i= 1,...,m. The lemma will ollow
om his.
The ing Z[Λ,1
(a−1)(a2−1)···(am−1) ] is ini ely gene a ed o e Z. The e-
o e he e exis s a su jec i e homomo phism φ o a ini e ield Fq. Se
α:= φ(a). Then αi6= 1 o i= 1,...,m and αq−1= 1. Le I⊂Z[Λ]
deno e he ideal gene a ed by he elemen s aq−1−1
ad−1 o d|q−1 and
1≤d≤m. This is a p ope ideal o Z[Λ]: ad−1/∈Isince φ(ad−1) 6= 0
and φsends I o 0 (φ(aq−1−1) = 0). The e exis s an elemen ∈T IT.
Indeed, suppose ha IT =T. Le mbe a maximal ideal o R:= Z[Λ]
con aining I, so mT =T. A e localiza ion wi h espec o S:= R m,
one inds ha B:= S−1Tis a ini ely gene a ed module o e he noe-
he ian local ing S−1Rsuch ha mB =B. This implies ha B= 0
(Nakayama’s lemma) [La]. Since he elemen s o Sa e no ze o di iso s
on T, one ob ains he con adic ion ha T= 0.
Finally, le Hbe a maximal commu a i e subg oup con aining h:=
aq−1X+ . Le d≥1 be minimal such ha ad∈λ(H). Since we ha e
also aq−1∈λ(H), hen d|q−1. I ollows ha Hcon ains an elemen
134 F. Lo ay, M. an de Pu , F. Reche
o he o m k:= adX+bwi h b∈Tsince aX ∈G. Now kq−1
d=h(i
no , he e exis s a non- i ial ansla ion in Hwhich is a con adic ion
since His commu a i e) and hus =aq−1−1
ad−1b. This implies ha d > m
and Hhas he equi ed p ope ies.
The answe o he second ques ion
We ha e o conside a g oup Ggene a ed by a ini e linea sub-
g oup hζnXio o de nand by a ansla ion subg oup {X+ ; ∈T}
whe e T⊂Cis a ini ely gene a ed Z[ζn]-module o ank . We no e ha
he module Thas no o sion and hus Tis a p ojec i e Z[ζn]-module.
Fo he compu a ion o # Fix(G)/G, we s a wi h wo examples.
Example 1. Assume ha n=pk+1 wi h pp ime and k≥0.As in he
beginning o he p oo o Theo em 2, one may assume wi hou loss o
gene ali y ha any elemen o Gw i es g=ζi
nX+ , ∈T. I g∈Ghas
a ixed poin (g6∈ T), hen a con enien i e a e g◦g◦ · · · ◦ ghas linea
pa ζpwi h he same ixed poin . The e o e, we ha e
Fix(G) = 1
1−ζp
·T
and we may conside
Fix(G)/T ≃T/(1 −ζp)T
as a module o e he ing Z[ζpk+1 ]/(1−ζp). We ha e o coun he numbe
o Λ-o bi s on T/(1 −ζp)T, whe e Λ = hζpk+1 i. Fi s , we ha e
Z[ζpk+1 ]/(1 −ζp)≃Z[X]/(Φpk+1 (X), Xpk−1),
whe e Φn=Xpk+1 −1
Xpk−1=X(p−1)pk+···+Xp+ 1 deno es he n h cyclo-
omic polynomial. The ideal is also gene a ed by pand Xpk−1. A e
subs i u ion X= 1 + Y, one inds ha
Z[ζpk+1 ]/(1 −ζp)≃Fp[Y]/(Ypk).
In pa icula , he ideal (1 −ζp) is con ained in only one maximal ideal,
say m. Le Sbe he mul iplica i e sys em S:= Z[ζpk+1 ] m. Then
S−1Tis a ee module o ank o e he local ing S−1Z[ζpk+1 ]. Since
T/(1−ζp)Tis isomo phic o S−1T/(1−ζp)S−1T, one has ha T/(1−ζp)T
is a ee module o ank o e Z[ζpk+1 ]/(1 −ζp) and we can iden i y
T/(1 −ζp)T≃Fp[Y]/(Ypk)
.
Coun ing Fixed Poin s 135
We ha e o coun he numbe o Λ-o bi s on Fp[Y]/(Ypk)
whe e Λ is
he cyclic g oup o o de pkgene a ed by (1 + Y) modulo (Ypk).
The se Fix(G)/T ≃Fp[Y]/(Ypk)
spli s in o he disjoin union o
C(ps) = nc∈Fp[Y]/(Ypk)
; he Λ-o bi o chas leng h pso
o s= 0,...,k. On he o he hand, one obse es ha he se o o bi s
ha ing leng h a mos psis gi en by s
i=0C(pi) = Ypk−sFp[Y]/(Ypk)
and hus
s
X
i=0
#C(pi) = # Ypk−sFp[Y]/(Ypk)
=p ps.
One concludes ha C(ps) con ains exac ly p o bi s i s= 0 and
p ps−p ps−1
pso bi s o s= 1,...,k. This leads o he o mula
# Fix(G)/G =p +
k
X
s=1
p ps−p ps−1
ps.
Example 2. Assume ha n=pk+1ql+1 wi h dis inc p imes p,qand
k, l ≥1.Le T⊂Cbe a ini ely gene a ed Z[ζpk+1ql+1 ]-module o
ank and le G⊂A (C) be he g oup gene a ed by ζpk+1ql+1 Xand
{X+ ; ∈T}. As in Example 1, he ixed poin o a non- i ial elemen
g∈G Tis also he ixed poin o a con enien i e a e o gha ing ζp
o ζqas linea pa . The e o e,
Fix(G) = 1
1−ζp
T[1
1−ζq
T.
The in e sec ion o hese wo se s is T. Indeed, a
1−ζp=b
1−ζqimplies
(1 −ζp)b= (1 −ζq)a. Since he image o (1 −ζq) in Z[ζpk+1ql+1 ]/(1 −ζp)
is in e ible, one has ha ais di isible in Tby (1 −ζp). Le Np,Nq
deno e he espec i e numbe s o Λ-o bi s on he se o he non-ze o
elemen s o 1
1−ζpT/T and 1
1−ζqT/T. Then
# Fix(G)/G = 1 + Np+Nq.
Now we concen a e on he coun ing o Np. The na u al homomo phism
Z[ζpk+1 ]⊗Z[ζql+1 ]−→ Z[ζpk+1ql+1 ]
is an isomo phism. The same a gumen s as in Example 1 yield
Z[ζpk+1ql+1 ]/(1 −ζp)≃Fp[Y]/(Ypk)⊗Fp[X]/(Φql+1 ).
