Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Conjugacy classes of pseudo-rotations and distortion
elements in groups of homeomorphisms of surfaces
Emmanuel Militon
École Polytechnique
10 avril 2014
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
1
Pseudo-rotations
2
Distortion elements in groups of homeomorphisms of surfaces
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
1
Pseudo-rotations
2
Distortion elements in groups of homeomorphisms of surfaces
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Lemma
Let
f ∈ Homeo0 (S1 )
α. Then f
Rα .
with rotation number
conjugates arbitrarily close to the rotation
Emmanuel Militon
has
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
e = R × [0, 1].
A = R/Z × [0, 1], A
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
e = R × [0, 1].
A = R/Z × [0, 1], A
Denition
A homeomorphism f in Homeo0 (A) is called a pseudo-rotation if
e →A
e of f such that, for any
there exists α in R and a lift f˜ : A
e
point x of A,
p1 (f˜n (x))
= α,
n→+∞
n
lim
where p1 : Ã = R × [0, 1] → R is the projection.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
e = R × [0, 1].
A = R/Z × [0, 1], A
Denition
A homeomorphism f in Homeo0 (A) is called a pseudo-rotation if
e →A
e of f such that, for any
there exists α in R and a lift f˜ : A
e
point x of A,
p1 (f˜n (x))
= α,
n→+∞
n
lim
where p1 : Ã = R × [0, 1] → R is the projection.
The class of α in R/Z is the angle of the pseudo-rotation f .
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
f2 = R2 .
T2 = R/Z × R/Z, T
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
f2 = R2 .
T2 = R/Z × R/Z, T
Denition
A homeomorphism f in Homeo0 (T2 ) is called a pseudo-rotation if
f2 → T
f2 of f such that, for any
there exists α in R2 and a lift f˜ : T
2
point x of R ,
f˜n (x)
= α.
n→+∞
n
lim
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
f2 = R2 .
T2 = R/Z × R/Z, T
Denition
A homeomorphism f in Homeo0 (T2 ) is called a pseudo-rotation if
f2 → T
f2 of f such that, for any
there exists α in R2 and a lift f˜ : T
2
point x of R ,
f˜n (x)
= α.
n→+∞
n
lim
The class of α in T2 is the angle of the pseudo-rotation f .
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Theorem (Béguin-Crovisier-Le Roux-Patou, M.)
Let
f
be a homeomorphism in
Homeo0 (A)
. The homeomorphism
is a pseudo-rotation of the annulus of angle
conjugates arbitrarily close to the rotation
Emmanuel Militon
α
if and only if
f
Rα .
pseudo-rotations and distortion elements
has
f
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Theorem (Béguin-Crovisier-Le Roux-Patou, M.)
Let
f
be a homeomorphism in
Homeo0 (A)
. The homeomorphism
is a pseudo-rotation of the annulus of angle
conjugates arbitrarily close to the rotation
Theorem (Kwapisz)
Let
f
be a homeomorphism in
torus of angle
the rotation
α
if and only if
f
f
if and only if
f
. Let
α ∈ R2 /Z2
be
is a pseudo-rotation of the
has conjugates arbitrarily close to
Rα .
Emmanuel Militon
f
has
Rα .
Homeo0 (T2 )
totally irrational. The homeomorphism
α
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Theorem (Béguin-Crovisier-Le Roux-Patou, M.)
Let
f
be a homeomorphism in
Homeo0 (A)
. The homeomorphism
is a pseudo-rotation of the annulus of angle
conjugates arbitrarily close to the rotation
Theorem (Kwapisz)
Let
f
be a homeomorphism in
torus of angle
the rotation
α
if and only if
f
f
if and only if
f
. Let
α ∈ R2 /Z2
be
is a pseudo-rotation of the
has conjugates arbitrarily close to
Rα .
Is it still true without hypothesis on α ?
Emmanuel Militon
f
has
Rα .
Homeo0 (T2 )
totally irrational. The homeomorphism
α
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let S be a compact surface and S̃ be its universal cover. Let
D ⊂ S̃ be a fundamental domain for the action of the group of
deck transformations. The surface S̃ is endowed with a "natural"
distance.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let S be a compact surface and S̃ be its universal cover. Let
D ⊂ S̃ be a fundamental domain for the action of the group of
deck transformations. The surface S̃ is endowed with a "natural"
distance.
Denition
A homeomorphism f in Homeo0 (S) is called a pseudo-rotation if
lim
n→+∞
diam(f˜n (D)) = 0.
Emmanuel Militon
n
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let S be a compact surface with ∂S 6= ∅ and which is dierent
from the annulus or the Möbius band.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let S be a compact surface with ∂S 6= ∅ and which is dierent
from the annulus or the Möbius band.
Theorem (M.)
Let
f ∈ Homeo0 (S).
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let S be a compact surface with ∂S 6= ∅ and which is dierent
from the annulus or the Möbius band.
Theorem (M.)
Let
f ∈ Homeo0 (S).
The homeomorphism
f
is a pseudo-rotation if
and only if it has conjugates arbitrarily close to the identity.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let S be a compact surface with ∂S 6= ∅ and which is dierent
from the annulus or the Möbius band.
Theorem (M.)
Let
f ∈ Homeo0 (S).
The homeomorphism
f
is a pseudo-rotation if
and only if it has conjugates arbitrarily close to the identity.
Is it still true when ∂S = ∅ ?
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
1
Pseudo-rotations
2
Distortion elements in groups of homeomorphisms of surfaces
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Denition
Let G be a nitely generated group and S be a nite generating set
of G . An element g of G is distorted or a distortion element if
lS (g n )
=0
n→+∞
n
lim
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Denition
Let G be a nitely generated group and S be a nite generating set
of G . An element g of G is distorted or a distortion element if
lS (g n )
=0
n→+∞
n
lim
lS (g n )
= 0.
n→+∞
n
⇔ lim inf
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Denition
Let G be a nitely generated group and S be a nite generating set
of G . An element g of G is distorted or a distortion element if
lS (g n )
=0
n→+∞
n
lim
lS (g n )
= 0.
n→+∞
n
⇔ lim inf
Proposition
ϕ : G → H be
ϕ(g ) is distorted.
Let
a group morphism. If
Emmanuel Militon
g ∈G
is distorted then
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Denition
Let G be a nitely generated group and S be a nite generating set
of G . An element g of G is distorted or a distortion element if
lS (g n )
=0
n→+∞
n
lim
lS (g n )
= 0.
n→+∞
n
⇔ lim inf
Proposition
ϕ : G → H be
ϕ(g ) is distorted.
Let
a group morphism. If
g ∈G
is distorted then
Denition
Let G be any group. An element g of G is distorted if it belongs to
a nitely generated subgroup of G in which it is distorted.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Lemma
Let
S
be a compact surface. If
Homeo0 (S)
, then
f
f
is a distortion element in
is a pseudo-rotation.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let M be a compact manifold.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let M be a compact manifold.
Proposition
Let
f ∈ Homeo0 (M).
If the homeomorphism
arbitrarily close to a distortion element in
distorted in
Homeo0 (M)
f
has conjugates
Homeo0 (M)
, then
f
.
Emmanuel Militon
pseudo-rotations and distortion elements
is
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Let M be a compact manifold.
Proposition
Let
f ∈ Homeo0 (M).
If the homeomorphism
arbitrarily close to a distortion element in
distorted in
Homeo0 (M)
f
has conjugates
Homeo0 (M)
, then
f
is
.
Corollary
Let
S
be a compact orientable surface with
Homeo0 (S)
∂S 6= ∅.
An element of
is a distortion element if and only if it is a
pseudo-rotation.
Emmanuel Militon
pseudo-rotations and distortion elements
Pseudo-rotations
Distortion elements in groups of homeomorphisms of surfaces
Theorem
Let
S
be a compact orientable surface and
Suppose
lim
n→+∞
Then
f
is distorted in
f ∈ Homeo0 (S).
diam(f˜n (D)) = 0.
n
log (n)
Homeo0 (S)
Emmanuel Militon
.
pseudo-rotations and distortion elements