EXTENSIONS OF AMENABLE GROUPS BY RECURRENT GROUPOIDS 1. Introduction M. Day

25 Pages·2014·0.43 MB·English

EXTENSIONS OF AMENABLE GROUPS BY RECURRENT
GROUPOIDS
KATEJUSCHENKO,VOLODYMYRNEKRASHEVYCH,MIKAELDELASALLE
Abstract. Weshowthatamenabilityofagroupactingbyhomeomorphisms
can be deduced from a certain local property of the action and recurrency of
theorbitalSchreiergraphs. Thisappliestoawideclassofgroups,amenability
of which was an open problem, as well as unifies many known examples to
one general proof. In particular, this includes Grigorchuk’s group, Basilica
group, groups associated with Fibonacci and Penrose tilings, the topologi-
cal full groups of Cantor minimal systems, groups acting on rooted trees by
boundedautomorphisms,groupsgeneratedbyfiniteautomataoflinearactiv-
itygrowth,andgroupsnaturallyappearinginholomorphicdynamics.
1. Introduction
M.DayintroducedtheclassEGofelementaryamenablegroups in[Day57]asthe
class of all groups that can be constructed from finite and abelian using operations
ofpassingtoasubgroup,quotient,groupextensions,anddirectlimits(thefactthat
the class of amenable groups is closed under these operations was already proved
by J. von Neumann). He notes that at that time no examples of amenable groups
that do not belong to the class EG were known.
The first example of an amenable group not belonging to EG was the Grig-
orchuk group of intermediate growth [Gri83]. An example of a group which can
not be constructed from groups of sub-exponential growth (which, in some sense,
canbealsoconsideredasan“easycase”ofamenability)istheBasilicagroupintro-
ducedin[GZ˙02]. Itsamenabilitywasprovedin[BV05]usingasymptoticproperties
of random walks on groups. These methods were then generalized in [BKN10]
and [AAV13] for a big class of groups acting on rooted trees. The commutator
subgroups of the topological full groups of Cantor minimal subshifts are known
to be finitely generated infinite simple, [Mat], thus they don’t belong to EG. The
amenability of this groups was proved in [JM12].
A common feature of all known examples of non-elementary amenable groups
is that they are defined as groups of homeomorphisms of the Cantor set (or con-
structed from such groups).
The aim of this paper is to show a general method of proving amenability for
a wide class of groups acting on topological spaces. This class contains many new
examples of groups, whose amenability was an open question. It also contains all
of the mentioned above non-elementary amenable groups as easy examples where
our main Therem 1.1 applies.
V.NekrashevychwassupportedbyNSFgrantDMS1006280. M.delaSallewassupportedby
ANRgrantsOSQPIandNEUMANN.
1
2 KATEJUSCHENKO,VOLODYMYRNEKRASHEVYCH,MIKAELDELASALLE
Weshowthatamenabilityofagroupofhomeomorphismscanbededucedfroma
combinationoflocaltopologicalinformationaboutthehomeomorphismsandglobal
information about the orbital Schreier graphs. Namely, we prove the following
amenabilitycondition(seeTheorem3.1). IfGisagroupactingbyhomeomorphisms
on a topological space X, then by [[G]] we denote the topological full group of the
action, i.e., the group of all homeomorphisms h of X such that for every x ∈ X
there exists a neighborhood of x such that restriction of h to that neighborhood is
equal to restriction of an element of G. For x∈X the group of germs of G at x is
the quotient of the stabilizer of x by the subgroup of elements acting trivially on a
neighborhood of x.
Theorem1.1. LetGandH begroupsofhomeomorphismsofacompacttopological
space X, and G is finitely generated. Suppose that the following conditions hold.
(1) The topological full group [[H]] is amenable.
(2) For every element g ∈ G, the set of points x ∈ X such that g does not
coincide with an element of H on any neighborhood of x is finite.
(3) For every point x∈X the Schreier graph of the action of G on the orbit of
x is recurrent.
(4) For every x∈X the group of germs of G at x is amenable.
Then the group G is amenable. Moreover, the group [[G]] is amenable.
(InTheorem3.1Condition(3)isreplacedbyaslightlylessrestrictivestatement.)
A key tool of the proof of the theorem is the following fact (see Theorem 2.8).
Theorem 1.2. Let G be a finitely generated group acting transitively on a set X.
If the graph of the action of G on X is recurrent, then there exists a (Z/2Z)(cid:111) G-
X
invariant mean on (cid:76) Z/2Z.
X
The last two sections of the paper are devoted to showing different examples of
applications of Theorem 1.1. At first we consider the case when the group [[H]]
is locally finite. Examples of such groups [[H]] are given by block-diagonal direct
limits ofsymmetricgroupsdefinedbyaBrattelidiagram. Thecorrespondinggroups
G satisfying the conditions of Theorem 1.1 are generated by homeomorphisms of
bounded type.
The class of groups generated by homeomorphisms of bounded type includes
groups whose amenability was an open problem, e.g., groups of arbitrary bounded
automorphisms of rooted trees. We present here some new concrete examples of
groups generated by homeomorphisms of bounded type: a group associated with
the Fibonacci tiling, and a group naturally defined using the Penrose tilings.
The last section describes some examples of application of Theorem 1.1 in the
case when H is not locally finite. For example, one can use Theorem 1.1 twice:
proveamenabilityof[[H]]usingit,andthenconstructnewamenablegroupsGusing
H. This way one gets a simple proof of the main result of [AAV13]: that groups
generated by finite automata of linear activity growth are amenable. Moreover,
our technique answers a well known question on amenability of finite automata of
quadratic growth in affirmative.
TwootherexamplesfromSection5arenew,andaregroupsnaturallyappearing
in holomorphic dynamics. One is a holonomy group of the stable foliation of the
Julia set of a H´enon map, the other is the iterated monodromy group of a mating
of two quadratic polynomials.
EXTENSIONS OF AMENABLE GROUPS BY RECURRENT GROUPOIDS 3
Acknowledgments. We thank Omer Angel, Laurent Bartholdi, Rostislav Grig-
orchukandPierredelaHarpeforveryusefulandnumerouscommentsonaprevious
version of this paper. In a preliminary version of the paper our results were not
stated in terms of random walks but rather in term of the validity of certain in-
equalities. U.Bader, B.HuaandA.Valettesuggestedtolookataconnectionwith
recurrence of random walks and B. Hua pointed out [Woe00, Theorem 3.24] to us.
We thank them for this very fruitful suggestion and for other comments which are
valuable to us.
2. Amenable and recurrent G-sets
2.1. Amenable actions.
Definition 2.1. Let G be a discrete group. An action of G on a set X is said to
be amenable if there exists an invariant mean on X. Here an invariant mean is
a map µ from the set of all subsets of X to [0,1] such that µ is finitely additive,
µ(X)=1, and µ(g(A))=µ(A) for all A⊂X and g ∈G.
A group G is amenable if and only if its action on itself by left multiplication is
amenable. Note that the above definition of amenability is different from another
definition of amenability of an action, due to Zimmer, see [Zim84].
Thefollowingcriteriaofamenabilityofactionsgeneralizethecorrespondingclas-
sical criteria of amenability of groups, see [Gre69] Theorems 3.5 and 4.1.
Theorem 2.2. Let G be a discrete group acting on a set X. Then the following
conditions are equivalent.
(1) The action of G on X is amenable.
(2) Reiter’s condition. For every finite subset S ⊂ G and for every (cid:15) > 0
there exists a non-negative function φ ∈ (cid:96)1(X) such that (cid:107)φ(cid:107) = 1 and
1
(cid:107)g·φ−φ(cid:107) <(cid:15) for all g ∈S.
1
(3) Følner’s condition. For every finite subset S ⊂ G and for every (cid:15) > 0
(cid:80)
there exists a finite subset F ⊂X such that |gF∆F|<(cid:15)|F|.
g∈S
Note that if G is finitely generated, it is enough to check conditions (2) and
(3) for one generating set S. If the condition (3) holds, then we say that F is an
(S,(cid:15))-Følner set, and that F is (cid:15)-invariant with respect to the elements of S.
Thefollowingpropositioniswellknownandfollowsfromthefactthatagroupis
amenableifandonlyifitadmitsanactionwhichisamenableandZimmeramenable,
see [Ros73].
Proposition 2.3. Let G be a group acting on a set X. If the action is amenable
and for every x ∈ X the stabilizer G of x in G is an amenable group, then the
x
group G is amenable.
2.2. Schreier graphs. Let G be a finitely generated group with finite symmetric
generating set S.
If G acts transitively on X the Schreier graph Γ(X,G,S) is the graph with the
set of vertices identified with X, the set of edges is S ×X, where an edge (s,x)
connects x to s(x).
If G acts (not necessarily transitively) on a set X, the orbital Schreier graph at
x∈X is the Schreier graph of the action of G on the G-orbit of x.
4 KATEJUSCHENKO,VOLODYMYRNEKRASHEVYCH,MIKAELDELASALLE
2.3. Recurrent actions. Let G be a finitely generated group acting transitively
on a set X. Choose a measure µ on G such that support of µ is a finite generating
set of G and µ(g) = µ(g−1) for all g ∈ G. Consider the Markov chain on X with
(cid:80)
transition probability from x to y equal to p(x,y)= µ(g).
g∈G,g(x)=y
The Markov chain is called recurrent if the probability of ever returning to x
0
after starting at x is equal to 1 for some (and hence for every) x ∈X.
0 0
Itiswellknown(see[Woe00,Theorems3.1,3.2])thatrecurrenceofthedescribed
Markov chain does not depend on the choice of the measure µ, if the measure is
symmetric, and has finite support generating the group. We say that the action of
GonX isrecurrent ifthecorrespondingMarkovchainisrecurrent. Everyrecurrent
action is amenable, see [Woe00, Theorem 10.6].
A transitive action of G on X is recurrent if and only if the simple random walk
on the Schreier graph Γ(X,G,S) is recurrent.
The following fact is also a corollary of [Woe00, Theorem 3.2].
Lemma 2.4. If the transitive action of a finitely generated group G on a set X is
recurrent, and H < G is a subgroup, then the action of H on every orbit of X is
also recurrent.
The following theorem is a straightforward corollary of the Nash-Williams crite-
rion (see [NW59] or [Woe00, Corollary 2.20]).
Theorem 2.5. Let Γ be a connected graph of uniformly bounded degree with set
of vertices V. Suppose that there exists an increasing sequence of finite subsets
(cid:83)
F ⊂V such that F =V, the subsets ∂F are pairwise disjoint, and
n n≥1 n n
(cid:88) 1
=∞,
|∂F |
n
n≥1
where ∂F is the set of vertices of F adjacent to the vertices of V \F . Then the
n n n
simple random walk on Γ is recurrent.
We will also use a characterization of transience of a random walk on a locally
finiteconnectedgraph(V,E)intermsofelectricalnetwork. Thecapacity ofapoint
x ∈V is the quantity defined by
0
 
 1/2
 (cid:88) 
cap(x0)=inf  |a(x)−a(x(cid:48))|2
 (x,x(cid:48))∈E 
where the infimum is taken over all finitely supported functions a : V → C with
a(x )=1. We will use the following
0
Theorem2.6([Woe00],Theorem2.12). Thesimplerandomwalkonalocallyfinite
connected graph (V,E) is transient if and only if cap(x )>0 for some (and hence
0
for all) x ∈V.
0
2.4. A mean on (cid:76) Z invariant with respect to Z (cid:111) G. LetGbeadiscrete
X 2 2 X
groupactingtransitivelyonasetX. Let{0,1}X bethesetofallsubsetsofX con-
sidered as an abelian group with multiplication given by the symmetric difference
of sets. It is naturally isomorphic to the Cartesian product ZX, where Z =Z/2Z.
2 2
The group G acts naturally on the group ZX by automorphisms.
2
EXTENSIONS OF AMENABLE GROUPS BY RECURRENT GROUPOIDS 5
Denote by P (X) the subgroup of ZX = {0,1}X which consists of all finite
f 2
subsets of X, i.e., the subgroup (cid:76)Z of ZX. It is obviously invariant under the
2 2
X
action of G.
Consider the restricted wreath product Z (cid:111) G ∼= G(cid:110)P (X). Its action on
2 X f
P (X) is given by the formula
f
(g,E)(F)=g(E∆F)
for E,F ∈P (X) and g ∈G.
f
The Pontryagin dual of P (X) is the compact group ZX, with the duality given
f 2
by the pairing φ(E,ω) = exp(iπ (cid:80) ω ), E ∈ P (X), ω ∈ ZX. Fix a point p ∈ X
j f 2
j∈E
and denote by L (ZX,µ) the Hilbert space of functions on ZX with the Haar
2 2 2
probability measure µ. Denote by A ={(ω ) ∈ZX :ω =0} the cylinder set
p x x∈X 2 p
which fixes ω as zero.
p
The following lemma was proved in [JM12, Lemma 3.1].
Lemma 2.7. Let G acts transitively on a set X and choose a point p ∈ X. The
following are equivalent:
(i) There exist a net of unit vectors {f } ∈ L ({0,1}X,µ) such that for every
n 2
g ∈G
(cid:107)g·f −f (cid:107) →0 and (cid:107)f ·χ (cid:107) →1.
n n 2 n Ap 2
(ii) The action of Z (cid:111) G on P (X) is amenable.
2 X f
(iii) The action of G on P (X) admits an invariant mean giving full weight to the
f
collection of sets containing p.
We say that a function f ∈ L (ZX,µ) is p.i.r. if it is a product of independent
2 2
random variables, i.e., there are functions fx ∈ RZ2 such that f(ω) = (cid:81) fx(ωx).
x∈X
In other words, if we consider L (ZX,µ) as the infinite tensor power of the Hilbert
2 2
space L (Z ,m) with unit vector 1, where m({0}) = m({1}) = 1/2, the condition
2 2
p.i.r. means that f is an elementary tensor in L (ZX,µ).
2 2
Theorem 2.8. Let G be a finitely generated group acting transitively on a set X.
There exists a sequence of p.i.r. functions {f } in L (ZX,µ) that satisfy condition
n 2 2
(i) in Lemma 2.7 if and only if the action of G on X is recurrent.
Proof. Denote by (X,E) the Schreier graph of the action of G on X with respect
to S. Suppose that the simple random walk on (X,E) is recurrent. By Theorem
2.6 there exists a = (a ) a sequence of finitely supported functions such that
n x,n x
a =1 and (cid:80) |a −a |2 →0. Replacing all values a that are greater
x0,n x∼x(cid:48) x,n x(cid:48),n x,n
than 1 (smaller than 0) by 1 (respectively 0) does not increase the differences
|a −a |, hence we may assume that 0≤a ≤1. For 0≤t≤1 consider the
x,n x(cid:48),n x,n
unit vector ξ ∈L ({0,1},m)
t 2
√ √
(ξ (0),ξ (1))=( 2cos(tπ/4), 2sin(tπ/4)).
t t
6 KATEJUSCHENKO,VOLODYMYRNEKRASHEVYCH,MIKAELDELASALLE
(cid:78)
Define f = ξ and f = f . We have to show that (cid:104)gf ,f (cid:105) → 1
x,n 1−ax,n n x∈X x,n n n
for all g ∈Γ. It is sufficient to show this for g ∈S. Then
(cid:89) (cid:89) π
(cid:104)gf ,f (cid:105)= (cid:104)f ,f (cid:105)= cos (a −a )≥
n n x,n gx,n 4 x,n gx,n
x x
(cid:89)e−π162(ax,n−agx,n)2 ≥e−π162 (cid:80)x∼x(cid:48)|ax,n−ax(cid:48),n|2
x
and the last value converges to 1. We used that cos(x)≥e−x2 if |x|≤π/4.
Let us prove the other direction of the theorem. Define the following pseudo-
metric on the unit sphere of L ({0,1},m) by
2
(cid:112)
d(ξ,η)= inf (cid:107)ωξ−η(cid:107)= 2−2|(cid:104)ξ,η(cid:105)|.
ω∈C,|ω|=1
Assume that there is a sequence of p.i.r. functions {f } in L (ZX,µ) that sat-
n 2 2
(cid:81)
isfy condition (i) of Lemma 2.7. Write f (ω) = f (ω ). We can assume
n x∈X n,x x
that the product is finite. Replacing f by f /(cid:107)f (cid:107) we can assume that
n,x n,x n,x
(cid:107)fn,x(cid:107)L2(Z2,m) = 1. Define ax,n = d(fx,n,1). It is straightforward that (ax,n)x∈X
has finite support and
√
lima =d( 2δ ,1)>0.
n x0,n 0
Moreover for every g ∈G
|(cid:104)gfn,fn(cid:105)|=(cid:89)|(cid:104)fn,x,fn,gx(cid:105)|=(cid:89)(1−d(fn,x,fn,gx)2/2)≤e−(cid:80)xd(fn,x,fn,gx)2/2,
x x
which by assumption goes to 1 for every g ∈S. By definition of the Schreier graph
and the triangle inequality for d,
(cid:88) (cid:88)(cid:88) (cid:88)(cid:88)
|a −a |2 = |a −a |2 ≤ d(f ,f )2 →0.
x,n x(cid:48),n x,n gx,n x gx
(x,x(cid:48))∈E g∈S x g∈S x
This proves that cap(x )=0 in (X,E), and hence by Theorem 2.6 that the simple
0
random walk on (X,E) is recurrent. (cid:3)
3. Amenability of groups of homeomorphisms
3.1. Groupoids. Let X be a topological space. A germ of homeomorphism of
X is an equivalence class of pairs (g,x) where x ∈ X and g is a homeomorphism
betweenaneighborhoodofxandaneighborhoodofg(x),wheretwogerms(g ,x )
1 1
and (g ,x ) are equal if x = x , and if g and g coincide on a neighborhood
2 2 1 2 1 2
of x . The set of all germs of the action of G on X is a groupoid. Denote by
1
o(g,x)=xandt(g,x)=g(x)theorigin andthetarget ofthegerm. Acomposition
(g ,x )(g ,x ) is defined if g (x ) = x , and then it is equal to (g g ,x ), which
1 1 2 2 2 2 1 1 2 2
makessensebecauseg g isdefinedonaneighborhoodofx . Theinverseofagerm
1 2 2
(g,x) is the germ (g,x)−1 =(g−1,g(x)).
A groupoid of germs of homeomorphisms on X is a set of germs of homeomor-
phisms of X that is closed under composition and inverse, and that contains all
germs (Id ,x) for x∈X.
X
IfagroupGactsbyhomeomorphismsonX,itsgroupoidofgermsisthegroupoid
all all germs (g,x) for g ∈G and x∈X.
For a given groupoid G of germs of homeomorphisms on X, and for x ∈ X, the
isotropy group, or group of germs G is the group of all germs γ ∈ G such that
x
o(γ) = t(γ) = x. If G is the groupoid of germs of the action of a group G on X,
EXTENSIONS OF AMENABLE GROUPS BY RECURRENT GROUPOIDS 7
thenthegroupofgermsG isthequotientofthestabilizerG ofxbythesubgroup
x x
of elements of G that act trivially on a neighborhood of x.
The topological full group of a groupoid of germs G, denoted [[G]] is the set of all
homeomorphisms F :X −→X such that all germs of F belong to G.
3.2. Amenability of groups.
Theorem 3.1. Let G be a finitely generated group of homeomorphisms of a topo-
logical space X, and G be its groupoid of germs. Let H be a groupoid of germs of
homeomorphisms of X. Suppose that the following conditions hold.
(1) The group [[H]] is amenable.
(2) For every generator g of G the set of points x ∈ X such that (g,x) ∈/ H
is finite. We say that x ∈ X is singular if there exists g ∈ G such that
(g,x)∈/ H.
(3) For every singular point x∈X, the action of G on the orbit of x is recur-
rent.
(4) The groups of germs G are amenable.
x
Then the group G is amenable.
Proof. After replacing H by H∩G, we may assume that H⊂G. Let S be a finite
symmetric generating set of G. Let Σ be the set of points x ∈ X such that there
existsg ∈S suchthat(g,x)∈/ H. LetV betheunionoftheG-orbitsoftheelements
ofΣ. Byassumption2everyG-orbitisaunionofafinitenumberofH-orbits. Since
the set Σ is finite, the set V is a union of a finite number of H-orbits.
Lemma 3.2. The set V contains all singular points of X.
Proof. Letg =g g ···g bearepresentationofgasaproductofgeneratorsg ∈S.
1 2 n i
Then the germ (g,x) is equal to the product of the germs
(g ,g ···g (x))·(g ,g ···g (x))···(g ,x)
1 2 n 2 3 n n
IfallthesegermsbelongtoH, then(g,x)belongstoH. Therefore, (g,x)∈/ H only
if x∈Σ∪g−1Σ∪(g g )−1Σ∪···(g ···g )−1Σ. (cid:3)
n n−1 n 2 n
Let A ⊂ V be an H-orbit transversal. For every v ∈ V there exists a unique
element of A that belongs to the same H-orbit as v. Let us denote it by α(v).
Choose a germ δ ∈H such that o(δ )=α(v) and t(δ )=v. For g ∈G and v ∈V
v v v
denote by g the element of G defined by
v
(1) g =δ−1 (g,v)δ ,
v g(v) v
and note that it satisfies the cocycle relation (gg(cid:48)) =g g(cid:48).
v g(cid:48)(v) v
Let G| (resp. H| ) be the set of germs γ ∈ G (resp. γ ∈ H) with the target
A A
and the origin in A. Note that H| is the disjoint union of the isotropy groups
A
H for a ∈ A, and that g ∈ G| for all g ∈ G and v ∈ V. Consider the quotient
a v A
Z =G| /H| ofG| definedbytherightactionofH| ,i.e.,twogermsγ ,γ ∈G|
A A A A 1 2 A
areequivalentifthereexistsγ ∈Hsuchthatγ =γ γ. Notethatthent(γ )=t(γ )
2 1 1 2
and o(γ )=o(γ ), hence the maps t:Z −→A and o:Z −→A are well defined.
1 2
Let P be the set of functions φ:V −→Z such that t(φ(v))=α(v) for all v ∈V
and of finite support, i.e., such that the values of φ(v) are trivial (i.e., belong to
H ) for all but a finite number of values v ∈V.
α(v)
8 KATEJUSCHENKO,VOLODYMYRNEKRASHEVYCH,MIKAELDELASALLE
Forφ∈P, g ∈G, andv ∈V, defineg(φ)(v)=g ·(φ(g−1v)). Byassumption
g−1v
2 of the theorem, g(φ) belongs to P and this defines an action of G on P by the
cocycle relation.
Proposition 3.3. There exists a G-invariant mean on P.
(cid:83)
Proof. If we decompose V = V as a finite union of G-orbits, we get a decom-
i i
position of P as a direct product of P where P are the restrictions of elements of
i i
P to V , and G acts diagonally. It is therefore enough to prove the proposition for
i
the case when G acts transitively on V.
For every pair of elements a,b ∈ A, choose an element f ∈ G such that
a,b
o(f ) = a and t(f ) = b. We also assume that f is the identity of G . For
a,b a,b a,a a
(cid:96)
every γ ∈ G| consider the element γ ∈ G (disjoint union) defined by γ =
A (cid:101) a∈A a (cid:101)
(cid:96)
f γ ∈G . We also denote by · the induced map G| /H| → G /H .
t(γ),o(γ) o(γ) (cid:101) A A a∈A a a
(cid:96)
Foreveryφ∈P,considerthemapψ: V → G /H definedbyψ(v)=φ(cid:103)(v).
a∈A a a
The map φ (cid:55)→ ψ allows to identify P with the set P(cid:101) of functions ψ from V to
(cid:96)
G /H such that ψ(v)=H for all but finitely many v’s.
a∈A a a α(v)
One easily checks that the action of G on P(cid:101) using this identification is given by
(2) (g·ψ)(gv)=f g f−1 ψ(v),
α(gv),o(ψ(v)) v α(v),o(ψ(v))
where g is given by (1). If the germ (g,v) belongs to H, then α(v) = α(gv)
v
and g ∈ H . If additionally ψ(v) is trivial (i.e. is equal to H ), then so is
v α(v) α(v)
(g·ψ)(gv) by our choice of f =1.
a,a
ByLemma2.7andTheorem2.8thereexistsaG-invariantmeanonP (V)giving
f
full weight to the collection of sets containing a given point p∈V. Note that since
the mean is G-invariant, finitely additive, and the action of G on V is transitive,
themeangivesfullweighttothecollectionofsetscontaininganygivenfinitesubset
of V. In particular, it gives full weight to the collection of sets containing Σ.
It follows then from Theorem 2.2 that for every (cid:15)>0 there exists a finite subset
F of P (V) which is (cid:15)-invariant under the action of elements of S, such that every
f
element of F contains Σ. Moreover, since G preserves cardinalities of elements of
P (V), we can choose F consisting of sets of the same cardinality N.
f
Fix x ∈A. Let us assume at first that G /H is infinite. Let R be the finite
0 x0 x0
subset of G defined by
x0
R={f g f−1 ,v ∈∪ B,g ∈S}.
α(gv),x0 v α(v),x0 B∈F
SincethegroupG isamenable,forevery(cid:15)>0thereexistsasubsetF ofG /H
x0 x0 x0
such that |γF∆F| ≤ (cid:15)/N for all γ ∈ R. Since G /H is infinite we may assume
x0 x0
that F does not contain the trivial element H .
x0
(cid:96)
Let F(cid:98) be the set of functions V −→ a∈AGa/Ha such that there exists B ∈F
such that φ(v) = Hα(v) for v ∈/ B, and φ(v) ∈ F for v ∈ B. Then F(cid:98) is split into
a disjoint union of sets F(cid:98)B of functions with support equal to B ∈ F. We use
herethefactthatF doesnotcontainthetrivialelementH ofG /H . Foreach
x0 x0 x0
B ∈F we have
(cid:12) (cid:12)
(cid:12)(cid:12)F(cid:98)B(cid:12)(cid:12)=|F||B| =|F|N,
hence
(cid:12) (cid:12)
(cid:12)F(cid:98)(cid:12)=|F|·|F|N.
(cid:12) (cid:12)
EXTENSIONS OF AMENABLE GROUPS BY RECURRENT GROUPOIDS 9
The number of sets B ∈ F such that g(B) ∈/ F for some g ∈ S is not larger than
(cid:15)|F|. Itfollowsthatthenumberoffunctionsφ∈F(cid:98) withsupportequaltosuchsets
is not larger than
(cid:12) (cid:12)
(cid:15)|F|·|F|N−1 =(cid:15)(cid:12)F(cid:98)(cid:12).
(cid:12) (cid:12)
Let g ∈ S. Suppose that B,g(B) ∈ F, and let ψ ∈ F(cid:98)B. Take v ∈/ B. Then
ψ(v)=H , and by our assumption that Σ⊂B, we have g ∈H . By (2) the
α(v) v α(v)
support of g(ψ) is therefore a subset of g(B). It follows that g(ψ) does not belong
to F(cid:98) if and only if there exists v ∈ B such that fα(gv),x0gvfα−(1v),x0ψ(v) ∈/ F. It
follows that the cardinality of the set of elements ψ ∈F(cid:98)B such that g(ψ)∈/ F(cid:98) is at
most
(cid:88)
|f g f−1 F \F||F|N−1 <(cid:15)|F|N.
α(gv),x0 v α(v),x0
v∈B
It follows that the cardinality of the set of functions φ ∈ F(cid:98) such that g(φ) ∈/ F(cid:98)
for some g ∈S is not greater than
(cid:12) (cid:12)
((cid:15)+(cid:15)·|S|)·(cid:12)F(cid:98)(cid:12).
(cid:12) (cid:12)
Since (cid:15) is an arbitrary positive number, it follows that the action of G on P is
amenable.
ThecasewhenG /H isfinitecanbereducedtotheinfinitecase,forinstance,
x0 x0
by the following trick. Replace G by G ×Z, and define P˜ as the set of maps
a a
V −→ (cid:96) (G ×Z)/H of finite support, where support is defined in the same
a∈A a a
way as before. Here H is considered to be the subgroup of H ×{0} < G ×Z.
a a a
Define the action of G on P˜ by the same formulae (1) and (2) as the action of G
on P. Then, by the same arguments as above, there exists a G-invariant mean on
P˜. The projection (cid:96) G ×Z−→(cid:96)a∈AG induces a surjective G-equivariant
a∈A a a
map P˜ −→P, which implies that P has a G-invariant mean. (cid:3)
In order to prove amenability of G, it remains to show that for every φ∈P the
stabilizerG ofφinGisamenable,seeProposition2.3. Wewilluseamodification
φ
of an argument of Y. de Cornulier from [dC13, Proof of Theorem 4.1.1]. Suppose
that it is not amenable. It follows from Lemma 2.4 that for every v ∈V the action
of every subgroup of G on the orbit of v has an invariant mean. Consequently,
by Proposition 2.3, the stabilizer G of v in G is non-amenable. It follows by
φ,v φ
induction, that the intersection K of the pointwise fixator of the support of φ with
G isnon-amenable. SinceK fixesallpointsy ∈suppφ,wehaveahomomorphism
φ
from K to the direct product of the isotropy groups G for y ∈suppφ. If g belongs
y
to the kernel of this homomorphism, then all germs of g at points of V belong to
H, hence g ∈[[H]], by Lemma 3.2. It follows that K is an extension of a subgroup
of [[H]] by a subgroup of a finite direct product of isotropy groups G . But this
y
implies that K is amenable, which is a contradiction. (cid:3)
Remark. NotethatsinceconditionsontheelementsofGinTheorem3.1arelocal,
it follows from the theorem that [[G]] is amenable if X is compact.
4. Homeomorphisms of bounded type
We start with a description of the groupoids H such that the topological full
group [[H]] belongs to the class of locally finite groups, which is in some sense the
10 KATEJUSCHENKO,VOLODYMYRNEKRASHEVYCH,MIKAELDELASALLE
most basic class of amenable groups. We will use then such groupoids as base for
application of Theorem 3.1 and construction of non-elementary amenable groups.
4.1. Bratteli diagrams. A Bratteli diagram D=((V ) ,(E ) ,o,t) is defined
i i≥1 i i≥1
by two sequences of finite sets (V ) and (E ) , and sequences of maps
i i=1,2,... i i=1,2,...
o : E −→ V , t : E −→ V . We interpret the sets V as sets of vertices of the
i i i i+1 i
diagram partitioned into levels. Then E is the set of edges connecting vertices of
i
the neighboring levels V and V . See [Bra72] for their applications in theory of
i i+1
C∗-algebras. Apath of length n, wherenisanaturalnumberor∞, inthediagram
D is a sequence of edges e ∈ E , 1 ≤ i ≤ n, such that t(e ) = o(e ) for all i.
i i i i+1
Denote by Ω (D) = Ω the set of paths of length n. We will write Ω instead of
n n
Ω .
∞
(cid:81)
The set Ω is a closed subset of the direct product E , and thus is a
i≥1 i
compact totally disconnected metrizable space. If w = (a ,a ,...,a ) ∈ Ω
1 2 n n
is a finite path, then we denote by wΩ the set of all paths beginning with w.
Let w = (a ,a ,...,a ) and w = (b ,b ,...,b ) be elements of Ω such that
1 1 2 n 2 1 2 n n
t(a ) = t(b ). Then for every infinite path (a ,a ,...,a ,e ,...), the sequence
n n 1 2 n n+1
(b ,b ,...,b ,e ,e ,...) is also a path. The map
1 2 n n+1 n+2
(3) T :(a ,a ,...,a ,e ,...)(cid:55)→(b ,b ,...,b ,e ,e ,...)
w1,w2 1 2 n n+1 1 2 n n+1 n+2
is a homeomorphism w Ω−→w Ω.
1 2
DenotebyT thesetofgermsofthehomeomorphismsT . Itisnaturally
w1,w2 w1,w2
identified with the set of all pairs ((e ) ,(f ) ) ∈ Ω2 such that e = a and
i i≥1 i i≥1 i i
f =b for all 1≤i≤n, and e =f for all i>n.
i i i i
Let T(D) (or just T) be the groupoid of germs of the semigroup generated by
the transformations of the form T . It can be identified with the set of all
w1,w2
pairs of cofinal paths, i.e., pairs of paths (e ) , (f ) such that e = f for all
i i≥1 i i≥1 i i
i big enough. The groupoid structure coincides with the groupoid structure of an
equivalencerelation: theproduct(w ,w )·(w ,w )isdefinedifandonlyifw =w ,
1 2 3 4 2 3
andthenitisequalto(w ,w ). Hereo(w ,w )=w andt(w ,w )=w . Itfollows
1 4 1 2 2 1 2 1
from the definition of topology on a groupoid of germs that topology on T is given
bythebasisofopensetsoftheformT . WecallT thetail equivalence groupoid
w1,w2
oftheBrattelidiagramD. Moreonthetailequivalencegroupoids,theirproperties,
and relation to C∗-algebras, see [ER06].
Let us describe the topological full group [[T]]. By compactness of Ω, for every
g ∈ [[T]] there exists n such that for every w ∈ Ω there exist a path w ∈ Ω
1 n 2 n
such that g(w) = T (w) for all w ∈ w Ω. Then we say that depth of g is at
w1,w2 1
most n. For every v ∈ V denote by Ω the set of paths w ∈ Ω ending in v.
n+1 v n
Then the group of all elements g ∈ [[T]] of depth at most n is naturally identified
(cid:81)
with the direct product Symm(Ω ) of symmetric groups on the sets Ω .
v∈V(cid:81)n+1 v v
Namely, if g = (π ) ∈ Symm(Ω ) for π ∈ Symm(Ω ), then g acts
v v∈Vn+1 v∈Vn+1 v v v
on Ω by the rule g(w w)=π (w )w, where w ∈Ω and w w ∈Ω. Let us denote
0 v 0 0 v 0
the group of elements of [[T]] of depth at most n by [[T]] . We obviously have
n
(cid:83)
[[T]] ≤ [[T]] and [[T]] = [[T]] . In particular, [[T]] is locally finite (i.e.,
n n+1 n≥1 n
all its finitely generated subgroups are finite).
The embedding [[T]] (cid:44)→ [[T]] is block diagonal with respect to the direct
n n+1
(cid:81) (cid:81)
productdecompositions Symm(Ω )and Symm(Ω ). Formoreon
v∈Vn+1 v v∈Vn+2 v
such embeddings of direct products of symmetric and alternating groups and their
inductive limits see [LP05, LN07].