MOLINO’S DESCRIPTION AND FOLIATED
HOMOGENEITY
JES´
US A. ´
ALVAREZ L´
OPEZ AND RAM´
ON BARRAL LIJ´
O
Abs ac . The opological Molino’s desc ip ion o equicon inuous o-
lia ed spaces, s udied by he i s au ho and Mo ei a Galicia, gi es
condi ions o educe hei s udy o he pa icula case whe e he holo-
nomy pseudog oup can be ep esen ed by a pseudog oup on some local
g oup Ggene a ed by some o i s local le ansla ions (a G- olia ed
space). Tha desc ip ion is sha pened in his pape by in oducing a o-
lia ed ac ion o a compac opological g oup on he esul ing G- olia ed
space, like in he case o Riemannian olia ions. Mo eo e a C∞ e sion
is also s udied. The i iali y o his compac g oup cha ac e izes com-
pac minimal G- olia ed spaces, which a e also cha ac e ized by hei
olia ed homogenei y in he C∞case. We also gi e an example whe e
he p ojec ion o he Molino’s desc ip ion is no a p incipal bundle, and
ano he example o posi i e opological codimension whe e he olia ed
homogenei y canno be checked by only compa ing pai s o lea es—in
he case o ze o opological codimension, weak solenoids wi h his p op-
e y we e gi en by Fokkink and O e s eegen, and la e by Dye , Hu de
and Lukina.
Con en s
1. In oduc ion 2
2. P elimina ies 5
3. Molino’s desc ip ion 19
4. Folia ed homogeneous olia ed spaces 24
5. C∞Molino’s desc ip ion 26
6. Righ local ans e se ac ions 27
7. C∞G- olia ed spaces a e C∞ olia ed homogeneous 32
8. Examples and open p oblems 32
Re e ences 37
Da e: Feb ua y 14, 2019.
1991 Ma hema ics Subjec Classi ica ion. 57R30.
Key wo ds and ph ases. Folia ed space, equicon inuous, s ongly quasi-analy ic,
Molino’s desc ip ion, olia ed homogeneous.
The au ho s a e pa ially suppo ed by MICINN, g an MTM2014-56950-P, and Xun a
de Galicia, g an 2015 GPC GI-1574.
1
2 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
1. In oduc ion
A desc ip ion o ce ain compac minimal equicon inuous olia ed spaces
was gi en by he i s au ho and Mo ei a Galicia [10]. I can be conside ed
as a opological e sion o he Molino’s desc ip ion o Riemannian olia ions
on compac mani olds [37], in he minimal case. This ga e ano he con-
i ma ion ha equicon inuous olia ed spaces should be conside ed as he
opological Riemannian olia ions, as asse ed by Ghys [37, Appendix E].
Tha desc ip ion educes he s udy o such olia ed spaces o he pa icu-
la case o G- olia ed spaces, which a e he olia ed spaces whose holonomy
pseudog oup can be ep esen ed by a pseudog oup on some local g oup G
gene a ed by some o i s local le ansla ions. The classical example o
G- olia ed spaces a e Lie olia ions, which a e used in he o iginal Molino’s
heo y o desc ibe Riemannian olia ions. Acco ding o he ole played by
Molino’s heo y in he s udy o Riemannian olia ions, i s opological e sion
should ha e in e es ing applica ions; o ins ance, i was al eady used in [10]
o s udy he g ow h o he lea es.
Dye , Hu de and Lukina also ga e an analogue o Molino’s desc ip ion
o equicon inuous ma chbox mani olds [18, 19], which is he case o compac
connec ed minimal olia ed spaces o opological codimension ze o; i.e., wi h
o ally disconnec ed local ans e sals. The ad an age o hei cons uc ion
is ha i wo ks wi hou any addi ional condi ion, bu hei desc ip ion is
unique jus when ou hypo heses a e ul illed.
Ou i s goal is o show he ollowing sligh sha pening o he main esul
o he opological Molino’s heo y (Sec ion 3). The e minology and no a ion
used he e a e ecalled in Sec ion 2.
Theo em A (C . [10, Theo em A]).Suppose ha a olia ed space X≡
(X, F)is compac , minimal, equicon inuous and s ongly quasi-analy ic, and
he closu e o i s holonomy pseudog oup is also s ongly quasi-analy ic. Then
he e is a local g oup G, a compac opological g oup H, a compac minimal
G- olia ed space 
X0≡(
X0,
F0), a olia ed map ˆπ0∶
X0→X, and a ee
olia ed igh H-ac ion on 
X0such ha he es ic ions o ˆπ0 o he lea es
o 
X0a e he holonomy co e ings o he lea es o X, and ˆπ0induces a
homeomo phism 
X0/H→X.
P ecisely, ou new con ibu ion in Theo em A is he exis ence o Hsa -
is ying he s a ed p ope ies. I His he ep esen a i e o he holonomy
pseudog oup o Xon a space Tinduced by he choice o a good olia ed
a las, and we ix some u0∈T, hen His he g oup o ge ms a u0o he
maps gin he closu e Hwi h u0∈domgand g(u0)=u0. Following he
cons uc ion o 
X0in [10], we ge a compa ible compac opology on Hand
a igh olia ed H-ac ion on 
X0sa is ying he s a emen o Theo em A.
We also show ha he cons uc ion o (G, H, 
X0,ˆπ0)is independen o he
choices in ol ed up o an ob ious equi alence ela ion (P oposi ion 3.1), and
he e o e (G, H, 
X0,ˆπ0)is called he Molino’s desc ip ion o X; in pa icula ,
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 3
Gis called he s uc u al local g oup acco ding o [37, 10], and His called he
disc iminan g oup acco ding o [18]. Unde he hypo hesis o Theo em A,
we also p o e he ollowing addi ional p ope ies:
●Xis a G- olia ed space o some local g oup Gi and only i i s
disc iminan g oup is i ial (P oposi ion 3.2).
●The e is a subg oup in Hisomo phic o he holonomy g oup o
e e y lea (P oposi ion 3.4).
●I Xis C∞, hen i s Molino’s desc ip ion becomes C∞in a unique
ob ious sense (P oposi ion 5.1).
●The map ˆπ0may no be a ibe bundle (an example is gi en in
Sec ion 8.2). This is he only missing p ope y when compa ing
wi h he Riemannian olia ion case.
Ou second goal is o cha ac e ize G- olia ed spaces using a p ope y
called olia ed homogenei y. A olia ed space X≡(X, F)is called oli-
a ed homogeneous i he g oup Homeo(X, F)o i s olia ed ans o ma ions
ac s ansi i ely on i sel (a olia ed e sion o homogenei y). This no ion
was s udied by Cla k and Hu de in he case o ma chbox mani olds [15],
whe e homogenei y and olia ed homogenei y a e equi alen no ions because
Homeo(X)=Homeo(X, F)since he lea es a e he pa h connec ed compo-
nen s. Cla k and Hu de ha e shown ha a ma chbox mani old is equicon-
inuous i and only i i is a weak solenoid (an in e se limi o a owe o
co e ing maps be ween closed connec ed mani olds), and i is homogeneous
i and only i i is a McCo d solenoid ( he co e ing maps can be chosen o
be egula ), also called s ong solenoid. Since McCo d solenoids a e ans-
e sely modeled by le ansla ions on p o ini e g oups, hey a e pa icula
cases o G- olia ed spaces. Fo his eason, he men ioned Molino’s de-
sc ip ion o Dye , Hu de and Lukina is a p ocedu e o cons uc McCo d
solenoids om weak solenoids. On he o he hand, acco ding o he o iginal
Molino’s heo y [37], among minimal Riemannian olia ions on closed man-
i olds, he homogeneous ones a e he Lie olia ions ( he G- olia ions o Lie
g oups G). Thus, gene alizing he case o ma chbox mani olds and minimal
Riemannian olia ions on closed mani olds, i makes sense o ask whe he
any compac minimal olia ed space is olia ed homogeneous i and only i
i is a G- olia ed space. We gi e he ollowing answe s.
Theo em B. I a olia ed space Xis compac , minimal and olia ed homo-
geneous, hen i sa is ies hypo heses o Theo em A and is a G- olia ed space
o some local g oup G.
Theo em C. Suppose ha a olia ed space Xis compac , minimal and C∞.
Then he ollowing condi ions a e equi alen :
(i) Xis C∞ olia ed homogeneous.
(ii) Xis olia ed homogeneous.
(iii) Xsa is ies he hypo heses o Theo em A and is a G- olia ed space
o some local g oup G.
4 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
He e, a olia ed space is said o be C∞when i has a olia ed a las whose
changes o coo dina es a e C∞along he lea es, and hei lea wise pa ial
de i a i es o a bi a y o de a e con inuous (on he ambien space). O he
ela ed concep s a e de ined in he same way, like C∞ olia ed maps, C∞
di eomo phisms, (lea wise) angen space, (lea wise) Riemannian me ics,
(lea wise) Riemannian olia ed spaces, e c. Fo C∞ olia ed spaces, he
concep o C∞ olia ed homogenei y can be de ined like olia ed homogenei y
using C∞ olia ed di eomo phisms.
Theo em B ollows wi h an adap a ion o an a gumen o Cla k and Hu -
de [15, Theo em 5.2], using ha he canonical le ac ion o Homeo(X, F)
on Xis mic o- ansi i e by a heo em o E os [20, 43].
To p o e Theo em C, i is enough o show “(iii) ⇒(i)” by Theo em B.
Assuming (iii), we ge he so-called s uc u al igh local ans e se ac ion,
which has i s own in e es ; o ins ance, i was in oduced and used in [7]
o Lie olia ions. I is he unique “ olia ed igh local ac ion up o lea wise
homo opies” o Gon X, which co esponds o he local igh ansla ions on
G ia olia ed cha s (P oposi ion 6.6 and Sec ion 6.3). I s cons uc ion uses
a pa i ion o uni y subo dina ed o a olia ed a las and he lea wise cen e o
mass o some (lea wise) Riemannian me ic o me ge he ob ious igh local
ans e se ac ions on he domains o olia ed cha s. The s uc u al igh
local ans e se ac ion gi es (i) because we always ha e lea wise homogenei y
(P oposi ion 7.1).
In Theo em C, ou p oo o “(iii) ⇒(i)” needs he C∞s uc u e o X
because we use he lea wise cen e o mass as an auxilia y ool. O cou se, i
could be possible o a oid he C∞condi ion and show “(iii) ⇒(ii)” di ec ly
wi h o he ools, bu ha p ocedu e would ce ainly equi e mo e wo k.
Since he e exis lea es wi hou holonomy, and since he (di e en iable)
quasi-isome y ype o he lea es is independen o he choice o a (lea wise)
Riemannian me ic on X, i ollows ha Xis no olia ed homogeneous i
he e is a lea wi h holonomy, o i he e is a pai o non-quasi-isome ic
lea es. The ecip ocal s a emen is no ue in gene al. Fokkink and O e -
s eegen [23, Theo em 35] cons uc ed an example o a non-homogeneous
weak solenoid all o whose lea es a e simply connec ed, and he e o e i
has no holonomy, and i s lea es a e quasi-isome ic o each o he because
weak solenoids a e suspension olia ed spaces. Dye , Hu de and Lukina
cons uc ed mo e examples o such weak solenoids [19, Theo em 10.7]. In
Sec ion 8.3, we gi e an example o a compac olia ed space Xsa is ying
he condi ions o Theo em A, which is no olia ed homogeneous and has no
holonomy, whose lea es a e quasi-isome ic o each o he , and wi h locally
connec ed local ans e sals ( hus i is no a weak solenoid).
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 5
2. P elimina ies
See [39, Chap e II], [24] and [13, Chap e 11] o he needed p elimina ies
on olia ed spaces and in e es ing examples, and [27, 28, 29] o he p elimi-
na ies on pseudog oups. We mainly ollow [10, Sec ions 2 and 4A], which in
u n ollows [4, 5, 6]. Some ideas a e also aken om [15, 9, 8]. The needed
basic concep s and ools a e ecalled he e o he eade ’s con enience, and
a ew new obse a ions a e also made.
In he whole pape , unless o he wise s a ed, spaces a e assumed o be
locally compac and Polish, and maps a e assumed o be con inuous. In
pa icula , his applies o olia ed spaces, opological g oups, local g oups
and pa ial maps.
2.1. Pseudog oups. Fo spaces Tand T′, he no a ion φ∶T↣T′is used
o a pa ial map. We will only conside he case whe e i s domain, dom φ,
is open in T. The ge m o φa any u∈dom φwill be deno ed by γ(φ, u).
I φis an open embedding, we may iden i y φwi h he homeomo phism
φ∶domφ→imφo an open subse o T o an open subse o T′, whose
in e se can be conside ed as a pa ial map wi h open domain, φ−1∶T′↣T;
in pa icula , when T=T′, such a φis called a local ans o ma ion o T.
Gi en ano he space T′′, le Φ and Ψ be amilies o pa ial maps T↣T′
and T′↣T′′, espec i ely, wi h open domains. We use he no a ion ΨΦ =
{ψφ ∣φ∈Ψ, ψ ∈Ψ}; in pa icula , Φn=Φ⋯Φ (n imes) i T=T′and
n∈Z+. I Φ consis s o open embeddings, le Φ−1={φ−1∣φ∈Φ}.
Recall ha a pseudog oup Hon Tis a amily o local ans o ma ions o
T ha con ains idT, and is closed by he ope a ions o composi e, in e sion,
es ic ion o open se s and union. I is said ha His gene a ed by S⊂H
i Hcan be ob ained om Susing he abo e ope a ions. By conside ing
a pseudog oup as a di ec gene aliza ion o a g oup o ans o ma ions, he
basic dynamical concep s ha e ob ious gene aliza ions o pseudog oups, like
o bi s,sa u a ion, ( opological) ansi i i y and minimali y. The o bi space
is deno ed by T/H. The H-sa u a ion o any A⊂Tis deno ed by H(A),
and he o bi o any u∈Tby H(u). Fo any open V⊂T, he es ic ion
H∣V∶={h∈H∣domh, imh⊂V}is a pseudog oup.
Gi en ano he pseudog oup H′on T′, a mo phism Φ∶H→H′is a maximal
collec ion o pa ial maps T↣T′wi h open domain such ha H′ΦH⊂Φ,
T=⋃φ∈Φdomφ, and, o all φ, ψ ∈Φ and u∈dom φ∩dom ψ, he e is some
h′∈H′so ha φ(u)∈domh′and γ(h′φ, u)=γ(ψ, u). Le Φ0be a amily
o pa ial maps T↣T′wi h open domain such ha T=H(⋃φ∈Φdom φ),
and he e is a subse So gene a o s o Hsuch ha , i φ, ψ ∈Φ0,h∈Sand
u∈domφ∩domψh, hen he e is some h′∈H′so ha φ(u)∈dom h′and
γ(h′φ, u)=γ(ψh, u). Then he e is a unique mo phism Φ ∶H→H′con ain-
ing Φ0, which is said o be gene a ed by Φ0. Fo ins ance, idTgene a es a
mo phism idH∶H→Hconsis ing o all possible unions o maps in H; in
pa icula , H⊂idH. Fo ano he pseudog oup H′′ on T′′ and a mo phism
Ψ∶H′→H′′, he amily ΨΦ gene a es a mo phism H→H′′, which may

6 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
be also deno ed by ΨΦ wi h some abuse o no a ion. In his way, he mo -
phisms o pseudog oups o m a ca ego y PsG . The e is a canonical unc o
Top →PsG , assigning he pseudog oup gene a ed by idT, also deno ed by
T, o e e y opological space T, and assigning he mo phism gene a ed by
φ, also deno ed by φ, o e e y map φ∶T→T′. A mo phism Φ ∶H→H′
is an isomo phism o PsG i and only i i is gene a ed by a amily Φ0o
open embeddings such ha Φ−1
0gene a es a mo phism H′→H, which is he
in e se Φ−1in PsG .
Wi h he e minology o Hae lige [27, 28, 29], an ´e al´e mo phism Φ∶
H→H′is a maximal amily o homeomo phisms o open subse s o T o
open subse s o T′such ha H′ΦH⊂Φ, T=⋃φ∈Φdom φand ΦΦ−1⊂H′.
I mo eo e Φ−1is an ´e al´e mo phism, hen Φ is called an equi alence, and
he pseudog oups Hand H′a e said o be equi alen . I Φ0is a amily o
homeomo phisms o open subse s o T o open subse s o T′such ha T=
H(⋃φ∈Φdomφ)and Φ0HΦ−1
0⊂H′, hen he e is a unique ´e al´e mo phism Φ ∶
H→H′con aining Φ0, which is said o be gene a ed by Φ0. Any equi alence
gene a es an isomo phism in PsG , and, ice e sa, any isomo phism in PsG
is gene a ed by a unique equi alence. Hence isomo phism and equi alences
a e equi alen concep s. Equi alen pseudog oups a e conside ed o ha e he
same dynamics. Fo ins ance, His equi alen o H∣V o any open V⊂T
ha mee s all H-o bi s. In ac , Φ ∶H→H′is an equi alence i and only i
G=H∪H′∪Φ∪Φ−1is a pseudog oup on T⊔T′such ha Tand T′mee
all G-o bi s, G∣T=Hand G∣T′=H′.
The ge ms γ(h, u), o h∈Hand u∈dom h, o m a opological g oupoid
H, equipped wi h he shea opology and he ope a ion induced by com-
posi e. I s uni subspace can be iden i ied wi h T. In ac , His an ´e al´e
g oupoid ( he sou ce and a ge maps, s, ∶H→T, a e local homeomo -
phisms). Gi en x∈T, he g oup o elemen s o γ∈Hwi h s(γ)= (γ)=xis
called he ge m g oup o Ha x.
Le us ecall he ollowing de ini ions o p ope ies ha Hmay ha e:
Compac gene a ion: This means ha he e is a ela i ely compac
open U⊂T, which mee s all o bi s, such ha H∣Uis gene a ed
by a ini e se , E={h1,...,hk}, and e e y hihas an ex ension
˜
hi∈Hwi h domhi⊂dom ˜
hi. This Eis called a sys em o compac
gene a ion o Hon U.
(S ong) equicon inui y: This means ha he e a e an open co e
{Ti}o Tand a me ic diinducing he opology o e e y Ti, and
His gene a ed by some subse S⊂H, wi h S2⊂S=S−1(Sis
symme ic and closed by composi es1), such ha , o e e y >0,
he e is some δ>0 so ha
di(x, y)<δÔ⇒dj(h(x), h(y))<
1The e m pseudo∗g oup was used in [10] when hese condi ions a e sa is ied. This
e m was in oduced in [36] o a amily ha mo eo e con ains idTand is also closed by
es ic ions o open subse s.
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 7
o all h∈S, indices i, j, and x, y ∈Ti∩h−1(Tj∩im h).
S ong quasi-analy ici y: This means ha His gene a ed by some
subse S⊂H, wi h S2⊂S=S−1, such ha , i any h∈Sis he
iden i y on some non-emp y open subse o i s domain, hen h=
iddom h.
S ong local eeness: This means ha His gene a ed by some sub-
se S⊂H, wi h S2⊂S=S−1, such ha , i any h∈S ixes some
poin in i s domain, hen h=iddom h. Equi alen ly, his means ha
His s ongly quasi-analy ic and all o i s ge m g oups a e i ial.
These p ope ies a e in a ian by equi alences. I compac gene a ion holds
wi h some U, hen i also holds wi h any o he ela i ely compac open
subse o T ha mee s all o bi s. Le Pdeno e any o he abo e las h ee
p ope ies. I Pholds wi h S, hen i also holds wi h i s localiza ion,
Sloc ={h∣O∣h∈S, O is open in dom h}.
Mo eo e we can add idT o Si desi ed (ob aining S2=S). I His com-
pac ly gene a ed and sa is ies P, hen, o e e y ela i ely compac open
U⊂T ha mee s all o bi s, we can choose a sys em o compac gene a ion
Eo Hon Usuch ha H∣Ualso sa is ies Pwi h S=⋃∞
n=1En. The ollowing
esul lis s some needed non-elemen a y p ope ies.
P oposi ion 2.1 ([4, P oposi ion 8.9, and Theo ems 11.1 and 12.1], [42]
and [5, Theo ems 3.3 and 5.2]).Suppose ha His compac ly gene a ed,
equicon inuous and s ongly quasi-analy ic. Then he ollowing holds:
(i) Assume ha Hsa is ies he condi ion o compac gene a ion wi h U,
E={h1,...,hk}and ˜
h1,...,˜
hk. Fo e e y h=hin⋯hi1∈⋃∞
n=1En,
le ˜
h=˜
hin⋯˜
hi1. Then he e is a ini e amily Vo open subse s o
Tco e ing Usuch ha , o any h∈⋃∞
n=1Enand V∈V, we ha e
V⊂dom ˜
hi V∩domh≠∅.
(ii) Suppose ha Hsa is ies he equicon inui y condi ion wi h a se
S. Then C(O, T )∩Sloc consis s o local ans o ma ions o all
small enough open subse s O⊂T, whe e he closu e is aken in he
compac -open opology, and he pseudog oup Hgene a ed by such
ans o ma ions is equicon inuous. Mo e p ecisely, Hsa is ies he
equicon inui y condi ion wi h he se Sde e mined by he condi ion
C(O, T )∩S=C(O, T )∩Sloc o all Oas abo e.
(iii) The o bi closu es a e minimal se s, and he e o e His ansi i e i
and only i i is minimal.
In P oposi ion 2.1-(ii), he pseudog oup His called he closu e o H.
2.2. Rela ion o pseudog oups wi h local g oups and local ac ions.
The gene al de ini ion o local g oup is a he in ol ed [34], bu , in he
locally compac case, a local g oup Gcan be conside ed as neighbo hood
o he iden i y elemen ein some opological g oup [16, 17]. Two such
neighbo hoods in he same opological g oup de ine equi alen local g oups;
8 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
hus i can be said ha , up o equi alences, a local g oup is he “ge m” o
a opological g oup a he iden i y elemen . Fo he sake o simplici y, he
amily o open neighbo hoods o ein Gwill be deno ed by N(G, e). Gi en
ano he local g oup G′wi h iden i y elemen e′, a local homomo phism o G
o G′is a pa ial map wi h open domain, σ∶G↣G′, such ha e∈domσ,
σ(e)=e′, and σ(gh)=σ(g)σ(h) o all g, h ∈domσsuch ha he p oduc s
gh and σ(g)σ(h)a e de ined wi h gh ∈domσ. Two local homomo phisms
o G o G′a e equi alen when hey ha e he same ge m a e. I he e is a
local homomo phism τ∶G′↣Gsuch ha τσ and στ a e equi alen o idG
and idG′, hen σis called a local isomo phism. The e m sublocal g oup will
be used o a subspace H⊂Gsuch ha (H∩V)2,(H∩V)−1⊂H o some
V∈N(G, e); in pa icula , e∈H, bu H∩Vis no equi ed o be closed
in V(con a y o [26, De ini ion 2.10.]). A sublocal g oup becomes a local
g oup wi h he induced s uc u e, bu i may no be locally compac , and he
inclusion map o any sublocal g oup is a local homomo phism. A igh local
ac ion o Gon Tis a pa ial map wi h open domain, χ∶T×G↣T, whe e
T×{e}⊂domχand χ(u, e)=u o all u∈T, and such ha , o all g, h ∈G
and u∈T, i he p oduc gh is de ined and (u, g),(u, gh),(χ(u, g), h)∈
domχ, hen χ(χ(u, g), h)=χ(u, gh). Two igh local ac ions o Gon T
a e equi alen when hey ag ee a ound T×{e}. I Tis compac , we can
assume domχ=T×O o some O∈N(G, e). Fo any open V⊂T, he
es ic ion χ∶χ−1(V)∩(V×G)→Vis a igh local ac ion o Gon V, called
he es ic ion o χ o V. Gi en an open co e {Ti}o Tand a igh local
ac ion χio Gon e e y Tisuch ha he es ic ions o χiand χj o Ti∩Tj
a e equi alen , i is easy o check ha he e is a unique igh local ac ion o
Gon T, up o equi alences, whose es ic ion o e e y Tiis equi alen o χi.
Conside ano he igh local ac ion χ′o G′on T′. A pa ial map wi h
open domain, φ∶T↣T′, is called locally equi a ian i he e is some open
neighbo hood Σ o dom φ×{e}in dom χ∩(φ×idG)−1(dom χ′)such ha
χ(Σ)⊂domφand φχ(u, g)=χ′(φ(u), g) o all (u, g)∈Σ. No e ha
composi es, es ic ions o open se s and unions o locally equi a ian pa ial
maps wi h open domain a e locally equi a ian , as well as hei in e ses
whene e de ined. A amily o pa ial maps T↣T′wi h open domain is
called locally equi a ian when all o i s elemen s a e locally equi a ian .
Local an i-homomo phisms,le local ac ions, hei equi alences and co -
esponding locally equi a ian maps a e simila ly de ined.
Fo ins ance, any ini e dimensional me izable locally compac local g oup
is indeed locally isomo phic o he di ec p oduc o a Lie g oup and a com-
pac ze o-dimensional opological g oup [34, Theo em 107] (co ec ed ac-
co ding o [26], o using [16, 17] and [38, Sec ion IV.4.9]). As a conc e e
example, we can conside he p oduc o any local Lie g oup and any coun -
able amily o ini e g oups. By Ado’s heo em, he equi alence classes o
local Lie g oups and hei local homomo phisms co espond one- o-one o
ini e dimensional eal Lie algeb as and hei homomo phisms. A ypical
example o igh local ac ion o a local g oup Gon i sel is gi en by i s
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 9
local igh ansla ions, and any local le ansla ion o Gbecomes locally
equi a ian .
P oposi ion 2.2 ([5, Theo ems 3.3 and 5.2], [10, Lemma 2.36, Theo em 2.38
and Rema k 21]).The ollowing holds:
(i) Suppose ha His minimal, compac ly gene a ed, equicon inuous
and s ongly quasi-analy ic. Then His s ongly locally ee i and
only i His equi alen o a pseudog oup on some local g oup G
gene a ed by he le local ac ion by local le ansla ions o a ini ely
gene a ed dense sublocal g oup Γ⊂G.
(ii) Le Gand G′be he pseudog oups on local g oups Gand G′gene a ed
by he le local ac ions by local le ansla ions o espec i e ini ely
gene a ed dense sublocal g oups Γand Γ′. Le Φ∶G→G′be a
mo phism such ha G(e)↦G′(e′)by he induced map G/G→G/G′.
Then Φis gene a ed by a local homomo phism G↣G′ ha es ic s
o a local homomo phism Γ↣Γ′.
P oposi ion 2.3. Le Φ∶H→H′be an equi alence be ween compac ly
gene a ed pseudog oups. Le χbe a igh local ac ion o Gon Tsuch ha
His locally equi a ian . Then he e is a unique igh local ac ion χ′o G
on T′, up o equi alences, such ha Φand H′a e locally equi a ian .
P oo . Le Ebe a sys em o compac gene a ion o Hon a ela i ely compac
open U⊂T, and le ˜
hbe an ex ension o e e y h∈Ewi h dom h⊂dom ˜
h.
The e is a subse Φ0⊂Φ such ha {dom φ×im φ∣φ∈Φ0}co e s U×T′,
{imφ∣φ∈Φ0}is locally ini e in T′, and e e y φ∈Φ0has an ex ension
˜
φ∈Φ wi h domφ⊂dom ˜
φ. W i e {φi}={φh ∣h∈E, φ ∈Φ0}, and
le ˜
φi=˜
φ˜
hi φi=φh o h∈Eand φ∈Φ0. Mo eo e le Ui=dom φi,
U′
i=imφi,
Ui=dom ˜
φi,
U′
i=im ˜
φi,
Uij =˜
φ−1
j(
U′
i∩
U′
j)=dom ˜
φ−1
i˜
φjand

U′
ij =φj(
Ui∩
Uj)=dom ˜
φi˜
φ−1
j. The ollowing asse ion is easy o check.
Claim 1.{φi}gene a es Φ and {φiφ−1
j}gene a es H′.
Le Ω =domχ, and le Σij be an open neighbo hood o 
Uij ×{e}in Ω ∩
(˜
φ−1
i˜
φj×idG)−1(Ω)such ha χ(Σij)⊂
Uij and ˜
φ−1
i˜
φjχ(u, g)=χ(˜
φ−1
i˜
φj(u), g)
o all (u, g)∈Σij. Le
Ω′
0={(u′, g)∈T′×G∣u′∈U′
i∩U′
j⇒(˜
φ−1
j(u′), g)∈Σij,∀i, j }.
Claim 2.Ω′
0is open in T′×G.
Take some (u′, g)∈Ω′
0. Le Ibe he se o indices isuch ha u′∈U′
i, and
le I′be he se o pai s o indices, (i, j), such ha u′∈U′
i∩U′
j, which a e
ini e se s because {U′
i}is locally ini e in T′. Then, using ha U′
i⊂
U′
i, e e y
˜
φiis a homeomo phism, and Σij is an open neighbo hood o (˜
φ−1
j(u′), g)in

Uij ×G o all (i, j)∈I′, i ollows ha he e a e open neighbo hoods, Vo
u′in T′and Po gin G, such ha V∩U′
i=∅i i/∈I, and ˜
φ−1
j(V)×P⊂Σij
o all (i, j)∈I′. Thus V×P⊂Ω′
0.
16 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
de ined by adding he condi ion p′
aiφ=p′
aiψon e e y Ki o he abo e de ini-
ion o Nk
F(φ, U,U′,K,E); using (3), his ex a condi ion can be also w i en
as φ2
aii=ψ2
aiion pi(Ki) o all i. The weak plaquewise C opology is simi-
la ly de ined by equi ing he condi ions only o ini e amilies o indices i.
The subindex “WP/SP” will be added o he no a ion o indica e ha he
weak/s ong plaquewise C opology is conside ed in a amily o C olia ed
maps. No e ha , i wo olia ed maps a e close enough in C
SP(X, F;X′,F′),
hen hey induce he same mo phism H→H′; in ac , hey a e lea wisely
homo opic i =∞, as ollows by aking basic open se s Nk
P(φ, U,U′,K,E)
as abo e whe e he plaques o he olia ed cha s in U′a e con ex balls in
he lea es o a gi en Riemannian me ic on X′, and hen using geodesic
segmen s o de ine homo opies.
Wi h he s ong plaquewise C opology, we can con inue he di ec ex-
ensions o esul s abou spaces o C maps be ween mani olds.
P oposi ion 2.6. The ollowing p ope ies hold:
(i) Emb (X, F;X′,F′)is open in C
SP(X, F;X′,F′) o 1≤ ≤∞.
(ii) Fo 1≤ ≤∞, he se o closed C olia ed embeddings is open in
C
SP(X, F;X′,F′).
(iii) Di eo (X, F;X′,F′)is open in C
SP(X, F;X′,F′) o 1≤ ≤∞.
(i ) Cs(X, F;X′,F′)is dense in C
SP(X, F;X′,F′) o 0≤ <s≤∞.
( ) Di eos(X, F;X′,F′)is dense in Di eo
SP(X, F;X′,F′) o 1≤ <
s≤∞.
( i) I 1≤ <∞, any C olia ed space is C di eomo phic o a C∞
olia ed space.
( ii) I 1≤ <s≤∞, wo Cs olia ed spaces a e Csdi eomo phic i and
only i hey a e C di eomo phic.
P oo . Adap he p oo s o [32, Theo ems 2.1.4, 2.1.6, 2.2.6, 2.2.7, 2.2.9
and 2.2.10, and Co olla y 2.1.6]. 
Like in he case o mani olds, i easily ollows om P oposi ion 2.6-(i )
ha , o 0 ≤ <s≤∞, i he e is a C lea wise homo opy be ween Cs
olia ed maps, hen he e is a Cslea wise homo opy be ween hem.
The abo e openness s a emen s a e s onge wi h he s ong olia ed
C opology, whe eas he denseness s a emen s a e s onge o he s ong
plaquewise C opology. The e is no e sion o P oposi ion 2.6-(i) wi h he
s ong olia ed C opology ( o ins ance, conside he case o compac spaces
olia ed by poin s). Howe e we can p o e a weake o m o ha s a emen
by using ce ain subspaces C
SF(X, F;X′,F′)de ined as ollows. A olia ed
map φ∶X→X′is called a ans e se embedding ( espec i ely, ans e se
equi alence) i he induced mo phism Φ ∶H→H′is gene a ed by embed-
dings ( espec i ely, Φ is an isomo phism). Obse e ha F′(φ(X))=X′i
φis a ans e se equi alence. A subse M⊂C(X, F;X′,F′)o ans e se
embeddings ( espec i ely, ans e se equi alences) is called uni o m i he e
a e some olia ed a lases, Uo Xand U′o X′like in Sec ion 2.3, such

MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 17
ha , o all φ∈M, he maps φ2
ai in (3) a e embeddings ( espec i ely, open
embeddings). No e ha , i hese p ope ies hold wi h Uand U′, hen hey
hold wi h all ine a lases. Fo example, Emb(X, F;X′,F′)consis s o uni-
o m ans e se embeddings, and Homeo(X, F;X′,F′)consis s o uni o m
ans e se equi alences.
P oposi ion 2.7. Fo 1≤ ≤∞, le M⊂C
SF(X, F;X′,F′)be a uni o m
subspace o ans e se embeddings. Then Emb (X, F;X′,F′)∩Mis open
in M.
P oo . I is enough o p o e he case =1. Fo any φ∈Emb1(X, F;X′,F′)∩
M, conside a basic open se N1∶=N1
F(φ, U,U′,K,E)in C1
SF(X, F;X′,F′)
as abo e. We can assume ha K(and he e o e U) co e s X, and U′co e s
X′. A e e inemen s, we can choose U,U′and Ksuch ha he maps ψ2
ai
a e embeddings o all ψ∈M, and he in e io s Vi∶=˚
Kico e X. Take
an open co e {Wi}o Xwi h Wi⊂Vi o all i. By [32, Lemma 1.3], we
can choose Esuch ha he maps ψ∶p−1
i(u)∩Vi→p′−1
ai(ψ2
aii(u)) a e C1
embeddings o u∈pi(Vi)and ψ∈N1. Hence ψ∶Vi→X′is a C1 olia ed
embedding o all ψ∈N1∩M.
Now, we adap he inal pa o he p oo o [32, Theo em 1.4] as ollows.
Since φis an embedding, we ge disjoin open subse s V′
i, W ′
i⊂X′ o e e y
isuch ha φ(Wi)⊂W′
iand φ(X∖Vi)⊂V′
i. Then i is easy o ind a
neighbo hood N0o φin CSF(X, F;X′,F′)so ha ψ(Wi)⊂W′
iand ψ(X∖
Vi))⊂V′
i o all ψ∈N0. We inally ob ain N0∩N1∩M⊂Emb1(X, F;X′,F′).

P oposi ion 2.8. Fo 1≤ ≤∞, le M⊂C
SF(X, F;X′,F′)be a uni o m
subspace o ans e se equi alences. Then Di eo (X, F;X′,F′)∩Mis open
in M.
P oo . We adap he p oo s o [32, Co olla y 1.6 and Theo em 1.6]. The se
M′={φ∈P op (X, F;X′,F′)∣Txφis su jec i e ∀x∈X}
is closed in C
SF(X, F;X′,F′)by P oposi ion 2.5-(i),(ii). On he o he hand,
Emb (X, F;X′,F′)∩Mis open in Mby P oposi ion 2.7. Thus he esul
ollows because Emb (X, F;X′,F′)∩M′=Di eo (X, F;X′,F′).
Acco ding o P oposi ion 2.6-( i),( ii), we will only conside ei he (C0)
olia ed spaces o C∞ olia ed spaces om now on.
P oposi ion 2.9. Le φ∶X→X′be a olia ed map. Suppose ha X′is
equipped wi h a C∞s uc u e. Then he e is a mos one C∞s uc u e on
Xsuch ha φis C∞and Txφis an isomo phism o all x∈X.
P oo . Conside wo C∞s uc u es on X, and ake C∞ olia ed cha s, ξ1∶
U1→B1×T1o he i s C∞s uc u e on X,ξ2∶U2→B2×T2o he second
C∞s uc u e on X, and ξ′∶U′→B′×T′o he C∞s uc u e on X′. We
18 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
can assume ha U2⊂U1and φ(U1)⊂U′. Then
ξ′φξ−1
1( 1, u1)=(g′
1( 1, u1), h′
1(u1)),
ξ′φξ−1
2( 2, u2)=(g′
2( 2, u2), h′
2(u2)),
ξ1ξ−1
2( 2, u2)=(g12( 2, u2), h12(u2)),
o ( k, uk)∈Bk×Tk,k=1,2, whe e g′
k∶Bk×Tk→B′has pa ial de i a i es
o a bi a y o de wi h espec o k, con inuous on Bk×Tk, and g12 ∶B2×
T2→B1is con inuous. Mo eo e he di e en ial map o g′
1wi h espec o 1
is an isomo phism a any poin . The e o e, by he in e se unc ion heo em,
we can assume ha g′
1(⋅, u1)∶B1→g′
1(B1×{u1})is a C∞di eomo phism o
all u1∈T1. I s in e se unc ion is deno ed by ¯g′
1(⋅, u1)∶g′
1(B1×{u1})→B1.
Fo any small ball B′
0⊂B′, le T10 ⊂T1be he open subse ha consis s o
he poin s u1∈T1such ha B′
0⊂g′
1(B1×{u1}). I also ollows om he
in e se unc ion heo em ha he pa ial de i a i es o a bi a y o de o
¯g′
1(⋅, u1)∶B′
0→B1depend con inuously on u1. Since
g12( 2, u2)=¯g1(g2( 2, u2), h12(u2))
on B2×h21(T10), he unc ion g12 ∶B2×h21(T10)→B1has pa ial de i a i es
o a bi a y o de wi h espec o 2, con inuous on B2×h21(T10).
2.5. Cen e o mass. In Sec ion 6.2, we will use he cen e o mass o a
mass dis ibu ion on a Riemannian mani old M[35], [14, Sec ion IX.7].
Le Ω ⊂Mbe a compac submani old wi h bounda y wi h dim Ω =dim M.
Fo 0 ≤ ≤∞, le C(Ω)be he se o unc ions ∈C +2(Ω)such ha g ad is
an ou wa d poin ing ec o ield on ∂Ω and Hess is posi i e de ini e on he
in e io ˚
Ω o Ω. No e ha C(Ω)is open in he Banach space C +2(Ω)wi h
he no m ∥∥C +2,Ω,g, and he e o e i is a C∞Banach mani old. Mo eo e
C(Ω)is p ese ed by he ope a ions o sum and p oduc by posi i e numbe s.
Any ∈C(Ω)a ains i s minimum alue a a unique poin mΩ( )∈˚
Ω,
de ining a unc ion mΩ∶C(Ω)→˚
Ω.
Lemma 2.10 ([3, Lemma 10.1 and Rema k 11-(ii)]).The map mΩis C .
Suppose ha Mis connec ed and comple e. Le (A, µ)be a p obabili y
space, Ba con ex open ball o adius >0 in M, and ∶A→Ba measu able
map, which is called a mass dis ibu ion on B. Conside he C∞ unc ion
P ,µ ∶B→Rde ined by
P ,µ(x)=1
2∫Ad(x, (a))2µ(a).
P oposi ion 2.11 (H. Ka che [35, Theo em 1.2]).We ha e he ollowing:
(i) g adP ,µ is an ou wa d poin ing ec o ield on he bounda y ∂B.
(ii) I δ>0is an uppe bound o he sec ional cu a u es o Min B,
and 2 <π/2√δ, hen HessP ,µ is posi i e de ini e on B.
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 19
I he hypo heses o P oposi ion 2.11 a e sa is ied, hen P ,µ ∈C(B), and
he e o e C ,µ ∶=mB(P ,µ)∈Bis de ined and called he cen e o mass o
(wi h espec o µ). This poin is independen o he choice o Bsa is ying
he abo e condi ions. The ollowing is a consequence o Lemma 2.10.
Co olla y 2.12 ([3, Co olla y 10.3]; c . [35, Co olla y 1.6]).The ollowing
p ope ies hold:
(i) C ,µ depends con inuously on and he me ic enso o M.
(ii) I Ais he Bo el σ-algeb a o a me ic space, hen C ,µ depends
con inuously on µin he weak-∗ opology.
Conside he ollowing pa icula case. Le Nbe a C∞mani old, φ=
(φ1,...,φk)∶N→MkaC∞map, and λ=(λ1,...,λk)a ini e C∞pa i ion
o uni y o N. Fo e e y x∈N, conside he p obabili y measu e µφ,λ,x =
∑k
i=1λi(x)δφi(x), whe e δydeno es he Di ac mass a e e y y∈M. Suppose
ha , o all x∈N, he poin s φ1(x),...,φk(x)lie in a ball Bxo Msa is ying
he condi ions o P oposi ion 2.11. Then we can de ine cen e o mass Cφ,λ,x
o idBxwi h espec o µφ,λ,x, which is independen o he choice o Bx. The
ollowing sha pening o Co olla y 2.12 also ollows om Lemma 2.10.
Co olla y 2.13. The map N→M,x↦Cφ,λ,x, is C∞.
3. Molino’s desc ip ion
Conside he no a ion o Sec ion 2.3 in he es o he pape .
P oo o Theo em A. Mos o he p ope ies s a ed in his heo em we e
al eady p o ed in [10, Theo em A]. I only emains o p o e he pa con-
ce ning H. Fo his pu pose, we ha e o ecall he cons uc ion o G,
X0,

F0and ˆπ0. We can assume ha Xsa is ies he condi ions o equicon inui y
and s ong quasi-analy ici y wi h he same se S, and ha Hsa is ies he
condi ions o equicon inui y and s ong quasi-analy ici y wi h he induced
se S. Le Sc-o be he space Swi h he es ic ion o he compac -open
opology on he se o pa ial maps T↣Twi h open domain [1]. Conside
he subspace
Sc-o ∗T={(g, u)∈S×T∣u∈domg}⊂Sc-o ×T ,
and equip he se 
To all ge ms o maps in S(o H) wi h he inal opology
induced by he ge m map γ∶Sc-o ∗T→
T( his is no he es ic ion o
he shea opology). Conside he es ic ions s, ∶
T→To he sou se and
a ge maps. The space 
Tis locally compac and Polish, and ˆπ∶=(s, )∶

T→T×Tis con inuous and p ope .
Fix some poin u0∈Ti0⊂T. Then he subspace 
T0∶=s−1(u0)⊂
Tis
locally compac and Polish. This de ini ion is di e en om he one gi en
in [10, Sec ion 3D], whe e 
T0= −1(u0)was conside ed. This change can be
made because he in e sion o local ans o ma ions de ines a homeomo -
phism o Sc-o [10, P oposi ion 3.1], and he e o e he ge m in e sion de ines
20 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
a homeomo phism o 
T, which becomes a opological g oupoid by [1, P opo-
si ion 10]. The es o de ini ions and a gumen s o [10, Sec ions 3D–3G]
mus be changed acco dingly. Fo ins ance, ake ˆπ0= ∶
T0→T(ins ead o
ˆπ0=s, used in [10]), which is open, con inuous and p ope , and i s ibe s a e
homeomo phic o each o he [10, Sec ion 3D]. We ha e 
T0≡⊔i
Ti,0, whe e

Ti,0=ˆπ−1
0(Ti).
No e ha H∶=ˆπ−1
0(u0)=ˆπ−1(u0, u0)becomes a compac Polish g oup
since 
Tis a opological g oupoid. Mo eo e he ge m p oduc de ines a
con inuous ee igh ac ion o Hon 
T0whose o bi s a e clea ly equal o
he ibe s o ˆπ0∶
T0→T. Thus his map induces a con inuous bijec ion

T0/H→T. In ac his bijec ion is a homeomo phism, as easily ollows by
using also ha His compac , 
T0is locally compac , and Tis Hausdo .
Fo any h∈H, de ine ˆ
h∶ˆπ−1
0(domh)→ˆπ−1
0(imh)by ˆ
h(γ(g, u0)) =
γ(hg, u0) o g∈Swi h u0∈domgand g(u0)∈domh(ins ead o ˆ
h(γ(g, u))=
γ(gh−1, h(u)) o u∈dom g∩dom hwi h g(u)=u0, used in [10]). The maps
ˆ
ha e local ans o ma ions o 
T0sa is ying hˆπ0=ˆπ0ˆ
h,
idT=id
T0,
hh′=ˆ
h
h′
and ˆ
h−1=
h−1[10, Sec ions 3E]. Mo eo e i is easy o see ha e e y ˆ
his
H-equi a ian (no e ha domˆ
hand im ˆ
ha e H-in a ian ). Le 
H0be he
pseudog oup on 
T0gene a ed by 
S0={ˆ
h∣h∈S}. The e is a local g oup
Gand some dense ini ely gene a ed sublocal g oup Γ ⊂Gsuch ha 
H0is
equi alen o he pseudog oup gene a ed by he local ac ion o Γ on Gby lo-
cal le ansla ions [10, P oposi ion 3.41]— his was p o ed by checking ha

H0is compac ly gene a ed, equicon inuous and s ongly locally ee, and i s
closu e is also s ongly locally ee, and hen applying P oposi ion 2.2-(i).
Fu he mo e ˆπ0gene a es a mo phism 
H0→H.
Le ˇ
Ui,0=Ui×
Ti,0×{i}≡Ui×
Ti,0, equipped wi h he p oduc opology,
and conside he opological sum
ˇ
X0∶=⊔
i(Ui×
Ti,0)=⋃
i
ˇ
Ui,0,
and he closed subspaces

Ui,0∶={(x, γ, i)∈ˇ
Ui,0∣pi(x)=ˆπ0(γ)}⊂ˇ
Ui,0,
X0∶=⋃
i
Ui,0⊂ˇ
X0.
No e ha 
X0is he opological sum o he spaces 
Ui,0. Conside he equi a-
lence ela ion “∼” on 
X0de ined by (x, γ, i)∼(y, δ, j)i x=yand γ=
hji(δ).
Le 
X0be he co esponding quo ien space, le q∶
X0→
X0be he quo-
ien map, le [x, γ, i]=q(x, γ, i), le 
Ui,0=q(
Ui,0), and le ˜pi,0∶
Ui,0→
Ti,0
deno e he es ic ion o ˇpi,0∶ˇ
Ui,0≡Ui×
Ti,0→
Ti,0, which induces a map
ˆpi,0∶
Ui,0→
Ti,0. Mo eo e a map ˆπ0∶
X0→Xis de ined by ˆπ0([x, γ, i])=x.
Obse e ha 
Ui,0=ˆπ−1
0(Ui). Then 
X0is compac and Polish, {
Ui,0,ˆpi,0,
hij}
is a de ining cocycle o a minimal olia ed s uc u e 
F0on 
X0, ˆπ0is con in-
uous and open, he ibe s o ˆπ0a e homeomo phic o each o he , and he
es ic ion o ˆπ0 o he lea es o 
X0a e he holonomy co e ings o he lea es
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 21
o X[10, Sec ion 4B]. In he p oo o hese p ope ies, i was used ha e e y
es ic ion q∶
Ui,0→
Ui,0is a homeomo phism.
Since e e y 
Ti,0is H-in a ian , we ge an induced ee igh ac ion o H
on e e y ˇ
Ui,0≡Ui×
Ti,0, ac ing as he iden i y on he ac o Ui, yielding a
igh H-ac ion on ˇ
X0by union. This es ic s o a ee igh ac ion o H
on 
X0, p ese ing e e y 
Ui,0, because he H-o bi s in 
T0a e equal o he
ibe s ˆπ0∶
T0→T. Since mo eo e e e y 
hij is H-equi a ian , we ge an
induced igh ac ion on 
X0, gi en by [x, γ, i]⋅σ=[x, γσ, i] o [x, γ, i]∈
X0
and σ∈H. This ac ion is also ee because e e y es ic ion q∶
Ui,0→
Ui,0
is a homeomo phism, and i is easy o see ha i s o bi s equal he ibe s o
ˆπ0∶
X0→X. Finally no e ha e e y map ˆpi,0∶
Ui,0→
Ti,0is H-equi a ian ,
and he e o e Hac s on 
X0by olia ed ans o ma ions. 
In he es o his sec ion, assume ha Xsa is ies he hypo heses o
Theo em A. Conside s uc u es (G, H, 
X0,ˆπ0)sa is ying he condi ions o
i s s a emen , whe e 
X0is conside ed as a olia ed space and H-space. I
desi ed, we may also add a ini ely gene a ed dense sublocal g oup Γ ⊂
G o he no a ion, (G, Γ, H, 
X0,ˆπ0), so ha he holonomy pseudog oup o

X0is ep esen ed by he pseudog oup gene a ed by he le local ac ion
o Γ on Gby local le ansla ions. I is said ha wo such s uc u es,
(G, Γ, H, 
X0,ˆπ0)and (G′,Γ′, H′,
X′
0,ˆπ′
0), a e equi alen i he e a e a local
isomo phism ψ∶G↣G′ ha es ic s o a local isomo phism Γ ↣Γ′,
an isomo phism χ∶H→H′, and a olia ed χ-equi a ian homeomo phism
φ∶
X0→
X′
0such ha ˆπ0=ˆπ′
0φ( he condi ion on Γ and Γ′is omi ed i Γ and
Γ′a e no conside ed). In his case, (ψ, χ, φ)is called an equi alence. This
no ion o equi alence is na u al because i clea ly means ha he desc ip ions
o he olia ed space Xgi en by (G, Γ, H, 
X0,ˆπ0)and (G′,Γ′, H′,
X′
0,ˆπ′
0)a e
essen ially he same, gi ing ise o equi alen in a ian s o X. Fo ins ance,
G, Γ and Hha e he same algeb aic and opological p ope ies as G′, Γ′
and H′, and ˆπ0is a p incipal bundle p ojec ion i and only is ˆπ′
0is also a
p incipal bundle p ojec ion.
P oposi ion 3.1 (C . [10, P oposi ions 3.43, 4.12 and 4.13]).All s uc u es
(G, Γ, H, 
X0,ˆπ0)cons uc ed in he p oo o Theo em A a e equi alen .
P oo . We ha e o p o e ha he equi alence class o (G, Γ, H, 
X0,ˆπ0)is
independen o he choices o u0,Sand {Ui, pi, hij}. Mos o his is al eady
p o ed in [10, P oposi ions 3.43, 4.12 and 4.13]. We only ha e o check wha
conce ns H.
To begin wi h, ake ano he poin o u1∈Ti1⊂T, and le 
T1, ˆπ1,
S1,

H1,G1, Γ1and H1be cons uc ed like 
T0, ˆπ0,
S0,
H0,G0∶=G, Γ0∶=Γ
and H0∶=Hby using u1ins ead o u0. Now, o each h∈H, le us use
he no a ion ˆ
h0∶=ˆ
h∈
H0, and le ˆ
h1∶ˆπ−1
1(domh)→ˆπ−1
1(imh)be he
map in 
H1de ined like ˆ
h. In pa icula , he maps (
hij)1a e de ined like
he maps (
hij)0∶=
hij. The e is some 0∈Ssuch ha u0∈dom 0and

22 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
0(u0)=u1. Le θ∶
T0→
T1be de ined by θ(γ( , u0))=γ( −1
0, u1)(ins ead
o θ(γ( , x))=γ( 0 , x), like in [10]). This map is a homeomo phism, and
sa is ies ˆπ0=ˆπ1θ, dom ˆ
h1=θ(dom ˆ
h0)and ˆ
h1θ=θˆ
h0 o all h∈S, ob aining
ha θgene a es an equi alence Θ ∶
H0→
H1[10, P oposi ion 3.42]. Fo k=
0,1, le Gkbe he pseudog oup on Gkgene a ed by local le ansla ions by
elemen s o Γk. Via equi alences 
Hk→Gk, Θ co esponds o an equi alence
Θ′∶G0→G1. Since he local igh ansla ions o G1gene a e equi alences
o G1, we can assume ha he o bi s o he iden i y elemen s co espond by
he induced map G0/G0→G1/G1. By P oposi ion 2.2-(ii), i ollows ha
Θ′is gene a ed by a local isomo phism ψ∶G0↣G1 ha es ic s o a
local isomo phism Γ ↣Γ′. On he o he hand, he conjuga ion mapping,
γ( , u0)↦γ( 0 −1
0, u1), de ines an isomo phism χ∶H0→H1so ha θis
χ-equi a ian .
Now, de ine 
X1≡(
X1,
F1),[x, γ, i]1and ˆπ1∶
X1→Xlike 
X0≡(
X0,
F0),
[x, γ, i]0∶=[x, γ, i]and ˆπ0∶
X0→X, using 
T1, ˆπ1∶
T1→Tand he maps
(
hij)1ins ead o 
T0, ˆπ0∶
T0→Tand he maps (
hij)0. Acco ding o [10,
P oposi ion 4.12], a olia ed homeomo phism φ∶
X0→
X1is de ined by
φ([x, γ, i]0)=[x, θ(γ), i]1, which sa is ies ˆπ0=ˆπ1φand induces he equi a-
lence Θ ∶
H0→
H1. Mo eo e φis χ-equi a ian : o all [x, γ, i]0∈
X0and
σ∈H0,
φ([x, γ, i]0⋅σ)=φ([x, γσ, i]0)=[x, θ(γσ), i]1
=[x, θ(γ)χ(σ), i]1=[x, θ(γ), i]1⋅χ(σ).
All choices o Sde ine he same space 
T0by [10, P oposi ions 3.43], gi ing
ise o he same Molino’s desc ip ion.
To p o e he independence o {Ui, pi, hij}, i is enough o conside he
case whe e {Ui, pi, hij} e ines ano he de ining cocycle {U′
a, p′
a, h′
ab}. Le
H′be he co esponding ep esen a i e o he holonomy pseudog oup on
T′=⊔aT′
a. I Ui⊂U′
ai, he e is an induced open embedding φi∶Ti→T′
ai.
These maps gene a e an equi alence Φ ∶H→H′. In ac , h′
aiajφj=φihij.
Le u′
0=φi0(u0)∈T′
ai0⊂T′, and le S′⊂H′be a gene a ing subse such
ha S′2⊂S′=S′−1. We can also use {U′
a, p′
a, h′
ab},u′
0and S′ o de ine

T′
0, ˆπ′
0∶
T′
0→T′and 
H′
0like 
T0, ˆπ0∶
T0→Tand 
H0; in pa icula , he
gene a o s 
h′
ab o 
H′
0a e de ined like he gene a o s 
hij o 
H0. We ge
open embeddings ˆ
φi,0∶
Ti,0→
T′
ai,0de ined by ˆ
φi,0(γ(g, u0))=γ(φigφ−1
i0, u′
0),
which gene a e an equi alence 
Φ0∶
H0→
H′
0( his is a co ec ed e sion o
[10, P oposi ion 3.44]). Le (G′,Γ′, H′,
X′
0,ˆπ′
0)be he Molino’s desc ip ion
de ined wi h 
T′
0, ˆπ′
0∶
T′
0→T′and he maps 
h′
ab. Le us use he no a ion
[x, γ′, a]′ o he elemen o 
X′
0 ep esen ed by a e n (x, γ′, a). Le Gand
G′be he pseudog oups on Gand G′gene a ed by he local le ansla ions
by elemen s o Γ and Γ′. Via equi alences H→Gand H′→G′,
Φ0co e-
sponds o an equi alence 
Φ′
0∶G→G′. As abo e, we can assume ha he
o bi s o he iden i y elemen s co espond by he induced map G/G→G′/G′,
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 23
and he e o e, acco ding o P oposi ion 2.2-(ii), 
Φ′
0is gene a ed by a local
isomo phism ψ∶G↣G′ ha es ic s o a local isomo phism Γ ↣Γ′. Mo e-
o e ˆ
φi0,0 es ic s o an isomo phism χ∶H→H′so ha any map in 
Φ0is
χ-equi a ian . Finally, a canonical olia ed homeomo phism φ∶
X0→
X′
0is
well de ined by φ([x, γ, i])=[x, ˆ
φi,0(γ), ai]′[10, P oposi ion 4.13]. I is easy
o check ha φis H-equi a ian . 
By P oposi ion 3.1, he equi alence class o any s uc u e (G, Γ, H, 
X0,ˆπ0)
cons uc ed in he p oo o Theo em A can be called he Molino’s desc ip ion
o X. Acco ding o he discussion o [10, Sec ion 1.E], hese s uc u es a e
kind o a opological in e p e a ion o he o iginal Molino’s desc ip ion in
he case o a Riemannian olia ion. Tha simila i y can be indeed ealized
as an equi alence be ween he o iginal Molino’s desc ip ion and ou s in ha
case. Acco ding o Molino’s e minology, he local isomo phism class o Gis
called he s uc u al local g oup [10], and, wi h he e minology o [18, 19],

X0will be called he Molino space and H he disc iminan g oup.
P oposi ion 3.2. Xis a G- olia ed space o some local g oup Gi and
only i i s disc iminan g oup is i ial.
P oo . The “i ” pa o he s a emen is di ec ly gi en by Theo em A. To
p o e he “only i ” pa , assume Xis a G- olia ed space o some local g oup
G. Thus His s ongly locally ee, ob aining ha H={e}acco ding o he
de ini ion o Hgi en in he p oo o Theo em A. 
Fo e e y ˆx∈
X0, le 
Lˆxdeno e he lea o 
X0 h ough ˆx, and conside
he iden i y 
Lhol
x≡
Lˆxgi en by Theo em A.
Lemma 3.3. Fo x∈Xand ˆx≡[x, γ, i]∈ˆπ−1
0(x), le c∶I→Xbe a
lea wise pa h om x o some poin y, and le ˆcbe he unique li o c o

Lhol
x≡
Lˆxbeginning a ˆx. Then ˆc(1)≡[y, δγ, jβ], whe e δ=γ(hJ, pi(x)) o
any J=(j0,...,jβ)co e ing cwi h j0=i.
P oo . Take a pa i ion 0 = 0< 1<⋅⋅⋅< β+1=1 o Isuch ha c([ k, k+1])⊂
Ujk o k=0,...,β. Fo s∈I, he pa h cs( )∶=c(s )in Lis co e ed
by Js∶=(j0,...,jβs), whe e βs=min{k∈{0,...,β} ∣ k+1≥s}, and le
δs=γ(hJs, pi(x)). Then i is easy o see ha ˆc(s)=[c(s), δsγ, jβs].
Fix some poin x0∈p−1
i0(u0)⊂Ui0⊂X.
P oposi ion 3.4. Fo ˆx0∈ˆπ−1
0(x0), we ha e
Hol(Lx0, x0)={γ∈H∣
Lˆx0⋅γ=
Lˆx0},(4)
and he map 
Lhol
x0≡
Lˆx0↪
X0becomes equi a ian wi h espec o he homo-
mo phism Hol(Lx0, x0)↪H.
24 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
P oo . Conside he no a ion o he p oo o Theo em A. Obse e ha
Hol(Lx0, x0)is a subg oup o H:
Hol(Lx0, x0)={γ(h, u0)∣h∈H, u0∈domh, h(u0)=u0}
⊂H={γ(g, u0)∣g∈H, u0∈domg, g(u0)=u0}.
I γ=hol([c])∈Hol(Lx0, x0) o some [c]∈π1(Lx0, x0), hen γ=γ(h−1
I, u0)
o some I=(i0, i1,...,iα)co e ing cwi h iα=i0. Fo any y∈Lx0and
ˆy∈
Lˆx0∩ˆπ−1
0(y), we ha e ˆy≡[y, δ, i], whe e δ=γ(hJ, u0) o some admissible
sequence J=(j0,...,jβ), wi h j0=i0and jβ=i, which co e s a lea wise
pa h cy∶I→X om x0 o y. Then ˆy⋅γis he inal poin o he li o

Lhol
x≡
Lˆx0, beginning a ˆy, o he loop c−1
yccyin Lx0, based a y. Thus
ˆy⋅γ≡[y, δγδ−1δ, i0]=[y, δγ, i0]=[y, δ, i0]⋅γ ,
whe e he iden i y be ween hese elemen s o 
Lhol
xand 
Lˆx0is gi en by
Lemma 3.3, applied o ˆyand c−1
yccy, because J−1IJ is de ined and co -
e s c−1
yccy, and hJ−1IJ =hJhIh−1
J. This p o es he inclusion “⊂” in (4) and
he equi a iance o 
Lhol
x0≡
Lˆx0↪
X0. On he o he hand, since he igh
H-ac ion on 
X0is ee, olia ed and p ese es e e y ˆπ0- ibe , any elemen o
he igh hand side o (4) de ines a co e ing ans o ma ion o he es ic ion
ˆπ0∶
Lˆx0→Lx0, showing he inclusion “⊃” in (4). 
Acco ding o he p oo o P oposi ion 3.1, i ollows om P oposi ion 3.4
ha , o all x∈Xand ˆx∈ˆπ−1
0(x), he e is an isomo phism
Hol(Lx, x)≅{γ∈H∣
Lˆx⋅γ=
Lˆx}
so ha he map 
Lhol
x≡
Lˆx↪
X0becomes equi a ian wi h espec o he
induced injec i e homomo phism Hol(Lx, x)→H. Ne e heless his isomo -
phism is no canonical in gene al.
4. Folia ed homogeneous olia ed spaces
The olia ed space Xis called olia ed homogeneous when he canoni-
cal le ac ion o Homeo(X, F)on Xis ansi i e. Simila ly, i Xis C∞,
i is called C∞ olia ed homogeneous when he canonical le ac ion o
Di eo(X, F)on Xis ansi i e. A p io y, C∞ olia ed homogenei y is
s onge han olia ed homogenei y, bu we will see ha indeed hey a e
equi alen condi ions o compac minimal C∞ olia ed spaces (Sec ion 7).
Take any comple e me ic dinducing he opology o X, and le Dbe he
induced comple e me ic on Homeo(X)de ined by
D(φ, ψ)=sup
x∈X
d(φ(x), ψ(x))+sup
x∈X
d(φ−1(x), ψ−1(x)).
In his way, Homeo(X)becomes a comple ely me izable opological g oup,
and i s canonical le ac ion on Xis con inuous. Mo eo e i is easy o check
ha Homeo(X, F)is closed in Homeo(X), and he e o e Homeo(X, F)is
also a comple ely me izable opological g oup.
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 25
Suppose ha Xis compac . Then Dinduces he compac -open opology
on Homeo(X), as ollows om [11, Theo em 3], ob aining ha Homeo(X)
is also second coun able. So Homeo(X)is a Polish g oup, and Homeo(X, F)
a Polish subg oup. The e o e, by a heo em o E os [20, 43], i Xis o-
lia ed homogeneous, hen he canonical le ac ion o Homeo(X, F)on X
is mic o- ansi i e; i.e., o all x∈Xand any neighbou hood No idXin
Homeo(X, F), he se N⋅xis a neighbo hood o xin X.
P oo o Theo em B. Cla k and Hu de ha e p o ed ha any C∞homo-
geneous ma chbox mani old is equicon inuous [15, Theo em 5.2]. Indeed,
hei a gumen applies o any compac minimal olia ed homogeneous oli-
a ed space. Mo eo e he C∞s uc u e is no used in ha esul . Thus he
condi ions o ou s a emen a e enough o ge ha (X, F)is equicon inuous.
The es o he p oo uses he same main ool as in [15, Theo em 5.2], he
indica ed heo em o E os.
Le us p o e ha His s ongly locally ee. Since {Ui}is ini e, he e
is some >0 such ha d(Ui, X ∖
Ui)< o all i. Since he ac ion o
Homeo(X, F)on Xis mic o- ansi i e, he e is some δ>0 such ha , o
all x, y ∈Xwi h d(x, y)<δ, he e exis s some φ∈Homeo(X, F)so ha
D(φ, idX)<and φ(x)=y.
Since e e y Tihas compac closu e in 
Ti, we easily ge a ini e open co e
{Tia}o Tisuch ha he d-diame e o e e y σi(Tia)is smalle han δ. Le
Uia =ξ−1
i(Bi×Tia),ξia =ξi∣Uia ,
Uia =
Uiand ˜
ξia =˜
ξi. By using {Uia, ξia}and
{
Uia,˜
ξia}, a ying iand a, ins ead o {Ui, ξi}and {
Ui,˜
ξi}, i ollows ha we
can assume ha he d-diame e o e e y σi(Ti)is smalle han δ.
Take Sequal o he amily o he maps hI o admissible sequences I.
Suppose ha some hI∈S ixes a poin u∈dom hI. Thus I=(i0,...,iα)
wi h iα=i0. Le x=σi0(u)∈Ui0and le c∶I→Xbe a lea wise loop
in Lxbased a xand U-co e ed by I. Take any poin ∈dom hI, and le
y=σi0( )∈Ui0. Since he d-diame e o σi0(Ti0)is smalle han δ, acco ding
o ou applica ion o he E os heo em, he e is some φ∈Homeo(X, F)wi h
φ(x)=yand d(c( ), φc( ))< o all ∈I. Hence he lea wise pa h φc ∶I→
Xis 
U-co e ed by I. I ollows ha ˜
hI( )=pi0φc(1)=pi0φ(x)=pi0(y)= ,
ob aining hI( )= . This shows ha hI=iddom hI, and he e o e Hsa is ies
he condi ion o being s ongly locally ee wi h his S.
His s ongly quasi-analy ic because i is s ongly locally ee, and he e-
o e he hypo heses o Theo em A a e sa is ied. In pa icula , he closu e H
is de ined and gene a ed by he se Sinduced by he abo e S.
Now, le us sha pen he abo e a gumen o p o e ha His also s ongly
locally ee, and he e o e (X, F)is a G- olia ed space o some local g oup
Gby P oposi ion 2.2-(i). Fo any g∈Swi h O=dom g, he e is a sequence
o admissible sequences, Ik=(ik,0,...,ik,αk), such ha O⊂dom hIk o
all kand g=limkhIk∣Oin he compac -open opology. Thus i0∶=ik,0is
independen o k. Suppose ha g(u)=u o some u∈O, which means ha
u′
k∶=hIk(u)→uas k→∞. So we can assume ha ik,αk=i0 o all k. Le
32 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
7. C∞G- olia ed spaces a e C∞ olia ed homogeneous
Suppose ha Xis compac and C∞. Then he ollowing esul gua an ees
ce ain lea wise homogenei y.
P oposi ion 7.1. Le Lbe he lea o X, le Dbe a ela i ely compac
egula domain wi hou holonomy in L, and le c∶I→Dbe any C∞pa h.
Then, o any open neighbo hood Uo c(I)in X, he e is some C∞lea wise
di eo opy φ∶X×I→Xsuppo ed in Uwi h φ(c(0),⋅)=c.
P oo . Le Ebe a ela i ely compac open subse o Lsuch ha c(I)⊂E
and E⊂D∩U. By he homogenei y o L, he e is a di eo opy ψ∶L×I→L
suppo ed in Eso ha ψ(⋅,0)=idXand ψ(c(0),⋅)=c. Le Σ be a local
ans e sal o X h ough x. By he Reeb’s s abili y heo em o C∞ olia ed
spaces [4, P oposi ion 1.7], he e is a C∞ olia ed embedding h∶D×Σ→X
ha can be iden i ied wi h he iden i y on D×{x}≡Dand {x}×Σ≡Σ.
W i e h−1=(h′, h′′)∶imh→D×Σ. Take a compac ly suppo ed con inuous
unc ion ∶Σ→Iwi h h(E×supp )⊂Uand (x)=1. Then he s a emen
is sa is ied wi h he C∞ olia ed di eo opy φ∶X×I→Xde ined by
φ(x, )=⎧
⎪
⎪
⎨
⎪
⎪
⎩
h(ψ(h′(x), h′′(x)), h′′(x)) i x∈imh
xo he wise .
Co olla y 7.2. I he e is a C∞ igh ans e se local ac ion o Gon X
sa is ying (5), hen Xis C∞ olia ed homogeneous.
P oo . Apply (5) and P oposi ion 7.1. 
P oo Theo em C. By Theo em B, i is enough o p o e “(iii) ⇒(i).” Wi h
he no a ion o Sec ion 6.3, (G, G, µ)sa is ies (1) because
µ((Γ×µ({g}×Q))∩dom µ)=G
o all g∈Gand Q∈N(G, e)wi h {g}×Q⊂domµ. So (T, H, χ)also sa is-
ies (1) by Lemma 2.4, and he e o e (X, F, φ)sa is ies (5) by Lemma 6.8.
Thus Xis C∞ olia ed homogeneous by Co olla y 7.2 
8. Examples and open p oblems
8.1. Molino’s desc ip ion o equicon inuous suspensions. Le Tbe a
compac space wi h a ansi i e le ac ion o a compac opological g oup
G, which is quasi-analy ic in he sense ha any g∈Gis he iden i y elemen
e∈Gi i ac s as he iden i y on some non-emp y open se , and le H⊂G
be he iso opy g oup a some ixed poin u0∈T. Mo eo e le Γ ⊂Gbe a
dense subg oup isomo phic o π1(M)/π1(L) o some egula co e ing Lo
some closed connec ed mani old M. Thus we ha e a igh Γ-ac ion on Lby
co e ing ans o ma ions, and a le Γ-ac ion on Tde ined by he G-ac ion.
The induced diagonal Γ-ac ion on L×T, gi en by (y, u)⋅γ=(y⋅γ, γ−1⋅u),
is p ope ly discon inuous and olia ed, whe e L×Tis olia ed wi h lea es
L×{u}, o u∈T. The co esponding olia ed quo ien space, L×ΓT, is

MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 33
called he suspension o he Γ-ac ion on T, and he quo ien p ojec ion is a
olia ed co e ing map L×T→L×ΓT. The elemen in L×ΓTde ined by any
(y, u)∈L×Twill be deno ed by [y, u]. Mo eo e he co e ing p ojec ion
θ∶L→Minduces a ibe bundle p ojec ion ρ∶L×ΓT→M,ρ([y, u])=θ(y),
wi h ypical ibe T; in pa icula , L×ΓTis compac . No e ha he ibe s
o ρa e ans e se o he lea es; i.e., ρ∶L×ΓT→Mis a la bundle. Any
la bundle wi h compac o al space is gi en by a suspension.
Le us use he no a ion X≡(X, F) o L×ΓT. Le V={Vi, ζi}be an a las
o M, wi h ζi∶Vi→Bi o some con ac ible open subse Bi⊂Rn. Thus he
la bundle ρ∶X→Mis i ial o e e e y Vi; i.e., he e a e homeomo phisms
ψi∶Ui∶=ρ−1(Vi)→Vi×Tsuch ha ρ∶Ui→Vico esponds o he i s
ac o p ojec ion Vi×T→Viand he lea es o F∣Uico espond o he ibe s
o he second ac o p ojec ion Vi×T→T. We ge an induced olia ed
a las U={Ui, ξi}o X, whe e ξi=(ζi×idT)ψi∶Ui→Bi×T′
iwi h T′
i≡T.
Assuming ob ious condi ions on V, we ge ha Uis egula . Then Uinduces
a ep esen a i e H′o he holonomy pseudog oup o Xon T′=⊔iT′
i. Fo
any ixed index i0, since T′
i0≡Tmee s all H′-o bi s, by es ic ing H′ o T′
i0,
we ge a pseudog oup Hon Tequi alen o H′, which is gene a ed by he Γ-
ac ion on T. Thus Xis minimal, equicon inuous and s ongly quasi-analy ic
( ake S=Γ o check he las wo p ope ies o H). Mo eo e His gene a ed
by he G-ac ion on T, and he e o e His also s ongly quasi-analy ic. So X
sa is ies he condi ions o Theo em A.
Fix some u0∈T≡T′
i0, and conside he associa ed space 
T′
0wi h he
pseudug oup 
H′
0, and he associa ed ep esen a i e o he Molino’s desc ip-
ion, (G′, H′,
X′
0≡(
X′
0,
F′
0),ˆπ′
0), cons uc ed like in he p oo o Theo em A.
Then 
T0∶=
T′
i0,0mee s all 
H′
0-o bi s, ob aining ha 
H′
0is equi alen o i s
es ic ion 
H0∶=
H′
0∣
T0. Thus 
T0={γ(g, u0)∣g∈G}has he inal opology
induced by he map G→
T0,g↦γ(g, u0). This map is a con inuous bijec-
ion, and he e o e i is a homeomo phism because Gis compac and 
T0is
Hausdo . So 
T0≡G,
His gene a ed by he ac ion o Gon i sel by le
ansla ions, G′is locally isomo phic o G, and ˆπ0∶
T0≡G→Tis he o bi
map g↦g⋅u0. The composi e ρˆπ′
0∶
X′
0→Mis a ibe bundle wi h ypical
ibe 
T0≡G, and (
X′
0, ρˆπ′
0,
F′
0)is also a la bundle. Thus he e is a olia ed
homeomo phism o 
X′
0 o 
X0≡(
X0,
F0)∶=L×ΓG. Mo eo e
H′≡H∶={h∈G∣h⋅u0=u0},
he igh H′-ac ion on 
X′
0co esponds o he igh H-ac ion on 
X0gi en
by [y, g]⋅h=[y, gh], and he map ˆπ′
0∶
X′
0→Xco esponds o he map
ˆπ0∶
X0→Xde ined by ˆπ0([y, g])=[y, g⋅u0], which is induced by he olia ed
map idL׈π0∶L×G→L×T. Thus (G, H, 
X0,ˆπ0)is ano he ep esen a i e
o he Molino’s desc ip ion, which will be used in he nex examples.
I Mis C∞, i s C∞s uc u e can be li ed o a C∞s uc u e on L, which
in u n can be li ed o L×T, which inally gi e ise o a C∞s uc u e
on Xso ha he p ojec ion ρ∶X→Mis C∞and Tρ has isomo phic
34 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
es ic ions o he ibe s. This can be simila ly applied o 
X0, ob aining he
C∞s uc u e gi en by P oposi ion 5.1. The same p ocedu e can be applied
o any Riemannian me ic on M, ob aining induced Riemannian me ics on
Xand 
X0so ha he p ojec ions ρ∶X→Mand ˆπ0∶
X0→Xha e locally
isome ic es ic ions o he lea es.
The ollowing esul is well known. A p oo is included o comple eness.
P oposi ion 8.1. The ollowing p ope ies a e equi alen :
(i) The Γ-ac ion on Thas no ixed poin s.
(ii) Γ∩gHg−1={e} o all g∈G.
(iii) The canonical olia ed p ojec ion L×T→X es ic s o homeomo -
phisms be ween he lea es.
P oo . Le us p o e “(i) ⇔(ii)”. Gi en any γ∈Γ and u∈T, ake some g∈G
such ha u=g⋅u0. Then
γu =u⇔γg ⋅u0=g⋅u0⇔g−1γg ⋅u0=u0
⇔g−1γg ∈H⇔γ∈Γ∩gHg−1={e}⇔γ=e .
Le us p o e “(i) ⇔(iii)”. Fo all y, y′∈Land u∈T, we ha e [y, u]=
[y′, u]i and only i he e is some γ∈Γ such ha (y′, u)=(y⋅γ, γ−1⋅u),
which means γ=eand y′=y.
When he condi ions o P oposi ion 8.1 a e sa is ied, Xis s ongly locally
ee (in pa icula , i has no holonomy), and all lea es a e homeomo phic
o L. I mo eo e Mis C∞/Riemannian, hen L×T→X es ic s o
di eomo phisms/isome ies be ween he lea es, ob aining ha all lea es a e
di eomo phic/isome ic o L.
8.2. The map ˆπ0∶
X0→Xmay no be a p incipal bundle. Conside
he canonical inclusion SO(2)⊂SO(3), and he canonical ansi i e analy ic
ac ion o SO(3)on he sphe e S2≡SO(3)/SO(2). We ge an induced
ansi i e quasi-analy ic le ac ion o he compac opological g oup G∶=
SO(3)Non he compac space T∶=(S2)N. Fix u0∈S2whose iso opy
g oup is SO(2), and le ¯u0=(u0, u0,...)∈T. The o bi map SO(3)→S2,
g↦g⋅u0, is a non- i ial p incipal SO(2)-bundle, and he e o e i has no
global sec ions. Then, using he a gumen s o he i s and second examples
o [40, Sec ion 1], i easily ollows ha he o bi map G→T,(gi)↦(gi)⋅¯u0=
(gi⋅u0), has no local sec ions. Since Gis second coun able, connec ed,
compac and non-abelian, i con ains a dense subg oup Γ isomo phic o
he undamen al g oup o he closed o ien ed su ace Σ2o genus 2 [12,
Co olla y 8.3]. Le Lbe he uni e sal co e ing o Σ2, which is di eomo phic
o he plane. Conside he co esponding suspension olia ed space, X=
L×ΓT, which sa is ies he condi ions o Theo em A, and he co esponding
Molino’s desc ip ion (G, H, 
X0,ˆπ0)cons uc ed in Sec ion 8.1, whe e 
X0=
L×ΓG,H=SO(2)N, he igh H-ac ion on 
X0is gi en by [y, g]⋅h=[y, gh],
and he map ˆπ0∶
X0→Xis de ined by ˆπ0([y, g])=[y, g ⋅u0].
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 35
P oposi ion 8.2. The map ˆπ0∶
X0→Xhas no local sec ions, and he e o e
i canno be a p incipal H-bundle.
P oo . Since ˆπ0∶
X0→Xis induced by idL׈π0∶L×G→L×T, any local
sec ion o ˆπ0wi h small enough domain de ines a local sec ion o ˆπ0∶G→T.
Bu his map has no local sec ions. 
8.3. Folia ed homogenei y may no be old by he lea es.
P oposi ion 8.3. I Xis olia ed homogeneous, hen i is wi hou holo-
nomy, and all o i s lea es a e homeomo phic one ano he . I mo eo e
Xis C∞( espec i ely, compac and Riemannian), hen all o i s lea es a e
di eomo phic ( espec i ely, quasi-isome ically di eomo phic) o each o he .
P oo . Elemen a y, using ha he e always exis lea es wi hou holonomy
in he i s asse ion, and using ha he di e en iable quasi-isome y class
o he lea es is independen o he choice o he Riemannian me ic on Xin
he las asse ion (see e.g. [6, P oposi ion 10.5]). 
Le us exhibi an example whe e he ecip ocal o P oposi ion 8.3 does no
hold. To begin wi h, le G1and G2be second coun able, connec ed compac
opological g oups, and le G=G1×G2. Assume ha G1is non-abelian.
Le us use he no a ion g=(g1, g2) o he elemen s o G; in pa icula , we
use e=(e1, e2) o he iden i y elemen .
P oposi ion 8.4. The e exis s a subse P⊂G×G, which is bo h esidual
and o ull Haa measu e, such ha , o all (g, h)∈P, he subg oup ⟨g, h⟩is
dense in Gand eely gene a ed by gand h, and ⟨g, h⟩∩({e1}×G2)={e}.
P oo . By [12, P oposi ion 8.2], he e a e subse s, O⊂G×Gand O1⊂
G1×G1, which a e esidual and o ull Haa measu e, such ha , o all
(g, h)∈Oand (a, b)∈O1, he subg oup ⟨g, h⟩( espec i ely, ⟨a, b⟩) is dense
in G( espec i ely, G1) and eely gene a ed by gand h( espec i ely, aand
b). Then he s a emen is sa is ied wi h
P=O∩{(g, h)∈G×G∣(g1, h1)∈O1}.
Take G2=SO(3), and conside SO(2)⊂SO(3)and S2≡SO(3)/SO(2)
like in Sec ion 8.2. By P oposi ion 8.4, Ghas a dense subg oup Γ eely
gene a ed by wo elemen s such ha Γ ∩({e1}×SO(3))={e}. Hence he
i s ac o p ojec ion G1×SO(3)→G1 es ic s o an injec ion Γ →G1,
and Γ does no mee any conjuga e o {e1}×SO(2)in G(all o hem a e
con ained in {e1}×SO(3)). Conside he canonical le ac ion o Gand
Γ on T∶=G1×S2≡G/({e1}×SO(2)). The e is a egula co e ing L
o he closed o ien ed su ace o genus wo, Σ2, whose g oup o co e ing
ans o ma ions is isomo phic o Γ. Conside he co esponding suspension
olia ed space, X=L×ΓT, which sa is ies he condi ions o Theo em A,
and he co esponding Molino’s desc ip ion (G, H, 
X0,ˆπ0)cons uc ed in
Sec ion 8.1, whe e 
X0=L×ΓG,H=SO(2), he igh H-ac ion on 
X0
is gi en by [y, g]⋅h=[y, gh], and he map ˆπ0∶
X0→Xis de ined by
36 J.A. ´
ALVAREZ L ´
OPEZ AND R. BARRAL LIJ ´
O
ˆπ0([y, g])=[y, g⋅u0]. We can equip Σ2wi h C∞and Riemannian s uc u es,
and conside he induced C∞and Riemannian s uc u es on Xand 
X0.
Since H≠{e},Xis no olia ed homogeneous by Theo em C (o Theo-
em B and P oposi ion 3.2). Howe e his canno be seen by compa ing any
pai o lea es since all o hem a e isome ic o L, and Xhas no holonomy
by “(ii) ⇔(iii)” in P oposi ion 8.1.
This a gumen canno p oduce ma chbox mani olds because P oposi-
ion 8.4 equi es G o be connec ed o apply [12, P oposi ion 8.2]. Examples
wi h o ally disconnec ed local ans e sals a e gi en in [23, Theo em 35]
and [19, Theo em 10.7].
8.4. In e se limi s o minimal Lie olia ions. This example was sug-
ges ed by S. Hu de . Le (X, G)be he McCo d solenoid de ined as he
p ojec i e limi o a owe o non- i ial egula co e ings be ween closed
connec ed mani olds,
⋯→Mk
φk
ÐÐÐ→ Mk−1→⋯→M0.
Le Γk=π1(Mk), and conside he induced owe o homomo phisms be-
ween ini e g oups,
⋯→Γ0/Γk→Γ0/Γk−1→⋯→Γ0/Γ1,
whose in e se limi Kcon ains a canonical dense copy o Γ0. Then (X, G)
can be also desc ibed as he suspension olia ed space 
M0×Γ0K, whe e 
M0
is he uni e sal co e ing o M0. We ge induced maps ψk∶X→Mk, whose
es ic ions o he lea es a e co e ing maps. Suppose ha M0is equipped
wi h a minimal Lie G0- olia ion F0, o some simply connec ed Lie g oup
G0. Then e e y Mkcan be endowed wi h he minimal Lie G0- olia ion
Fk∶=(φ1⋯φk)∗F0. On e e y G-lea M, conside he pull-back o F0by
ψ0∶M→M0. These olia ions on all lea es o Gcan be combined o o m
a olia ed s uc u e Fon X, which is a “Lie G0-sub olia ed s uc u e” o
Gin an ob ious sense. We can w i e F=ψ∗
0F0, which equals ψ∗
kFk o
all k. Ex ending he no a ion o suspensions, we can also w i e (X, F)=
(
M0,
F0)×Γ0K, whe e 
F0is he li o F0. I easily ollows ha (X, F)is a
minimal G- olia ed space o G=G0×K.
8.5. Open p oblems.
8.5.1. S ong quasi-analy ici y o H.This p oblem was p oposed in [10]. I is
eally unknown o he au ho s i he s ong quasi-analy ici y o His needed
in Theo em A. Mo e p ecisely, assuming ha His a minimal compac ly
gene a ed equicon inuous s ongly quasi-analy ic pseudog oup, is Hs ongly
quasi-analy ic? I minimali y is no assumed, hen coun e examples can be
easily gi en. Bu he minimal case seems o be an in e es ing open p oblem.
Among he wild ma chbox solenoilds o [33] he e migh be coun e examples.
MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 37
8.5.2. Func o ali y, uni e sali y and uniqueness o he Molino’s desc ip-
ion. I would be desi able o ha e a uniqueness o he Molino’s desc ip-
ion s onge han P oposi ion 3.1, s a ing ha no only he s uc u es
(G, Γ, H, 
X0,ˆπ0)cons uc ed in he p oo o Theo em A, bu also all possi-
ble s uc u es (G, Γ, H, 
X0,ˆπ0)sa is ying he condi ions o i s s a emen a e
equi alen . This would ollow by showing a uni e sali y p ope y, which in
u n would ollow by exhibi ing i s unc o iali y wi h espec o some kind o
olia ed maps. Since he de ini ion o 
X0uses ge ms o maps in H, he unc-
o iali y o Molino’s desc ip ion could be achie ed by showing ha olia ed
maps be ween equicon inuous olia ed spaces induce mo phisms be ween he
closu es o hei holonomy pseudog oups. This would be an ex ension o he
case o Riemannian olia ions, sol ed in [9, 8]. Such unc io ali y, uni e -
sali y and uniqueness o he Molino’s desc ip ion is no e en p o ed in he
Riemannian olia ion case. A di ec consequence would be ha His ini e
i and only i Xis a i ually olia ed homogeneous olia ed space (a ini e
old co e ing o Xis olia ed homogeneous as olia ed space).
8.5.3. How la ge is he class o in e se limi s o minimal Lie olia ions?
Since any me izable locally compac local g oup o ini e opological di-
mension is locally isomo phic o he di ec p oduc o a Lie g oup and a
compac ze o-dimensional opological g oup [34, Theo em 107], i was asked
by S. Hu de whe he any compac minimal olia ed homogeneous olia ed
space o ini e “ opological codimension” can be ealized as in e se limi o
minimal Lie olia ions, like in Sec ion 8.4. This would gene alize he esul s
o [15] (see also [2]), whe e an a i ma i e answe is gi en o homogeneous
ma chbox mani olds ( he case o codimension ze o). I his is ue, using also
he Molino’s desc ip ion, i could be possible o p o e ha any equicon inu-
ous olia ed space sa is ying he condi ions o Theo em A is an in e se limi
o Riemannian olia ions.
8.5.4. Molino’s desc ip ions wi hou assuming s ong quasi-analy ici y. This
p oblem a ises om he Molino spaces cons uc ed by Dye , Hu de and
Lukina in [19] o equicon inuous ma chbox mani olds, whe e s ong quasi-
analy ici y is no needed. Thei Molino spaces a e also olia ed homoge-
neous, and hei lea es co e he lea es o he o iginal ma chbox, bu hey
may no be unique. Thus he ollowing ques ion makes sense. Does he e ex-
is his kind o Molino spaces o a bi a y compac minimal equicon inuous
olia ed spaces?
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