The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The...
Transcript of The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The...
![Page 1: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/1.jpg)
The Furstenberg boundary and C-star simplegroups
Emmanuel Breuillard, joint work with M. Kalantar, M. Kennedyand N. Ozawa
Universite Paris-Sud, Orsay, France
Lyon, June 29th, 2015
![Page 2: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/2.jpg)
Non abelian Fourier analysis
G a group.
A linear representation of G is a group homomorphism:
G → GL(V )
V is a vector space over C. It is called irreducible if 0 and V arethe only G -invariant subspaces.
... representations are key to the understanding of the group Gfrom the algebraic point of view ... and the also the analytic pointof view.
![Page 3: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/3.jpg)
Non abelian Fourier analysis
G a group.
A linear representation of G is a group homomorphism:
G → GL(V )
V is a vector space over C. It is called irreducible if 0 and V arethe only G -invariant subspaces.
... representations are key to the understanding of the group Gfrom the algebraic point of view ... and the also the analytic pointof view.
![Page 4: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/4.jpg)
Non abelian Fourier analysis
Example 1: if G = (R/Z,+) the circle group, or one-dimensionaltorus.
Irreducible representations of G are the one-dimensional charactersπn, for n ∈ Z defined by:
πn : G → GL1(C)
x 7→ e−2iπnx
![Page 5: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/5.jpg)
Non abelian Fourier analysis
Example 1: if G = (R/Z,+) the circle group, or one-dimensionaltorus.
Irreducible representations of G are the one-dimensional charactersπn, for n ∈ Z defined by:
πn : G → GL1(C)
x 7→ e−2iπnx
Fourier analysis tells us that functions on G can berepresented by linear combinations of characters.
![Page 6: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/6.jpg)
Non abelian Fourier analysis
Example 1: if G = (R/Z,+) the circle group, or one-dimensionaltorus.
Irreducible representations of G are the one-dimensional charactersπn, for n ∈ Z defined by:
πn : G → GL1(C)
x 7→ e−2iπnx
Namely, we have the Fourier inversion formula forf : R/Z→ C :
f (x) =∑n∈Z
fn(x) =∑n∈Z
f (n)πn(−x),
where f (n) :=∫R/Z f (x)πn(x)dx is the Fourier transform of f .
![Page 7: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/7.jpg)
Non abelian Fourier analysis
Example 2: Now assume that G is a finite group.
Irreducible representations of G are finite-dimensional, there is onefor each conjugacy class of G , and the Fourier inversion formulareads, for f : G → C :
f (x) =∑π∈G
fπ(x)dπ|G |
=∑π∈G
〈f (π), π(x)〉 dπ|G |
,
where
![Page 8: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/8.jpg)
Non abelian Fourier analysis
Example 2: Now assume that G is a finite group.
Irreducible representations of G are finite-dimensional, there is onefor each conjugacy class of G , and the Fourier inversion formulareads, for f : G → C :
f (x) =∑π∈G
fπ(x)dπ|G |
=∑π∈G
〈f (π), π(x)〉 dπ|G |
,
where
• dπ is an integer: the dimension of the representation spaceof π.
• G is the set of (equivalence classes of) irreduciblerepresentations of G ,
![Page 9: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/9.jpg)
Non abelian Fourier analysis
Example 2: Now assume that G is a finite group.
Irreducible representations of G are finite-dimensional, there is onefor each conjugacy class of G , and the Fourier inversion formulareads, for f : G → C :
f (x) =∑π∈G
fπ(x)dπ|G |
=∑π∈G
〈f (π), π(x)〉 dπ|G |
,
where
• f (π) := π(f ) =∑
x∈G f (x)π(x) is an operator on therepresentation space of π.
• the scalar product is 〈A,B〉 = Tr(AB∗).
![Page 10: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/10.jpg)
Non abelian Fourier analysis
Example 2 continued: cards shuffling
Suppose G acts transitively on a finite set X , i.e. G → Sym(X ),and let µ be a probability measure on G .
This gives rise to a random walk on X : jump from x ∈ X to gxwith probability µ(g).
![Page 11: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/11.jpg)
Non abelian Fourier analysis
Example 2 continued: cards shuffling
Suppose G acts transitively on a finite set X , i.e. G → Sym(X ),and let µ be a probability measure on G .
This gives rise to a random walk on X : jump from x ∈ X to gxwith probability µ(g).
Basic question: How fast does the walk approach equilibrium?
Answer: depends on the size of ‖π(µ)‖, for the irreduciblesubrepresentations π of `2(X ).
![Page 12: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/12.jpg)
Non abelian Fourier analysis
Example 2 continued: cards shuffling
Suppose G acts transitively on a finite set X , i.e. G → Sym(X ),and let µ be a probability measure on G .
This gives rise to a random walk on X : jump from x ∈ X to gxwith probability µ(g).
Indeed by Fourier inversion, for f : X → C.
∫Gf (gx)dµn(g) =
∑π∈G
〈f (·x)(π), π(µ)n〉 dπ|G |
=1
|X |∑y∈X
f (y) +∑
π∈G\{1}
〈f (·x)(π), π(µ)n〉 dπ|G |
![Page 13: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/13.jpg)
Non abelian Fourier analysis
What about more general (say locally compact separable) groups ?
→ restrict attention to unitary representations of G , i.e.(continuous) homomorphisms
G → U(H),
where H is a Hilbert space, and U(H) the group of unitaryisomorphisms of H.
Let G be the unitary dual of G (equivalence classes of irreducibleunitary representations). It has a canonical Borel structure(Mackey).
→ We would like to have a Fourier inversion formula for G .
![Page 14: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/14.jpg)
Non abelian Fourier analysis
What about more general (say locally compact separable) groups ?
→ restrict attention to unitary representations of G , i.e.(continuous) homomorphisms
G → U(H),
where H is a Hilbert space, and U(H) the group of unitaryisomorphisms of H.
Let G be the unitary dual of G (equivalence classes of irreducibleunitary representations). It has a canonical Borel structure(Mackey).
→ We would like to have a Fourier inversion formula for G .
![Page 15: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/15.jpg)
Non abelian Fourier analysis
What about more general (say locally compact separable) groups ?
→ restrict attention to unitary representations of G , i.e.(continuous) homomorphisms
G → U(H),
where H is a Hilbert space, and U(H) the group of unitaryisomorphisms of H.
Let G be the unitary dual of G (equivalence classes of irreducibleunitary representations). It has a canonical Borel structure(Mackey).
→ We would like to have a Fourier inversion formula for G .
![Page 16: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/16.jpg)
Non abelian Fourier analysis
What about more general (say locally compact separable) groups ?
→ restrict attention to unitary representations of G , i.e.(continuous) homomorphisms
G → U(H),
where H is a Hilbert space, and U(H) the group of unitaryisomorphisms of H.
Let G be the unitary dual of G (equivalence classes of irreducibleunitary representations). It has a canonical Borel structure(Mackey).
→ We would like to have a Fourier inversion formula for G .
![Page 17: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/17.jpg)
Non abelian Fourier analysis
→ We would like to have a Fourier inversion formula for G .
Works well for compact groups (Peter-Weyl), abelian locallycompact groups (Pontryagin dual), and more generally for the
groups of type 1 = groups for which G is countably separated.
For these groups we have a Fourier inversion formula, forf : G → C (nice enough):
f (x) =
∫Gfπ(x)dµ(π)
where fπ(x) := Tr(π(f )π(x)∗), and dµ is a Borel measure on G . Itis called the Plancherel measure and is unique.
![Page 18: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/18.jpg)
Non abelian Fourier analysis
→ We would like to have a Fourier inversion formula for G .
Works well for compact groups (Peter-Weyl), abelian locallycompact groups (Pontryagin dual), and more generally for the
groups of type 1 = groups for which G is countably separated.
For these groups we have a Fourier inversion formula, forf : G → C (nice enough):
f (x) =
∫Gfπ(x)dµ(π)
where fπ(x) := Tr(π(f )π(x)∗), and dµ is a Borel measure on G . Itis called the Plancherel measure and is unique.
![Page 19: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/19.jpg)
Non abelian Fourier analysis
Many groups are type 1 (compact, abelian locally compact,algebraic groups over local fields, etc)... but many are not.
In fact if G is a discrete countable group, G is type 1 if and only ifG is virtually abelian (Thoma 1964).
![Page 20: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/20.jpg)
Non abelian Fourier analysis
Many groups are type 1 (compact, abelian locally compact,algebraic groups over local fields, etc)... but many are not.
In fact if G is a discrete countable group, G is type 1 if and only ifG is virtually abelian (Thoma 1964).
![Page 21: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/21.jpg)
Weak containment
Recall: A unitary representation π is said to be weakly contained ina unitary representation σ, if matrix coefficients of π can beapproximated uniformly on compact sets by convex combinationsof matrix coefficients of σ. Notation: π ≺ σ.
![Page 22: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/22.jpg)
Weak containment
Recall: A unitary representation π is said to be weakly contained ina unitary representation σ, if matrix coefficients of π can beapproximated uniformly on compact sets by convex combinationsof matrix coefficients of σ. Notation: π ≺ σ.
matrix coefficient = a function on G on the formg 7→ 〈π(g)v ,w〉 for vectors v ,w ∈ Hπ)
• The support of the Plancherel measure is precisely the setof irreducible representations that are weakly contained in theregular representation λG , namely the action of G by lefttranslations on L2(G ,Haar).
![Page 23: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/23.jpg)
Weak containment
Recall: A unitary representation π is said to be weakly contained ina unitary representation σ, if matrix coefficients of π can beapproximated uniformly on compact sets by convex combinationsof matrix coefficients of σ. Notation: π ≺ σ.
• A consequence of the Fourier inversion formula is that wehave decomposed λG into irreducibles:
λG =
∫Xπxdm(x)
![Page 24: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/24.jpg)
Weak containment
Recall: A unitary representation π is said to be weakly contained ina unitary representation σ, if matrix coefficients of π can beapproximated uniformly on compact sets by convex combinationsof matrix coefficients of σ. Notation: π ≺ σ.
• G is amenable if the trivial representation of G (equivalentlyany irreducible rep.) is weakly contained in the regularrepresentation λG .
![Page 25: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/25.jpg)
Weak containment
Recall: A unitary representation π is said to be weakly contained ina unitary representation σ, if matrix coefficients of π can beapproximated uniformly on compact sets by convex combinationsof matrix coefficients of σ. Notation: π ≺ σ.
• G has Kazhdan’s property (T ) if the trivial representation ofG (equivalently any irreducible rep.) is weakly contained in nounitary representation without non-zero G -invariant vector.
![Page 26: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/26.jpg)
Weak containment
Recall: A unitary representation π is said to be weakly contained ina unitary representation σ, if matrix coefficients of π can beapproximated uniformly on compact sets by convex combinationsof matrix coefficients of σ. Notation: π ≺ σ.
• The condition of weak containment π ≺ σ is equivalent tothe condition ‖π(f )‖ 6 ‖σ(f )‖ for every f ∈ Cc(G ).
![Page 27: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/27.jpg)
Weak containment
Example 3: G is the free group on 2 generators.
![Page 28: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/28.jpg)
Weak containment
Example 3: G is the free group on 2 generators.
Given x ∈ G \ {1}, one can restrict f ∈ `2(G ) to each coset of the cyclicsubgroup 〈x〉 and perform ordinary Fourier transform on this cyclicsubgroup.
Get a decomposition:
λG =
∫R/Z
IndG〈x〉χtdt,
where χt : 〈x〉 ' Z→ GL1(C) is the character χt(xn) = e2iπnt .
![Page 29: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/29.jpg)
Weak containment
Example 3: G is the free group on 2 generators.
Get a decomposition:
λG =
∫R/Z
IndG〈x〉χtdt,
where χt : 〈x〉 ' Z→ GL1(C) is the character χt(xn) = e2iπnt .
Let CG (x) be the centralizer of x in G .
Mackey: for x , y ∈ G \ {1}, and s, t ∈ R/Z,
I IndG〈x〉χt is irreducible, and
I if CG (x) 6= CG (y), then IndG〈x〉χt is not equivalent to IndG
〈y〉χs .
![Page 30: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/30.jpg)
Weak containment
Example 3: G is the free group on 2 generators.
Mackey: for x , y ∈ G \ {1}, and s, t ∈ R/Z,
I IndG〈x〉χt is irreducible, and
I if CG (x) 6= CG (y), then IndG〈x〉χt is not equivalent to IndG
〈y〉χs .
So if x and y do not commute, we obtain 2 distinct decompositions ofλG = `2(G ) with disjoint supports on G !
λG =
∫R/Z
πt,xdt =
∫R/Z
πs,yds
where πt,x := IndG〈x〉χt .
![Page 31: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/31.jpg)
C ∗-simple groupsSuppose G is a countable discrete group.
DefinitionG is said to be C ∗-simple, if every unitary representation of G ,which is weakly contained in the regular representation λG isweakly equivalent to λG .
Remarks:
![Page 32: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/32.jpg)
C ∗-simple groupsSuppose G is a countable discrete group.
DefinitionG is said to be C ∗-simple, if every unitary representation of G ,which is weakly contained in the regular representation λG isweakly equivalent to λG .
Remarks:
It is the opposite of type 1, in a sense : only the trivial groupis type 1 and C ∗-simple among discrete groups.
[ non discrete C ∗-simple locally compact groups exist, butthey are totally disconnected (S. Raum 2015). ]
![Page 33: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/33.jpg)
C ∗-simple groupsSuppose G is a countable discrete group.
DefinitionG is said to be C ∗-simple, if every unitary representation of G ,which is weakly contained in the regular representation λG isweakly equivalent to λG .
Remarks:
It is equivalent to the simplicity (= no non-trivial closed∗-invariant bi-submodule) of the reduced C ∗-algebra C ∗λ(G )of the group G ,
C ∗λ(G ) = closure of the group algebra C[G ] when viewed as a
subalgebra of operators on `2(G ) acting by convolution.
![Page 34: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/34.jpg)
C ∗-simple groupsSuppose G is a countable discrete group.
DefinitionG is said to be C ∗-simple, if every unitary representation of G ,which is weakly contained in the regular representation λG isweakly equivalent to λG .
Remarks:
If G has a non-trivial normal amenable subgroup N, then G isnot C ∗-simple: λG/N = `2(G/N) is weakly contained in λG ,but not weakly equivalent.
(matrix coefficients of λG/N are N-invariant, while those ofλG are not)
![Page 35: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/35.jpg)
C ∗-simple groupsSuppose G is a countable discrete group.
DefinitionG is said to be C ∗-simple, if every unitary representation of G ,which is weakly contained in the regular representation λG isweakly equivalent to λG .
Remarks:
If G has a non-trivial normal amenable subgroup N, then G isnot C ∗-simple: λG/N = `2(G/N) is weakly contained in λG ,but not weakly equivalent.
(matrix coefficients of λG/N are N-invariant, while those ofλG are not)
So if G is C ∗-simple, its amenable radical Rad(G ) (= largestamenable normal subgroup) is trivial.
![Page 36: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/36.jpg)
C ∗-simple groupsSuppose G is a countable discrete group.
DefinitionG is said to be C ∗-simple, if every unitary representation of G ,which is weakly contained in the regular representation λG isweakly equivalent to λG .
Remarks:
So if G is C ∗-simple, its amenable radical Rad(G ) (= largestamenable normal subgroup) is trivial.
OPEN PROBLEM: Does the converse hold ?
![Page 37: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/37.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 38: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/38.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 39: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/39.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 40: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/40.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 41: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/41.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 42: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/42.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 43: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/43.jpg)
C ∗-simple groups
Examples of C ∗-simple groups:The following groups (after possibly moding out the amenableradical) are known to be C ∗-simple
I Non-abelian free groups (Powers 1974).
I Lattices in real semisimple algebraic groups(Bekka-Cowling-de la Harpe 1994).
I Mapping class groups, outer automorphims of free groups(Bridson - de la Harpe 2004).
I Relatively hyperbolic groups (Arzhantseva-Minasyan 2007)
I Linear groups (Poznansky 2008).
I Baumslag-Solitar groups (de la Harpe-Preaux 2011)
I Free Burnside groups of large odd exponent (Osin-Olshanskii2014).
![Page 44: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/44.jpg)
C ∗-simple groups
Not known: whether Thompson’s group T is C ∗-simple ?
It is known to be simple as an abstract group.
![Page 45: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/45.jpg)
C ∗-simple groups
Not known: whether Thompson’s group T is C ∗-simple ?
It is known to be simple as an abstract group.
![Page 46: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/46.jpg)
C ∗-simple groups
Proofs were based on Powers’ original idea:
Powers’ lemma: Assume that ∀ε > 0 and for every finite setF ⊂ G \ {1} one can find group elements g1, . . . , gk such that
‖λG (µx)‖ 6 ε,
for each x ∈ F , where
µx :=1
k
k∑1
δgixg−1i.
Then G is C ∗-simple.
![Page 47: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/47.jpg)
C ∗-simple groupsPowers’ lemma: Assume that ∀ε > 0 and for every finite setF ⊂ G \ {1} one can find group elements g1, . . . , gk such that
‖λG (µx)‖ 6 ε,
for each x ∈ F , where
µx :=1
2k
k∑1
δgixg−1i
+ δgix−1g−1i.
Then G is C ∗-simple.
![Page 48: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/48.jpg)
C ∗-simple groupsPowers’ lemma: Assume that ∀ε > 0 and for every finite setF ⊂ G \ {1} one can find group elements g1, . . . , gk such that
‖λG (µx)‖ 6 ε,
for each x ∈ F , where
µx :=1
2k
k∑1
δgixg−1i
+ δgix−1g−1i.
Then G is C ∗-simple.
For example: if the gi ’s can be chosen so that g1xg−11 , . . . , gkxg
−1k
generate a free subgroup, then (Kesten 1959),
‖λG (µx)‖ =
√2k − 1
k6 1/
√2k .
![Page 49: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/49.jpg)
C ∗-simple groupsPowers’ lemma: Assume that ∀ε > 0 and for every finite setF ⊂ G \ {1} one can find group elements g1, . . . , gk such that
‖λG (µx)‖ 6 ε,
for each x ∈ F , where
µx :=1
2k
k∑1
δgixg−1i
+ δgix−1g−1i.
Then G is C ∗-simple.
For linear groups one can use Random Matrix Products to achievethis (see Aoun’s thesis) : set gi = S i
n, where S1n , ...,S
kn are
independent random matrix products ; then ‖µx‖ 6 2/√k with
probability → 1 as n→∞.
![Page 50: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/50.jpg)
The Furstenberg boundary
![Page 51: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/51.jpg)
The Furstenberg boundary
Recently Merhdad Kalantar and Matt Kennedy found a newcriterion for C ∗-simplicity. It is phrased in dynamical terms.
![Page 52: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/52.jpg)
The Furstenberg boundary
Furstenberg (1973) introduced the following notion:
Definition (G -boundary)
A compact Hausdorff G -space X is called a G -boundary, if itis:
I minimal (every G -orbit is dense), and
I strongly proximal (every probability measure on X admitsa Dirac mass in the closure of its G -orbit).
![Page 53: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/53.jpg)
The Furstenberg boundary
Furstenberg (1973) introduced the following notion:
Definition (G -boundary)
A compact Hausdorff G -space X is called a G -boundary, if itis:
I minimal (every G -orbit is dense), and
I strongly proximal (every probability measure on X admitsa Dirac mass in the closure of its G -orbit).
He showed that there is a (unique up to isomorphism)universal boundary associated to every locally compact group,that is a G -boundary B(G ) = ∂FG , such that everyG -boundary is an equivariant image of ∂FG .
![Page 54: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/54.jpg)
The Furstenberg boundary
Furstenberg (1973) introduced the following notion:
Definition (G -boundary)
A compact Hausdorff G -space X is called a G -boundary, if itis:
I minimal (every G -orbit is dense), and
I strongly proximal (every probability measure on X admitsa Dirac mass in the closure of its G -orbit).
This universal boundary is now called the Furstenbergboundary of the group G .
![Page 55: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/55.jpg)
The Furstenberg boundary
Furstenberg (1973) introduced the following notion:
Definition (G -boundary)
A compact Hausdorff G -space X is called a G -boundary, if itis:
I minimal (every G -orbit is dense), and
I strongly proximal (every probability measure on X admitsa Dirac mass in the closure of its G -orbit).
This universal boundary is now called the Furstenbergboundary of the group G .
For example if G is a real semisimple Lie group, ∂FG = G/P,where P is a minimal parabolic subgroup. This notion wasimportant in Margulis’ proof of his super-rigidity theorem.
![Page 56: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/56.jpg)
The Furstenberg boundary
If G is amenable, then ∂FG is trivial. In fact the kernel of theG -action on ∂FG is precisely the amenable radical (Furman 2003).
If G is discrete and not amenable, ∂FG is huge (not metrizable).
Theorem (Kalantar-Kennedy 2014)
If G is discrete, then the Furstenberg boundary ∂FG is anextremally disconnected space (i.e. open sets have open closures).
idea:
![Page 57: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/57.jpg)
The Furstenberg boundary
If G is amenable, then ∂FG is trivial. In fact the kernel of theG -action on ∂FG is precisely the amenable radical (Furman 2003).
If G is discrete and not amenable, ∂FG is huge (not metrizable).
Theorem (Kalantar-Kennedy 2014)
If G is discrete, then the Furstenberg boundary ∂FG is anextremally disconnected space (i.e. open sets have open closures).
idea:
![Page 58: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/58.jpg)
The Furstenberg boundary
If G is amenable, then ∂FG is trivial. In fact the kernel of theG -action on ∂FG is precisely the amenable radical (Furman 2003).
If G is discrete and not amenable, ∂FG is huge (not metrizable).
Theorem (Kalantar-Kennedy 2014)
If G is discrete, then the Furstenberg boundary ∂FG is anextremally disconnected space (i.e. open sets have open closures).
idea:
• Andrew Gleason (1958) showed that extremallydisconnected compact Hausdorff spaces are precisely theprojective objects among compact Hausdorff spaces (recall: Xis projective if for given Y � Z , any map to Z lifts to Y .)
![Page 59: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/59.jpg)
The Furstenberg boundary
If G is amenable, then ∂FG is trivial. In fact the kernel of theG -action on ∂FG is precisely the amenable radical (Furman 2003).
If G is discrete and not amenable, ∂FG is huge (not metrizable).
Theorem (Kalantar-Kennedy 2014)
If G is discrete, then the Furstenberg boundary ∂FG is anextremally disconnected space (i.e. open sets have open closures).
idea:
• By duality X is projective iff C (X ) is injective as aC ∗-algebra.
![Page 60: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/60.jpg)
The Furstenberg boundary
If G is amenable, then ∂FG is trivial. In fact the kernel of theG -action on ∂FG is precisely the amenable radical (Furman 2003).
If G is discrete and not amenable, ∂FG is huge (not metrizable).
Theorem (Kalantar-Kennedy 2014)
If G is discrete, then the Furstenberg boundary ∂FG is anextremally disconnected space (i.e. open sets have open closures).
idea:
• The boundary map ∂FG → P(βG ) induces a G -equivariantretraction r := `∞(G ) = C (βG )� C (∂FG ). So injectivity ofC (∂FG ) follows from that of `∞G .
![Page 61: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/61.jpg)
The Furstenberg boundary
Corollary
If x ∈ ∂FG , then StabG (x) is amenable.
idea: the composition ex ◦ r is a StabG (x)-invariant positive functional.
Question: is StabG (x) trivial ?
Theorem (Kalantar-Kennedy, 2014)
G is C ∗-simple if and only if it acts freely on ∂FG .
Remarks:• In an extremally disconnected compact Hausdorff G -space, fixpoints sets Fix(g), g ∈ G , are clopen (= closed and open).
• So G acts freely on ∂FG if and only if it acts topologically freely(i.e. Fix(g) has empty interior).
• Consequence: If there exists some G -boundary on which G actstopologically freely, then G is C ∗-simple.
![Page 62: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/62.jpg)
The Furstenberg boundary
Corollary
If x ∈ ∂FG , then StabG (x) is amenable.
idea: the composition ex ◦ r is a StabG (x)-invariant positive functional.
Question: is StabG (x) trivial ?
Theorem (Kalantar-Kennedy, 2014)
G is C ∗-simple if and only if it acts freely on ∂FG .
Remarks:• In an extremally disconnected compact Hausdorff G -space, fixpoints sets Fix(g), g ∈ G , are clopen (= closed and open).
• So G acts freely on ∂FG if and only if it acts topologically freely(i.e. Fix(g) has empty interior).
• Consequence: If there exists some G -boundary on which G actstopologically freely, then G is C ∗-simple.
![Page 63: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/63.jpg)
The Furstenberg boundary
Corollary
If x ∈ ∂FG , then StabG (x) is amenable.
idea: the composition ex ◦ r is a StabG (x)-invariant positive functional.
Question: is StabG (x) trivial ?
Theorem (Kalantar-Kennedy, 2014)
G is C ∗-simple if and only if it acts freely on ∂FG .
Remarks:• In an extremally disconnected compact Hausdorff G -space, fixpoints sets Fix(g), g ∈ G , are clopen (= closed and open).
• So G acts freely on ∂FG if and only if it acts topologically freely(i.e. Fix(g) has empty interior).
• Consequence: If there exists some G -boundary on which G actstopologically freely, then G is C ∗-simple.
![Page 64: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/64.jpg)
The Furstenberg boundary
Corollary
If x ∈ ∂FG , then StabG (x) is amenable.
idea: the composition ex ◦ r is a StabG (x)-invariant positive functional.
Question: is StabG (x) trivial ?
Theorem (Kalantar-Kennedy, 2014)
G is C ∗-simple if and only if it acts freely on ∂FG .
Remarks:• In an extremally disconnected compact Hausdorff G -space, fixpoints sets Fix(g), g ∈ G , are clopen (= closed and open).
• So G acts freely on ∂FG if and only if it acts topologically freely(i.e. Fix(g) has empty interior).
• Consequence: If there exists some G -boundary on which G actstopologically freely, then G is C ∗-simple.
![Page 65: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/65.jpg)
The Furstenberg boundary
Corollary
If x ∈ ∂FG , then StabG (x) is amenable.
idea: the composition ex ◦ r is a StabG (x)-invariant positive functional.
Question: is StabG (x) trivial ?
Theorem (Kalantar-Kennedy, 2014)
G is C ∗-simple if and only if it acts freely on ∂FG .
Remarks:• In an extremally disconnected compact Hausdorff G -space, fixpoints sets Fix(g), g ∈ G , are clopen (= closed and open).
• So G acts freely on ∂FG if and only if it acts topologically freely(i.e. Fix(g) has empty interior).
• Consequence: If there exists some G -boundary on which G actstopologically freely, then G is C ∗-simple.
![Page 66: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/66.jpg)
The Furstenberg boundary
Corollary
If x ∈ ∂FG , then StabG (x) is amenable.
idea: the composition ex ◦ r is a StabG (x)-invariant positive functional.
Question: is StabG (x) trivial ?
Theorem (Kalantar-Kennedy, 2014)
G is C ∗-simple if and only if it acts freely on ∂FG .
Remarks:• In an extremally disconnected compact Hausdorff G -space, fixpoints sets Fix(g), g ∈ G , are clopen (= closed and open).
• So G acts freely on ∂FG if and only if it acts topologically freely(i.e. Fix(g) has empty interior).
• Consequence: If there exists some G -boundary on which G actstopologically freely, then G is C ∗-simple.
![Page 67: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/67.jpg)
No amenable normalish subgroup criterion
Definition (Normalish subgroup)
A subgroup H 6 G is said to be normalish, if ∩g∈FgHg−1 isinfinite for every finite subset F ⊂ G .
Theorem (BKKO 2014)
If G has no normal finite subgroup and no amenable normalishsubgroup, then G is C ∗-simple.
![Page 68: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/68.jpg)
No amenable normalish subgroup criterion
Definition (Normalish subgroup)
A subgroup H 6 G is said to be normalish, if ∩g∈FgHg−1 isinfinite for every finite subset F ⊂ G .
Theorem (BKKO 2014)
If G has no normal finite subgroup and no amenable normalishsubgroup, then G is C ∗-simple.
• Linear groups without amenable radical satisfy the abovecriterion.
![Page 69: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/69.jpg)
No amenable normalish subgroup criterion
Definition (Normalish subgroup)
A subgroup H 6 G is said to be normalish, if ∩g∈FgHg−1 isinfinite for every finite subset F ⊂ G .
Theorem (BKKO 2014)
If G has no normal finite subgroup and no amenable normalishsubgroup, then G is C ∗-simple.
• Linear groups without amenable radical satisfy the abovecriterion.• So do groups with a positive `2-Betti number (Thom,Bader-Furman-Sauer), groups with non-vanishing boundedcohomology.
![Page 70: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/70.jpg)
No amenable normalish subgroup criterion
Definition (Normalish subgroup)
A subgroup H 6 G is said to be normalish, if ∩g∈FgHg−1 isinfinite for every finite subset F ⊂ G .
Theorem (BKKO 2014)
If G has no normal finite subgroup and no amenable normalishsubgroup, then G is C ∗-simple.
• Linear groups without amenable radical satisfy the abovecriterion.• So do groups with a positive `2-Betti number (Thom,Bader-Furman-Sauer), groups with non-vanishing boundedcohomology.• We recover this way essentially all previously known cases.
![Page 71: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/71.jpg)
No amenable normalish subgroup criterion
Definition (Normalish subgroup)
A subgroup H 6 G is said to be normalish, if ∩g∈FgHg−1 isinfinite for every finite subset F ⊂ G .
Theorem (BKKO 2014)
If G has no normal finite subgroup and no amenable normalishsubgroup, then G is C ∗-simple.
![Page 72: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/72.jpg)
No amenable normalish subgroup criterion
Definition (Normalish subgroup)
A subgroup H 6 G is said to be normalish, if ∩g∈FgHg−1 isinfinite for every finite subset F ⊂ G .
Theorem (BKKO 2014)
If G has no normal finite subgroup and no amenable normalishsubgroup, then G is C ∗-simple.
Point is: if G does not act topologically freely on ∂FG , thenStabG (x) is amenable and normalish.
![Page 73: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/73.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
![Page 74: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/74.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
Observation: If G has (CS), then G/Rad(G ) has no amenablenormalish subgroup, so it is C ∗-simple.
![Page 75: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/75.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
Observation: If G has (CS), then G/Rad(G ) has no amenablenormalish subgroup, so it is C ∗-simple.
indeed: • if H is normalish, then given an arbitrary finite setF ⊂ G/H, there is a non trivial element in G fixing each x ∈ F .
• if H is amenable then λG/H ≺ λG .
![Page 76: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/76.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
why (CS) ? Connes and Sullivan had conjectured in the early80’s that a countable dense subgroup G of connected Liegroup G acts amenably on it iff the Lie group G is solvable.
![Page 77: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/77.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
why (CS) ? Connes and Sullivan had conjectured in the early80’s that a countable dense subgroup G of connected Liegroup G acts amenably on it iff the Lie group G is solvable.This was shown by Carriere-Ghys in 1985 for dense subgroups of
SL2(R), and later by Zimmer in full generality.
![Page 78: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/78.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
why (CS) ? Connes and Sullivan had conjectured in the early80’s that a countable dense subgroup G of connected Liegroup G acts amenably on it iff the Lie group G is solvable.if G 6 G acts amenably on G, then (λG)|G ≺ λG , but (λG)|G not
discrete unless G is solvable.
![Page 79: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/79.jpg)
Connes-Sullivan propertyB.+ Ozawa (2015) : In fact linear groups, and groups with nontrivial bounded cohomology, verify a stronger property:
Definition: Say that a discrete group G has the Connes-Sullivanproperty (CS) if for every unitary representation π of G , if
π ≺ λ⇒ π is discrete.
i.e. if gn ∈ G s.t. π(gn)→ 1 in strong operator topology (i.e.π(gn)v → v for each v), then gn ∈ Rad(G ) eventually.
why (CS) ? Connes and Sullivan had conjectured in the early80’s that a countable dense subgroup G of connected Liegroup G acts amenably on it iff the Lie group G is solvable.Proof of (CS) for linear groups is a consequence of the strong Tits
alternative (Breuillard-Gelander 2006).
![Page 80: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/80.jpg)
Further consequences
Exploiting the KK dynamical criterion, we further show:
I if G has only countably many amenable subgroups and noamenable radical, then G is C ∗-simple.
I this applies to Tarski monster groups, or free Burnside groups.
I we get that C ∗-simplicity is invariant under group extensions.In fact if N C G , then G is C ∗-simple if and only if N andCG (N) are.
I we get that if G is C ∗-simple and X is a G -boundary, which isnot topologically free, then StabG (x) is non-amenable.
![Page 81: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/81.jpg)
Further consequences
Exploiting the KK dynamical criterion, we further show:
I if G has only countably many amenable subgroups and noamenable radical, then G is C ∗-simple.
I this applies to Tarski monster groups, or free Burnside groups.
I we get that C ∗-simplicity is invariant under group extensions.In fact if N C G , then G is C ∗-simple if and only if N andCG (N) are.
I we get that if G is C ∗-simple and X is a G -boundary, which isnot topologically free, then StabG (x) is non-amenable.
![Page 82: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/82.jpg)
Further consequences
Exploiting the KK dynamical criterion, we further show:
I if G has only countably many amenable subgroups and noamenable radical, then G is C ∗-simple.
I this applies to Tarski monster groups, or free Burnside groups.
I we get that C ∗-simplicity is invariant under group extensions.In fact if N C G , then G is C ∗-simple if and only if N andCG (N) are.
I we get that if G is C ∗-simple and X is a G -boundary, which isnot topologically free, then StabG (x) is non-amenable.
![Page 83: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/83.jpg)
Unique traceA trace on a C ∗-algebra A is a linear functional τ : A→ C suchthatI τ(1) = 1,I τ(xx∗) ∈ [0,+∞), andI τ(ab) = τ(ba).
For example if A := C ∗λ(G ) ⊂ B(`2(G )) is the reduced C ∗-algebraof the discrete group G , then setting
τ(λg ) = 1 iff g = 1,
we obtain a trace on C ∗λ(G ) called the canonical trace.
![Page 84: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/84.jpg)
Unique traceA trace on a C ∗-algebra A is a linear functional τ : A→ C suchthatI τ(1) = 1,I τ(xx∗) ∈ [0,+∞), andI τ(ab) = τ(ba).
For example if A := C ∗λ(G ) ⊂ B(`2(G )) is the reduced C ∗-algebraof the discrete group G , then setting
τ(λg ) = 1 iff g = 1,
we obtain a trace on C ∗λ(G ) called the canonical trace.
![Page 85: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/85.jpg)
Unique traceA trace on a C ∗-algebra A is a linear functional τ : A→ C suchthatI τ(1) = 1,I τ(xx∗) ∈ [0,+∞), andI τ(ab) = τ(ba).
For example if A := C ∗λ(G ) ⊂ B(`2(G )) is the reduced C ∗-algebraof the discrete group G , then setting
τ(λg ) = 1 iff g = 1,
we obtain a trace on C ∗λ(G ) called the canonical trace.
Question: How to construct traces on C ∗λ(G ) ?Is the canonical trace the only trace ?
![Page 86: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/86.jpg)
Unique traceA trace on a C ∗-algebra A is a linear functional τ : A→ C suchthatI τ(1) = 1,I τ(xx∗) ∈ [0,+∞), andI τ(ab) = τ(ba).
For example if A := C ∗λ(G ) ⊂ B(`2(G )) is the reduced C ∗-algebraof the discrete group G , then setting
τ(λg ) = 1 iff g = 1,
we obtain a trace on C ∗λ(G ) called the canonical trace.
Question: How to construct traces on C ∗λ(G ) ?Is the canonical trace the only trace ?
e.g. if G is finite, setting τ(λg ) = 1 for all g gives rise to anon-canonical trace. If G is amenable, similarly one can buildnon-canonical traces.
![Page 87: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/87.jpg)
Unique traceA trace on a C ∗-algebra A is a linear functional τ : A→ C suchthatI τ(1) = 1,I τ(xx∗) ∈ [0,+∞), andI τ(ab) = τ(ba).
For example if A := C ∗λ(G ) ⊂ B(`2(G )) is the reduced C ∗-algebraof the discrete group G , then setting
τ(λg ) = 1 iff g = 1,
we obtain a trace on C ∗λ(G ) called the canonical trace.
Powers’ lemma also yields uniqueness of traces for groups satisfyingthe assumptions of Powers’ lemma.
Open problem: are being C ∗-simple and have unique traceequivalent ?
![Page 88: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/88.jpg)
Unique trace
Theorem (BKKO 2015)
Every trace concentrates on the amenable radical. In particular, ifRad(G ) = 1, then the canonical trace is the only trace.
idea: Use injectivity of C (∂FG ) to extend a trace to a positive G -map of
the cross product C (∂FG )n G to C (∂FG ), then exploit the fact that no
g /∈ Rad(G ) acts trivially on ∂FG .
Application to Invariant Random Subgroups:
Theorem (Bader-Duchesne-Lecureux)
If µ is an ergodic IRS on a locally compact group G , then µ isconcentrated on the amenable radical, i.e. H 6 Rad(G ) for µalmost every H.
![Page 89: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/89.jpg)
Unique trace
Theorem (BKKO 2015)
Every trace concentrates on the amenable radical. In particular, ifRad(G ) = 1, then the canonical trace is the only trace.
idea: Use injectivity of C (∂FG ) to extend a trace to a positive G -map of
the cross product C (∂FG )n G to C (∂FG ), then exploit the fact that no
g /∈ Rad(G ) acts trivially on ∂FG .
Application to Invariant Random Subgroups:
Theorem (Bader-Duchesne-Lecureux)
If µ is an ergodic IRS on a locally compact group G , then µ isconcentrated on the amenable radical, i.e. H 6 Rad(G ) for µalmost every H.
![Page 90: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/90.jpg)
Unique trace
Theorem (BKKO 2015)
Every trace concentrates on the amenable radical. In particular, ifRad(G ) = 1, then the canonical trace is the only trace.
idea: Use injectivity of C (∂FG ) to extend a trace to a positive G -map of
the cross product C (∂FG )n G to C (∂FG ), then exploit the fact that no
g /∈ Rad(G ) acts trivially on ∂FG .
Application to Invariant Random Subgroups:
Theorem (Bader-Duchesne-Lecureux)
If µ is an ergodic IRS on a locally compact group G , then µ isconcentrated on the amenable radical, i.e. H 6 Rad(G ) for µalmost every H.
![Page 91: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/91.jpg)
Unique trace
Theorem (BKKO 2015)
Every trace concentrates on the amenable radical. In particular, ifRad(G ) = 1, then the canonical trace is the only trace.
idea: Use injectivity of C (∂FG ) to extend a trace to a positive G -map of
the cross product C (∂FG )n G to C (∂FG ), then exploit the fact that no
g /∈ Rad(G ) acts trivially on ∂FG .
Application to Invariant Random Subgroups:
Theorem (Bader-Duchesne-Lecureux)
If µ is an ergodic IRS on a locally compact group G , then µ isconcentrated on the amenable radical, i.e. H 6 Rad(G ) for µalmost every H.
We get a new proof in the discrete case as a consequence of uniquetrace:
![Page 92: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/92.jpg)
Unique trace
Theorem (BKKO 2015)
Every trace concentrates on the amenable radical. In particular, ifRad(G ) = 1, then the canonical trace is the only trace.
idea: Use injectivity of C (∂FG ) to extend a trace to a positive G -map of
the cross product C (∂FG )n G to C (∂FG ), then exploit the fact that no
g /∈ Rad(G ) acts trivially on ∂FG .
Application to Invariant Random Subgroups:
Theorem (Bader-Duchesne-Lecureux)
If µ is an ergodic IRS on a locally compact group G , then µ isconcentrated on the amenable radical, i.e. H 6 Rad(G ) for µalmost every H.
idea: (Tucker-Drob) setting τ(λg ) := Proba(g ∈ H) we obtain atrace on C ∗λ(G )...
![Page 93: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/93.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0). T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
![Page 94: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/94.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0). T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
![Page 95: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/95.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0).
T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
![Page 96: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/96.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0). T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
![Page 97: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/97.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0). T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
So we get that if T is C ∗-simple, then F is non-amenable !
![Page 98: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/98.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0). T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
In particular, the two statements:
1. Every discrete group without amenable radical is C ∗-simple.
2. Thompson’s group F is amenable.
![Page 99: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/99.jpg)
Thompson groupsThompson’s group F is the subgroup of piecewise linear
homeomorphisms of [0, 1] whose non-differentiable breakpoints are dyadic
rationals and slopes are powers of 2.
Thompson’s group T is the subgroup of Homeo(S1) generated byF and the translation x 7→ x + 1
2 in S1 ' R/Z.
F is the stabilizer of 0 in T , F = StabT (0). T is an abstractlysimple, finitely presented, group. The circle S1 ' R/Z is aboundary for T .
But the circle is not topologically free !
In particular, the two statements:
1. Every discrete group without amenable radical is C ∗-simple.
2. Thompson’s group F is amenable.
are incompatible!
![Page 100: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/100.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
![Page 101: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/101.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H.
That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
![Page 102: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/102.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
![Page 103: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/103.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)
indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
![Page 104: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/104.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
![Page 105: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/105.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
indeed: λG/H ≺ λG , but λG ⊀ λG/H .
![Page 106: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/106.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
This applies to G = T and H = F the Thompson groups...
![Page 107: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/107.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
This applies to G = T and H = F the Thompson groups...
PROBLEM: Find pairs G ,H as above with Rad(G ) = 1.
![Page 108: The Furstenberg boundary and C-star simple groupsgeometrie.math.cnrs.fr/Breuillard.pdf · The Furstenberg boundary and C-star simple groups Emmanuel Breuillard, joint work with M.](https://reader033.fdocuments.net/reader033/viewer/2022041804/5e53a1670a6ec8761d5071bf/html5/thumbnails/108.jpg)
Haagerup and Olesen observation
Suppose G is a discrete countable group, and H 6 G a subgroup.
Suppose that there are a, b in G \ {1} with disjoint support whenacting on G/H. That is : ∀x ∈ G/H, either ax = x or bx = x .
Then (1− a)(1− b) = 0 when acting by convolution on `2(G/H)indeed: f (x) + f (b−1a−1x) = f (a−1x) + f (b−1x) for all x ∈ G/H.
Consequence: if H is amenable, then G is not C ∗-simple !
This applies to G = T and H = F the Thompson groups...
PROBLEM: Find pairs G ,H as above with Rad(G ) = 1.
→ you will get a non C ∗-simple group with trivial amenable radical...[Added July 1st: Adrien Le Boudec has just shown that his new construction of Burger-Mozes-type groups with
singularities solve this problem (as well as act non topologically freely on a boundary with amenable stabilizers), and
thus give rise to the first examples of non C∗-simple discrete groups without amenable radical.]