Distortion elements of Diff^{∞}_{0} (M)

E. Militon April 16, 2012

1 Introduction

Distortion elements in non-conservative dieomorphism groups have been stud-
ied since Franks and Handel asked the following question: is a rotation of the
circle (or of the sphere) distorted ? Calegari and Freedman managed to give
a rst answer to this question. They proved that a rotation of the circle is
distorted in the groupDiff^{1}(S^{1})ofC^{1}-dieomorphisms. They also proved that
a rotation of the sphere is distorted in the group of C^{∞}-dieomorphisms of
the sphereS^{2}. Finally, they proved that, for any N, a homeomorphism of the
N-dimensional sphere is distorted. Avila then proved that a rotation of the
circle is distorted in the groupDiff^{1}(S^{1}). He actually proved that every recur-
rent orientation-preserving element inDiff^{∞}(S^{1})is distorted. In this article, we
generalize Avila's result to any manifold.

2 Statement of the results

Let us start by introducing some denitions and notations which will be useful.

Dénition 1 LetGbe a group. For a nite subset S⊂Gand for an element g ofGis in the subgroup < S >generated byS, we dene:

lS(g) = inf (

n∈N,∃(i)i=1...n∈ {±1}^{n},∃(si)i=1...n∈S^{n}, g=

n

Y

i=1

s^{}_{i}^{i}
)

. An elementg in Gis said to be distorted if there exists a nite subset S ⊂G such that:

g∈< S > .
lim_{n→+∞}^{l}^{S}^{(g}_{n}^{n}^{)} = 0.

We notice that, as the sequence (lS(g^{n}))_{n∈N} is subadditive, it suces to
show that:

lim inf

n→+∞

l_{S}(g^{n})
n = 0
in order to prove that the elementg is distorted.

Given a dierentiable manifoldM, we denote by:

• Homeo0(M) the group of compactly supported homeomorphisms of M compactly isotopic to the identity;

• forrinN∪{∞},Diff^{r}_{0}(M)the group of compactly supportedC^{r}-dieomorphisms
ofM compactly isotopic to the identity.

Notice thatHomeo0(M) = Diff^{0}_{0}(M).

Here, the support of a homeomorphismf ofM is the set:

supp(f) ={x∈M, f(x)6=x}.

We denote byd_{r}a distance which is compatible with the topology onDiff^{r}_{0}(M).
Dénition 2 A dieomorphismf in Diff^{∞}_{0} (M)is recurrent when:

lim inf

n→+∞d_{∞}(f^{n}, IdM) = 0.

The purpose of this article is to show the following theorems. The rst one generalizes Avila's result on dieomorphisms of the circle (see [1]).

Theorem 1 IfM is a compact connected manifold, every recurrent element in
Diff^{∞}_{0} (M)is distorted.

The method used to prove this theorem is the same as in [1]. With this
method, we can indeed prove that every sequence(f_{n})_{n∈}_{N}of recurrent elements
is simultaneously distorted which means that we can nd a nite set S such
that:

∀n∈N, lim inf

p→+∞

lS(f_{n}^{p})
p = 0.

With this method, we can also provide a new proof of Calegari and Freedman's result (see [4]).

Theorem 2 (Calegari-Freedman) Every element inHomeo_{0}(S^{n})is distorted.

Here also, it can be shown that every sequence of elements inHomeo0(S^{n})
is simultaneously distorted.

3 Proof of the theorems

The proof of these theorems relies on a generalisation of results by [1] to any manifold. The method and the notations are similar to those in [1] but the generalisation to higher dimensions make a crucial use of a local perfection result shown in the appendice of this paper. We will prove theorem 1 only when the dimension ofM is greater than1. The1-dimensional case is proved in [1].

The theorems will be a direct consequence of the following lemmas.

Lemma 1 LetM be a compact manifold with dimension greater than 1.
There exist sequences (n)_{n∈N} and (kn)_{n∈N} of positive real numbers such that
very sequence(hn)_{n∈N}of dieomorphisms in Diff^{∞}_{0} (M)with:

∀n∈N, d∞(hn, IdM)< n,

satises the following property: there exists a nite setS ⊂Diff^{∞}_{0} (M)such that,
for any integern,

• The dieomorphismh_{n} belongs to the subgroup generated byS.

• l_{S}(h_{n})≤k_{n}.

Lemma 2 LetN be a positive integer.

There exists a sequence (kn)_{n∈N} of positive real numbers such that, for every
sequence (hn)_{n∈N} of homeomorphisms in Homeo0(S^{N}), there exists a nite set
S⊂Homeo0(S^{N})such that, for any integern:

• the homeomorphism hn belongs to the group generated by S.

• l_{S}(h_{n})≤k_{n}.

Now, let us prove that these lemmas imply the theorems.

Proof of the theorems. Let f be a recurrent element in Diff^{∞}_{0} (M). Let us
consider a strictly increasing mapp:N→Nsuch that:

lim_{n→+∞}_{p(n)}^{k}^{n} = 0
d_{∞}(f^{p(n)}, IdM)≤n

,

where the sequences (n)_{n∈N} and (kn)_{n∈N} are given by lemma 1. This lemma
applied to the sequence(f^{p(n)})_{n∈N}shows thatf is distorted. Theorem 2 can be
shown the same way by using lemma 2.

4 Proof of lemmas 1 et 2

In order to prove these lemmas, we will need the following lemmas which will
be shown in the next section. Lemmas 3 and 4 are analogous to lemmas 1 and 2
but deal with commutator of dieomorphisms inR^{N}. Let us denote byB(0,2)
the open ball of center0 and radius 2. Iff andg are dieomorphisms ofR^{N},
we denote by[f, g]the dieomorphismf gf^{−1}g^{−1}.

Lemma 3 LetN be a positive integer and r be an element inN∪ {∞}.
There exist sequences(^{0}_{n})n∈Nand(k_{n}^{0})n∈Nof positive real numbers which satisfy
the following property. Let(f_{n})_{n∈}_{N}and(g_{n})_{n∈}_{N}be sequences of dieomorphisms
inDiff^{r}_{0}(R^{N})with support included inB(0,2) such that:

∀n∈N, dr([fn, gn], Id_{R}N)< ^{0}_{n}.

Then there exists a nite set S^{0} ⊂Diff^{r}_{0}(R^{N})such that, for any integer n, the
dieomorphism[f_{n}, g_{n}] belongs to the group generated byS^{0} andl_{S}^{0}([f_{n}, g_{n}])≤
k_{n}^{0}.

Whenr= 0, we have a stronger lemma.

Lemma 4 There exists a sequence(k_{n}^{0})_{n∈}_{N} of positive real numbers such that,
for any sequence(˜hn)_{n∈}_{N}of homeomorphisms inHomeo0(R^{N})with support in-
cluded in B(0,2), there exists a nite set S^{0} ⊂ Homeo0(R^{N}) such that, for
any integern, the homeomorphism˜hn belongs to the group generated byS^{0} and
lS^{0}(˜hn)≤k^{0}_{n}.

Proof of lemma 1 Denote byN the dimension of M (N ≥2). We consider a
nite open covering (U_{i})_{0≤i≤p} of M by open sets dieomorphic to R^{N} whose
closure is included in a chart. For any integeri between0 and p, we chose a
chartϕi ofM dened on a neighbourhood of the closure ofUi which satises:

ϕi(Ui)⊂B(0,2).

Denote byψa bijection fromN× {1, ..., p} × {1, . . . ,4N}ontoN.

We will now construct the sequence(n)_{n∈N}using the sequence(^{0}_{l})_{l∈N}given
by lemma 3 in the caser=∞.

By a local perfection lemma (see appendix), for every natural integern, we can
nd n so that, if a dieomorphism h in Diff^{∞}_{0} (M) satises d(h, IdM) < n,
then there exist two nite sequences(fk,l)0≤k≤p,1≤l≤3N and(gk,l)0≤k≤p,1≤l≤3N

of dieomorphisms inDiff^{∞}_{0} (M)with support included inB(0,2)such that:

h=Qp k=0

Q3N

l=1[fk,l, gk,l]

∀k, l∈[0, p]×[1,3N], d_{∞}(ϕ_{k}◦f_{k,l}◦ϕ^{−1}_{k} , Id_{R}N)< ^{0}_{ψ(n,k,l)}

∀k, l∈[0, p]×[1,3N], d_{∞}(ϕk◦gk,l◦ϕ^{−1}_{k} , Id_{R}N)< ^{0}_{ψ(n,k,l)}
.

Let us take a sequence(hn)_{n∈N}of dieomorphisms inDiff^{∞}_{0} (M)with:

∀n∈N, d∞(hn, IdM)< n.

By construction of the sequence(_{n})_{n∈}_{N}, there exist two nite sequences( ˜f_{n,k,l})
and(˜g_{n,k,l})associated to h_{n}. We consider then:

f_{ψ(n,k,l)}=ϕk◦f˜n,k,l◦ϕ^{−1}_{k}

gψ(n,k,l)=ϕk◦˜gn,k,l◦ϕ^{−1}_{k}

and we apply lemma 3 to the sequences (f_{n})_{n∈}_{N} et (g_{n})_{n∈}_{N} to conclude by
taking:

k_{n}=X

k,l

k^{0}_{ψ(n,k,l)}.

Proof of lemma 2 We use the following lemma which follows from a deep result by Kirby, Siebenmann and Quinn in the case of a dimension greater than 3(see [4] lemma 6.10, [10] and [11]):

Lemma 5 Consider two closed disks in S^{n} whose interiors cover S^{n}. Then
every homeomorphism hisotopic to the identity can be written as a product of
six dieomorphisms inHomeo0(S^{n})with support included in one of those disks.

Using this result, the same method as above with lemma 4 gives lemma 2.

5 Proof of lemmas 3 and 4

Here also, we follow the same method as in[1]. Proof of Lemma 3

Denote byF1a dieomorphism inDiff^{∞}_{0} (R^{N})which satises:

∀x∈B(0,2), F_{1}(x) =λx,
with0< λ <1.

LetF_{2} be a dieomorphism inDiff^{∞}_{0} (R^{N})supported inB(0,2)with:

∀x∈B(0,1), F2(x) =x+a, withkak<1.

LetF3 be a dieomorphism inDiff^{∞}_{0} (R^{N})supported inB(0,2)which satises:

• the points(F_{3}^{n}(0))_{n∈N}are pairwise distinct;

• the sequence(F_{3}^{n}(0))_{n∈N}converges tox0= (1,0, . . . ,0).

Consider a sequence of integers(ln)n∈Nwhich increases suciently fast so that:

• n6=n^{0}⇒F_{3}^{n}F_{1}^{l}^{n}(B(0,2))∩F_{3}^{n}^{0}F_{1}^{l}^{n}^{0}(B(0,2)) =∅.

• The diameter of the setF_{3}^{n}F_{1}^{l}^{n}(B(0,2))converges to 0.

ConsiderF_{n}=F_{3}^{n}F_{1}^{l}^{n},U_{n} =F_{3}^{n}F_{1}^{l}^{n}(B(0,2))andFˆ_{n} =F_{n}F_{2}F_{n}^{−1}.
For avery integern, take an open setV_{n} with:

• F_{3}^{n}(0)∈Vn⊂Un,

• Fˆn(Vn)∩Vn=∅,

and a sequence of integers(˜ln)_{n∈N}so that:

F_{3}^{n}F_{1}^{˜}^{l}^{n}(B(0,2))⊂V_{n}

Let us denote byF˜_{n} the dieomorphismF_{3}^{n}F_{1}^{˜}^{l}^{n}.

Take two sequences (f_{n})_{n∈}_{N} and (g_{n})_{n∈}_{N} of sieomorphisms in Diff^{r}_{0}(R^{N})
supported inB(0,2)and choose a sequence(^{0}_{n})_{n∈}_{N}which conserges suciently
fast to0so as to satisfy the following property: if, for every integern,dr(fn, Id)<

^{0}_{n} anddr(gn, Id)< ^{0}_{n}, then the dieomorphismsF4 andF5 dened by:

∀n∈N, F_{4|V}_{n}= ˜FnfnF˜_{n}^{−1}

∀n∈N, F_{5|V}_{n}= ˜FngnF˜_{n}^{−1}
F_{4|}_{R}N−S

n∈NV_{n}=F_{5|}_{R}N−S

n∈NV_{n} =Id_{R}N−S

n∈NV_{n}

areC^{r} inx0.
Notice that

A_{n} =F_{4}Fˆ_{n}F_{4}^{−1}Fˆ_{n}^{−1}

is supported inVn∪Fˆn(Vn), equalsF˜nfnF˜_{n}^{−1} on Vn andFˆnF˜nf_{n}^{−1}F˜_{n}^{−1}Fˆ_{n}^{−1} on
Fˆn(Vn). We can make an analogous statement for Bn = F5FˆnF_{5}^{−1}Fˆ_{n}^{−1} and
Cn =F_{4}^{−1}F_{5}^{−1}FˆnF5F4Fˆ_{n}^{−1}. Then:

AnBnCn= ˜Fn[fn, gn] ˜F_{n}^{−1}.
That is why:

[f_{n}, g_{n}] = ˜F_{n}^{−1}A_{n}B_{n}C_{n}F˜_{n}.
It suces to takeS={F_{1}, F_{2}, F_{3}, F_{4}, F_{5}}to end the proof.

Proof of lemma 4 Notice that, we need the sequence (n)_{n∈N} to make the
dieomorphismsF4 andF5 regular. In the case of C^{0} regularity, this problem
doesn't occur anymore, which allows us to show lemma 4 in the case where each
hn is a commutator. Hence, to conclude the proof of lemma 4, it suces to
show the following lemma.

Lemma 6 Every compactly supported homeomorphism of R^{N} isotopic to the
identity is a commutator.

Proof (taken from [3], proof of theorem 1.1.3) Denote byϕa homeomorphism
inHomeo0(R^{n})whose restriction toB(0,2) is:

B(0,2) → R^{N}
x 7→ ^{x}_{2} .

For any integer n, consider the set:

An =

x∈R^{N}, 1

2^{n+1} ≤ kxk ≤ 1
2^{n}

.

Lethbe a homeomorphism inHomeo0(R^{N}). As any dieomorphism inHomeo0(R^{N})
is conjugated to a homeomorphism supported in the interior ofA0, we may sup-
pose thathis supported in the interior ofA0. We dene the homeomorphismg
by:

• g=IdoutsideB(0,1).

• for anyi,g_{|A}_{i} =ϕ^{i}h_{i}ϕ^{−i}.

• g(0) = 0. Then we have:

h= [g, ϕ].

A Appendix: local perfection of Diff^{∞}_{0} (M)

In this appendix, we show the following result which was used during the proof of lemma 1:

Theorem 3 Let M be a compact connected n-dimensional manifold. Fix a
covering (Uk)_{0≤k≤p} of M by open sets whose closure is dieomorphic to the

unit closed ball in R^{n}. Then, for every neighbourhood Ω of the identity in
Diff^{∞}_{0} (M) there exists a neighbourhood Ω^{0} of the identity in Diff^{∞}_{0} (M) such
that, for any dieomorphismf in Ω^{0}, there exist nite sequences of dieomor-
phisms(fk,l)0≤k≤p,1≤l≤3n and(gk,l)0≤k≤p,1≤l≤3n in Ωsuch that:

f =

p

Y

k=0 3n

Y

l=1

[fk,l, gk,l]

and the dieomorphismsfk,l andgk,l are supported inUk.

This proof is an elementary realisation of Haller and Teichmann's idea to
write a dieomorphism as a product of dieomorphisms which preserve some
foliations (see [8]). This proof relies heavily on Herman's KAM theorem on
dieomorphisms of the circle. Notice that this property proves the perfectness
of the group Diff^{∞}_{0} (M), hence its simplicity: it gives a dierent proof of this
fact from Thurston's proof and Mather's proof (see [3] and [2]). The proof given
here also shows the local perfection ofDiff^{∞}_{0} (R^{n})forn≥2 but does not work
for the1-dimensional case.

By the fragmentation lemma (see [3] theorem 2.2.1 for a proof), for any
η >0, there existsα >0 such that, for every dieomorphismhinDiff^{∞}_{0} (M), if
d(h, IdM)< αthen there exists a nite sequence(hi)0≤i≤p of dieomorphisms
inDiff^{∞}_{0} (M)such that:

supp(hi)⊂Ui

h=Qp i=0hi

∀i∈[0, p]∩N, d(hi, IdM)< η .

Take two open setsU,V inR^{n}, whereUis a cube whose closure is included in
V and a neighbourhoodΩof the identity inDiff^{∞}_{0} (V). We will show the follow-
ing statement: there exists a neighbourhoodΩ^{0} of the identity inDiff^{∞}_{0} (U)such
that, for any dieomorphismf inΩ^{0} there exist dieomorphismsf_{1}, f_{2}, . . . , f_{3n}
inΩsuch that:

f = [f_{1}, f_{2}]◦[f_{3}, f_{4}]◦. . .◦[f_{3n−1}, f_{3n}].

To show this last property, the startegy will be as follows: we will start by writing a dieomorphism f close to the identity as a product of n dieomor- phisms which preserve each a foliation of U by straight lines. Each of these foliations can be extended as a foliation by circles of an annulus included inV. Then it will suce to apply Herman's theorem on dieomorphisms of the circle to conclude that each of the dieomorphisms appearing in this decomposition can be written as a product of three commutators of small dieomorphisms supported inV.

Let us detail this now. For any integer k between 1 and n, we denote by Fk the foliation by the straight lines parallel to thekth axis of coordinate. We

denote byDiff^{∞}_{0} (U, Fk)the group of compactly supported dieomorphisms ofU
compactly isotopic to the identity which preserve the foliationFk. Let us dene
by induction onka mapφk: Diff^{∞}_{0} (U)→Diff^{∞}_{0} (U, Fk)dened and continuous
on a neighbourhood of the identity such thatφk(IdU) =IdU and such that, for
a dieomorphismf suciently close te the identity:

pk◦f ◦φ1(f)^{−1}◦φ2(f)^{−1}◦. . .◦φk(f)^{−1}=pk,

where p_{k} is the kth projection of R^{n}. Suppose that φ_{1}, φ_{2}, . . . , φ_{k} have been
dened. For a dieomorphismf inDiff^{∞}_{0} (U)close to the identity and forxin
U, consider:

f◦φ1(f)^{−1}◦φ2(f)^{−1}◦. . .◦φk(f)^{−1}(x) = (x1, . . . , xk, fk+1(x), fk+2(x), . . . , fn(x)).

Then let us deneφk+1 the following way:

∀x∈U, φ_{k+1}(f)(x) = (x_{1}, . . . , x_{k}, f_{k+1}(x), x_{k+2}, . . . , x_{n})

forf suciently close to the identity. This ends the induction. The maps φk

areC^{∞}, map the identity to the identity and satisfy the following property:

f =φn(f)◦φ_{n−1}(f)◦. . .◦φ1(f).

Let us x an integerk. As planned, we consider an embedding:

ψk :R^{k−1}×R/Z×R^{n−k}→V

which maps(−1,1)^{k−1}×(−^{1}_{4},^{1}_{4})×(−1,1)^{n−k}intoU and the foliation by straight
lines parallel to thekth axis of coordinate of(−1,1)^{k−1}×(−^{1}_{4},^{1}_{4})×(−1,1)^{n−k}
on the foliation by straight lines parallel to thekth axis of coordinate ofU. The
dieomorphism ψ^{−1}_{k} ◦φk(f)◦ψk can be identied with a family (with n−1
parameters) of dieomorphisms of the circle. We write, for points (x, t, y) in
R^{k−1}×R/Z×R^{n−k}:

ψ_{k}^{−1}◦φk(f)◦ψk(x, t, y) = (x, g(x, y)(t), y),
whereg:R^{n−1}→Diff^{∞}_{0} (R/Z)is a continuous map with:

g(x, y) =Id
when(x, y)∈/ [−1,1]^{n−1}.

It remains to write g(x, y) as a product of commutators of elements in
Diff^{∞}_{0} (R/Z)close to the identity which depend in aC^{∞}way onxandy, whose
derivatives in thexandy directions are small and which are the identity when
(x, y)∈/ (−2,2)^{n−1}.

The following lemma was shown by Haller and Teichmann (see [9], example 3):

Lemma 7 For any neighbourhoodW^{0} of the identity inDiff^{∞}_{0} (R/Z), there ex-
ists a neighbourhoodW of the identity, dieomorphismsA,B andC inW^{0} and
C^{∞} maps

a, b, c:W →Diff^{∞}_{0} (R/Z)

which map the identity to the identity and satisfy, for any dieomorphismhin W:

h= [a(h), A][b(h), B][c(h), C].

Remark: the proof of this result relies on the KAM theorem by Herman on
dieomorphisms of the circle. Moreover, we may suppose thatA,B andC are
inP SL2(R)⊂Diff^{∞}_{0} (R/Z).

End of the proof of the theorem Denote by λ:R^{n−1} →R a C^{∞} map sup-
ported in(−2,2)^{n−1}which is equal to1on a neighbourhood of[−1,1]^{n−1}. Then
we choose a neighbourhoodW^{0} of the identity inDiff^{∞}_{0} (R/Z)such that the fol-
lowing property is satised. For any dieomorphismD in W^{0}, if we denote by
D˜ the map dened by:

∀(x, t, y)∈R^{k−1}×R/Z×R^{n−k}, D(x, t, y) = (x, λ(x, y)D(t) + (1˜ −λ(x, y))t, y),
then D˜ is a compactly supported dieomorphism of R^{k−1}×R/Z×R^{n−k} and
ψk◦D˜ ◦ψ_{k}^{−1} belongs to the neighbourhoodΩof the identity inDif f_{0}^{∞}(V).

When the dieomorphism f is suciently close to the identity, g(x, y) be-
longs to the neighbourhoodW of the identity given by the preceding lemma for
anyxandy. Then, for any(x, y)∈R^{k−1}×R^{n−k}:

g(x, y) = [a(g(x, y)), A][b(g(x, y)), B]◦[c(g(x, y)), C] and

g(x, y) = [a(g(x, y)),A(x, y)][b(g(x, y)),˜ B(x, y)]˜ ◦[c(g(x, y)),C(x, y)].˜
Indeed, on [−1,1]^{n−1}, we have A(x, y) =˜ A, B(x, y) =˜ B et C(x, y) =˜ C,
and outside [−1,1]^{n−1} both members of the equality equal the identity. That
is why the dieomorphism ψ_{k}^{−1}◦φk(f)◦ψk can be written as a product of 3
commutators of dieomorphisms inΩ, whenfis suciently close to the identity.

Thus a dieomorphism f close to the identity can be written as a product of 3ncommutators of dieomorphisms inΩ.

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