GROUPS ACTING ON 1-DIMENSIONAL SPACES AND SOME QUESTIONS
Andr´esNavas Universidad de Santiago de Chile
ICM Special Session “Dynamical Systems and Ordinary Differential Equations” Rio de Janeiro, August 2018 A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph).
Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator.
Groups
“Les math´ematiquesne sont qu’une histoire de groupes” (Poincar´e). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph).
Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator.
Groups
“Les math´ematiquesne sont qu’une histoire de groupes” (Poincar´e).
A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator.
Groups
“Les math´ematiquesne sont qu’une histoire de groupes” (Poincar´e).
A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Groups
“Les math´ematiquesne sont qu’une histoire de groupes” (Poincar´e).
A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph).
Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups
“Les math´ematiquesne sont qu’une histoire de groupes” (Poincar´e).
A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph).
Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Thinking from a group-theoretical perspec- tive can also give useful insight into “more classical” dynamics.
A General Principle
If a group (class of groups) acts nicely on a nice space, then the action should reveal so- me algebraic structure ( nice theorem). A General Principle
If a group (class of groups) acts nicely on a nice space, then the action should reveal so- me algebraic structure ( nice theorem).
Thinking from a group-theoretical perspec- tive can also give useful insight into “more classical” dynamics. The group of piecewise dyadic homeomorphisms of the binary Cantor• set is finitely presented and simple (Thompson’s group V ). It contains a copy of every finite group. • Its automorphism group is large but can been understood dynamically• (Bleak, Cameron, Lanoue, Yonah, N).
Groups acting on the Cantor set
Every group Γ acts on 0, 1 Γ by shifting coordinates. For infinite countable Γ, this is a Cantor{ } set.
Which groups can arise as groups of isometries (diffeomorphisms) of a Cantor set ? The group of piecewise dyadic homeomorphisms of the binary Cantor• set is finitely presented and simple (Thompson’s group V ). It contains a copy of every finite group. • Its automorphism group is large but can been understood dynamically• (Bleak, Cameron, Lanoue, Yonah, N).
Groups acting on the Cantor set
Every group Γ acts on 0, 1 Γ by shifting coordinates. For infinite countable Γ, this is a Cantor{ } set.
Which groups can arise as groups of isometries (diffeomorphisms) of a Cantor set ? Groups acting on the Cantor set
Every group Γ acts on 0, 1 Γ by shifting coordinates. For infinite countable Γ, this is a Cantor{ } set.
Which groups can arise as groups of isometries (diffeomorphisms) of a Cantor set ?
The group of piecewise dyadic homeomorphisms of the binary Cantor• set is finitely presented and simple (Thompson’s group V ). It contains a copy of every finite group. • Its automorphism group is large but can been understood dynamically• (Bleak, Cameron, Lanoue, Yonah, N). The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. F = f , g :[fg −1, f −1gf ] = [fg −1, f −2gf 2] = id .
...... 1 .... 1 ...... 3/4 ...... 5/8 ...... 1/2 ...... 1/2 ...... f ...... g ...... 1/4 ...... 0 ...... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1
Is Thompson’s group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha.
1
Richard Thompson’s groups
The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. F = f , g :[fg −1, f −1gf ] = [fg −1, f −2gf 2] = id .
...... 1 .... 1 ...... 3/4 ...... 5/8 ...... 1/2 ...... 1/2 ...... f ...... g ...... 1/4 ...... 0 ...... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1
Is Thompson’s group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha.
1
Richard Thompson’s groups
The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. Is Thompson’s group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha.
Richard Thompson’s groups
The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. F = f , g :[fg −1, f −1gf ] = [fg −1, f −2gf 2] = id .
...... 1 .... 1 ...... 3/4 ...... 5/8 ...... 1/2 ...... 1/2 ...... f ...... g ...... 1/4 ...... 0 ...... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1
1 Richard Thompson’s groups
The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. F = f , g :[fg −1, f −1gf ] = [fg −1, f −2gf 2] = id .
...... 1 .... 1 ...... 3/4 ...... 5/8 ...... 1/2 ...... 1/2 ...... f ...... g ...... 1/4 ...... 0 ...... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1
Is Thompson’s group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha.
1 YES for linear groups (Burnside). • The Burnside groups: • B(n) := a, b : w n = id for all w h i
- B(2), B(3), B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov).
The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside] The Burnside groups: • B(n) := a, b : w n = id for all w h i
- B(2), B(3), B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov).
The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]
YES for linear groups (Burnside). • - B(2), B(3), B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov).
The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]
YES for linear groups (Burnside). • The Burnside groups: • B(n) := a, b : w n = id for all w h i - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov).
The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]
YES for linear groups (Burnside). • The Burnside groups: • B(n) := a, b : w n = id for all w h i
- B(2), B(3), B(4) and B(6) are finite. - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov).
The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]
YES for linear groups (Burnside). • The Burnside groups: • B(n) := a, b : w n = id for all w h i
- B(2), B(3), B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov).
The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]
YES for linear groups (Burnside). • The Burnside groups: • B(n) := a, b : w n = id for all w h i
- B(2), B(3), B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). The Burnside groups
Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]
YES for linear groups (Burnside). • The Burnside groups: • B(n) := a, b : w n = id for all w h i
- B(2), B(3), B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov). 1 For Homeo+(S ), the answer is affirmative (exercise). • For other surfaces, see the work of Guelman-Liousse. • According to a theorem of Ker´ekj´art´o(based on the work of Brouwer),• every finite-order homeomorphism of the sphere is conjugate to a rotation (compare Kovacevic, N). Partial results by Hurtado (area preserving, uniform torsion, smooth• maps) and Conejeros (area preserving, uniform torsion, power of 2 exponents, no hypothesis of regularity).
Remarkable recent progress on higher dimensions (Brown-Fisher- Hurtado), but still a lack of a structure for diffeomorphisms groups.
A basic question in dimension 2
2 Let Γ Homeo+(S ) be a finitely-generated group in which every element⊂ has finite order. Must Γ be finite ? For other surfaces, see the work of Guelman-Liousse. • According to a theorem of Ker´ekj´art´o(based on the work of Brouwer),• every finite-order homeomorphism of the sphere is conjugate to a rotation (compare Kovacevic, N). Partial results by Hurtado (area preserving, uniform torsion, smooth• maps) and Conejeros (area preserving, uniform torsion, power of 2 exponents, no hypothesis of regularity).
Remarkable recent progress on higher dimensions (Brown-Fisher- Hurtado), but still a lack of a structure for diffeomorphisms groups.
A basic question in dimension 2
2 Let Γ Homeo+(S ) be a finitely-generated group in which every element⊂ has finite order. Must Γ be finite ?
1 For Homeo+(S ), the answer is affirmative (exercise). • According to a theorem of Ker´ekj´art´o(based on the work of Brouwer),• every finite-order homeomorphism of the sphere is conjugate to a rotation (compare Kovacevic, N). Partial results by Hurtado (area preserving, uniform torsion, smooth• maps) and Conejeros (area preserving, uniform torsion, power of 2 exponents, no hypothesis of regularity).
Remarkable recent progress on higher dimensions (Brown-Fisher- Hurtado), but still a lack of a structure for diffeomorphisms groups.
A basic question in dimension 2
2 Let Γ Homeo+(S ) be a finitely-generated group in which every element⊂ has finite order. Must Γ be finite ?
1 For Homeo+(S ), the answer is affirmative (exercise). • For other surfaces, see the work of Guelman-Liousse. • Partial results by Hurtado (area preserving, uniform torsion, smooth• maps) and Conejeros (area preserving, uniform torsion, power of 2 exponents, no hypothesis of regularity).
Remarkable recent progress on higher dimensions (Brown-Fisher- Hurtado), but still a lack of a structure for diffeomorphisms groups.
A basic question in dimension 2
2 Let Γ Homeo+(S ) be a finitely-generated group in which every element⊂ has finite order. Must Γ be finite ?
1 For Homeo+(S ), the answer is affirmative (exercise). • For other surfaces, see the work of Guelman-Liousse. • According to a theorem of Ker´ekj´art´o(based on the work of Brouwer),• every finite-order homeomorphism of the sphere is conjugate to a rotation (compare Kovacevic, N). Remarkable recent progress on higher dimensions (Brown-Fisher- Hurtado), but still a lack of a structure for diffeomorphisms groups.
A basic question in dimension 2
2 Let Γ Homeo+(S ) be a finitely-generated group in which every element⊂ has finite order. Must Γ be finite ?
1 For Homeo+(S ), the answer is affirmative (exercise). • For other surfaces, see the work of Guelman-Liousse. • According to a theorem of Ker´ekj´art´o(based on the work of Brouwer),• every finite-order homeomorphism of the sphere is conjugate to a rotation (compare Kovacevic, N). Partial results by Hurtado (area preserving, uniform torsion, smooth• maps) and Conejeros (area preserving, uniform torsion, power of 2 exponents, no hypothesis of regularity). A basic question in dimension 2
2 Let Γ Homeo+(S ) be a finitely-generated group in which every element⊂ has finite order. Must Γ be finite ?
1 For Homeo+(S ), the answer is affirmative (exercise). • For other surfaces, see the work of Guelman-Liousse. • According to a theorem of Ker´ekj´art´o(based on the work of Brouwer),• every finite-order homeomorphism of the sphere is conjugate to a rotation (compare Kovacevic, N). Partial results by Hurtado (area preserving, uniform torsion, smooth• maps) and Conejeros (area preserving, uniform torsion, power of 2 exponents, no hypothesis of regularity).
Remarkable recent progress on higher dimensions (Brown-Fisher- Hurtado), but still a lack of a structure for diffeomorphisms groups. – Actions on the real line: such an action comes from a left-order (folklore). A. Kokorin, V. Kopytov. Fully Ordered Groups, Wiley (1974). “For some unexplained reason, some people like to linearly order groups. Even more mysteriously, most of these people work in the Soviet Union. It would greatly help their cause, where they to add a few words by way of explanation of their activity.” (Gian-Carlo Rota).
– Actions on the circle without a finite orbit: such an action comes from a bounded-cohomology class with coefficients in Z having a representative taking only the values 0 and 1 (Poincar´e-Ghys).
Groups acting on 1-dimensional manifolds
Algebraic description of groups that act faithfully by orientation- preserving homeomorphisms: A. Kokorin, V. Kopytov. Fully Ordered Groups, Wiley (1974). “For some unexplained reason, some people like to linearly order groups. Even more mysteriously, most of these people work in the Soviet Union. It would greatly help their cause, where they to add a few words by way of explanation of their activity.” (Gian-Carlo Rota).
– Actions on the circle without a finite orbit: such an action comes from a bounded-cohomology class with coefficients in Z having a representative taking only the values 0 and 1 (Poincar´e-Ghys).
Groups acting on 1-dimensional manifolds
Algebraic description of groups that act faithfully by orientation- preserving homeomorphisms: – Actions on the real line: such an action comes from a left-order (folklore). – Actions on the circle without a finite orbit: such an action comes from a bounded-cohomology class with coefficients in Z having a representative taking only the values 0 and 1 (Poincar´e-Ghys).
Groups acting on 1-dimensional manifolds
Algebraic description of groups that act faithfully by orientation- preserving homeomorphisms: – Actions on the real line: such an action comes from a left-order (folklore). A. Kokorin, V. Kopytov. Fully Ordered Groups, Wiley (1974). “For some unexplained reason, some people like to linearly order groups. Even more mysteriously, most of these people work in the Soviet Union. It would greatly help their cause, where they to add a few words by way of explanation of their activity.” (Gian-Carlo Rota). Groups acting on 1-dimensional manifolds
Algebraic description of groups that act faithfully by orientation- preserving homeomorphisms: – Actions on the real line: such an action comes from a left-order (folklore). A. Kokorin, V. Kopytov. Fully Ordered Groups, Wiley (1974). “For some unexplained reason, some people like to linearly order groups. Even more mysteriously, most of these people work in the Soviet Union. It would greatly help their cause, where they to add a few words by way of explanation of their activity.” (Gian-Carlo Rota).
– Actions on the circle without a finite orbit: such an action comes from a bounded-cohomology class with coefficients in Z having a representative taking only the values 0 and 1 (Poincar´e-Ghys). Does there exist a finitely-generated, torsion-free group that cannot be realized as a group of homeomorphisms of a surface ?
Does there exists a group that cannot be realized as a group of homeomorphisms of a surface which is a limit of groups that do realize ?
Groups acting on 1-dimensional manifolds
Does there exist an algebraic characterization of groups that do act faithfully by homeomorphisms of a certain 2-manifold ? Does there exists a group that cannot be realized as a group of homeomorphisms of a surface which is a limit of groups that do realize ?
Groups acting on 1-dimensional manifolds
Does there exist an algebraic characterization of groups that do act faithfully by homeomorphisms of a certain 2-manifold ?
Does there exist a finitely-generated, torsion-free group that cannot be realized as a group of homeomorphisms of a surface ? Groups acting on 1-dimensional manifolds
Does there exist an algebraic characterization of groups that do act faithfully by homeomorphisms of a certain 2-manifold ?
Does there exist a finitely-generated, torsion-free group that cannot be realized as a group of homeomorphisms of a surface ?
Does there exists a group that cannot be realized as a group of homeomorphisms of a surface which is a limit of groups that do realize ? Theorem (Ghys-Sergiescu, Ghys) Thompson’s group T is globally structurally stable: up to topolo- gical semi-conjugacy, it has a unique action on the circle.
Does there exist a finitely-generated, structurally-stable group of homeomorphisms of the sphere ?
Structural stability
Up to topological semi-conjugacy, there is a unique action of B3 on the real line without global fixed points for which the generators −1 a=σ1σ2 and b =σ2 send the origin into positive real numbers. Does there exist a finitely-generated, structurally-stable group of homeomorphisms of the sphere ?
Structural stability
Up to topological semi-conjugacy, there is a unique action of B3 on the real line without global fixed points for which the generators −1 a=σ1σ2 and b =σ2 send the origin into positive real numbers.
Theorem (Ghys-Sergiescu, Ghys) Thompson’s group T is globally structurally stable: up to topolo- gical semi-conjugacy, it has a unique action on the circle. Structural stability
Up to topological semi-conjugacy, there is a unique action of B3 on the real line without global fixed points for which the generators −1 a=σ1σ2 and b =σ2 send the origin into positive real numbers.
Theorem (Ghys-Sergiescu, Ghys) Thompson’s group T is globally structurally stable: up to topolo- gical semi-conjugacy, it has a unique action on the circle.
Does there exist a finitely-generated, structurally-stable group of homeomorphisms of the sphere ? Theorem (Thurston) 1 Every nontrivial finitely-generated subgroup of Diff+([0, 1]) admits 1 a nontrivial homomorphism into R (i.e. Diff+([0, 1]) is locally in- dicable).
“Proof”: Take g log(Dg(0)). 7→
Local indicability does not hold for Homeo+([0, 1]) (even for the group of Lipschitz homeomorphisms). An example (also due to Thurston) is the lifting to PSLg (2, R) of the (2,3,7)-triangle sub- group of PSL(2, R).
The regularity enters into the game
This comes from codimension-1 foliations theory: Sacksteder, Plante, Thurston, Ghys, Tsuboi, Matsumoto, Cantwell-Conlon, Hurder... (inspired on Denjoy’s work). Local indicability does not hold for Homeo+([0, 1]) (even for the group of Lipschitz homeomorphisms). An example (also due to Thurston) is the lifting to PSLg (2, R) of the (2,3,7)-triangle sub- group of PSL(2, R).
The regularity enters into the game
This comes from codimension-1 foliations theory: Sacksteder, Plante, Thurston, Ghys, Tsuboi, Matsumoto, Cantwell-Conlon, Hurder... (inspired on Denjoy’s work).
Theorem (Thurston) 1 Every nontrivial finitely-generated subgroup of Diff+([0, 1]) admits 1 a nontrivial homomorphism into R (i.e. Diff+([0, 1]) is locally in- dicable).
“Proof”: Take g log(Dg(0)). 7→ The regularity enters into the game
This comes from codimension-1 foliations theory: Sacksteder, Plante, Thurston, Ghys, Tsuboi, Matsumoto, Cantwell-Conlon, Hurder... (inspired on Denjoy’s work).
Theorem (Thurston) 1 Every nontrivial finitely-generated subgroup of Diff+([0, 1]) admits 1 a nontrivial homomorphism into R (i.e. Diff+([0, 1]) is locally in- dicable).
“Proof”: Take g log(Dg(0)). 7→
Local indicability does not hold for Homeo+([0, 1]) (even for the group of Lipschitz homeomorphisms). An example (also due to Thurston) is the lifting to PSLg (2, R) of the (2,3,7)-triangle sub- group of PSL(2, R). Theorem (N, Bonatti-Monteverde-N-Rivas, Bonatti-Lodha- Triestino) Local indicability is not the only obstruction for C 1 embeddings.
Local indicability
A tiling of the hyperbolic disk induced by the action of the (2,3,7)-triangle group. Local indicability
A tiling of the hyperbolic disk induced by the action of the (2,3,7)-triangle group.
Theorem (N, Bonatti-Monteverde-N-Rivas, Bonatti-Lodha- Triestino) Local indicability is not the only obstruction for C 1 embeddings. Theorem (N) 1+α Every finitely-generated subgroup of Diff+ ([0, 1]) has either 1 polynomial or exponential growth. This is false for Diff+([0, 1]).
Exponential growth means that the number of elements that may• be written as products of no more than n generators grows exponentially as a function of n. The class of groups of polynomial growth coincides with that of almost-nilpotent• ones (Bass, Guivarch; Gromov). Does there exist a group of intermediate growth of C ω diffeomorphisms of the sphere ?
In class C 1+α
Notice that Diff1+α(M) is a group:
α Df (x) Df (y) C x y − ≤ | − | Exponential growth means that the number of elements that may• be written as products of no more than n generators grows exponentially as a function of n. The class of groups of polynomial growth coincides with that of almost-nilpotent• ones (Bass, Guivarch; Gromov). Does there exist a group of intermediate growth of C ω diffeomorphisms of the sphere ?
In class C 1+α
Notice that Diff1+α(M) is a group:
α Df (x) Df (y) C x y − ≤ | − | Theorem (N) 1+α Every finitely-generated subgroup of Diff+ ([0, 1]) has either 1 polynomial or exponential growth. This is false for Diff+([0, 1]). Does there exist a group of intermediate growth of C ω diffeomorphisms of the sphere ?
In class C 1+α
Notice that Diff1+α(M) is a group:
α Df (x) Df (y) C x y − ≤ | − | Theorem (N) 1+α Every finitely-generated subgroup of Diff+ ([0, 1]) has either 1 polynomial or exponential growth. This is false for Diff+([0, 1]).
Exponential growth means that the number of elements that may• be written as products of no more than n generators grows exponentially as a function of n. The class of groups of polynomial growth coincides with that of almost-nilpotent• ones (Bass, Guivarch; Gromov). In class C 1+α
Notice that Diff1+α(M) is a group:
α Df (x) Df (y) C x y − ≤ | − | Theorem (N) 1+α Every finitely-generated subgroup of Diff+ ([0, 1]) has either 1 polynomial or exponential growth. This is false for Diff+([0, 1]).
Exponential growth means that the number of elements that may• be written as products of no more than n generators grows exponentially as a function of n. The class of groups of polynomial growth coincides with that of almost-nilpotent• ones (Bass, Guivarch; Gromov). Does there exist a group of intermediate growth of C ω diffeomorphisms of the sphere ? a¯ b¯ c¯ d¯ y y y y ...... ••••...... ¯ ...... ¯ ...... a¯ ...... c¯ a¯ ...... d id ...... b ...... y...... y y...... y y...... y ...... ←→ ...... ••••...... ••••...... ••••...... •...... ••••...... •...... •...... •• ...... •...... •• ...... •• ......
The generators of the Grigorchuk group
1
The Grigorchuk group: G = a¯, b¯, c¯, d¯ h i The Grigorchuk group: G = a¯, b¯, c¯, d¯ h i
a¯ b¯ c¯ d¯ y y y y ...... ••••...... ¯ ...... ¯ ...... a¯ ...... c¯ a¯ ...... d id ...... b ...... y...... y y...... y y...... y ...... ←→ ...... ••••...... ••••...... ••••...... •...... ••••...... •...... •...... •• ...... •...... •• ...... •• ......
The generators of the Grigorchuk group
1 The Grigorchuk group: G = a¯, b¯, c¯, d¯ h i
Some formulae: for li 0, 1 , ∈ { }
a¯(l1, l2, l3,...) = (1 l1, l2, l3,...), − (l1, a¯(l2, l3,...)), l1 = 0, b¯(l1, l2, l3,...) = (l1, c¯(l2, l3,...)), l1 = 1, (l1, a¯(l2, l3,...)), l1 = 0, c¯(l1, l2, l3,...) = (l1, d¯(l2, l3,...)), l1 = 1, (l1, l2, l3,...), l1 = 0, d¯(l1, l2, l3,...) = (l1, b¯(l2, l3,...)), l1 = 1. This is a torsion-free group of intermediate growth. It has a faithful• action on a rooted tree for which every vertex different from the root has one ancestor and infinitely many descendants. The natural action of GM on the interval is C 1 smoothable. • Nonexistence of embeddings group of intermediate growth (as for • 1+α example GM) in Diff+ ([0, 1]) is established by using (nontrivial extensions) of classical techniques in 1-dimensional dynamics.
The Grigorchuk-Machi group: GM = a, b, c, d h i
Some formulae: for li Z, ∈ a(l1, l2, l3,...) = (1+ l1, l2, l3,...), (l1, a(l2, l3,...)), l1 even, b(l1, l2, l3,...) = (l1, c(l2, l3,...)), l1 odd, (l1, a(l2, l3,...)), l1 even, c(l1, l2, l3,...) = (l1, d(l2, l3,...)), l1 odd, (l1, l2, l3,...), l1 even, d(l1, l2, l3,...) = (l1, b(l2, l3,...)), l1 odd. The natural action of GM on the interval is C 1 smoothable. • Nonexistence of embeddings group of intermediate growth (as for • 1+α example GM) in Diff+ ([0, 1]) is established by using (nontrivial extensions) of classical techniques in 1-dimensional dynamics.
The Grigorchuk-Machi group: GM = a, b, c, d h i
Some formulae: for li Z, ∈ a(l1, l2, l3,...) = (1+ l1, l2, l3,...), (l1, a(l2, l3,...)), l1 even, b(l1, l2, l3,...) = (l1, c(l2, l3,...)), l1 odd, (l1, a(l2, l3,...)), l1 even, c(l1, l2, l3,...) = (l1, d(l2, l3,...)), l1 odd, (l1, l2, l3,...), l1 even, d(l1, l2, l3,...) = (l1, b(l2, l3,...)), l1 odd. This is a torsion-free group of intermediate growth. It has a faithful• action on a rooted tree for which every vertex different from the root has one ancestor and infinitely many descendants. Nonexistence of embeddings group of intermediate growth (as for • 1+α example GM) in Diff+ ([0, 1]) is established by using (nontrivial extensions) of classical techniques in 1-dimensional dynamics.
The Grigorchuk-Machi group: GM = a, b, c, d h i
Some formulae: for li Z, ∈ a(l1, l2, l3,...) = (1+ l1, l2, l3,...), (l1, a(l2, l3,...)), l1 even, b(l1, l2, l3,...) = (l1, c(l2, l3,...)), l1 odd, (l1, a(l2, l3,...)), l1 even, c(l1, l2, l3,...) = (l1, d(l2, l3,...)), l1 odd, (l1, l2, l3,...), l1 even, d(l1, l2, l3,...) = (l1, b(l2, l3,...)), l1 odd. This is a torsion-free group of intermediate growth. It has a faithful• action on a rooted tree for which every vertex different from the root has one ancestor and infinitely many descendants. The natural action of GM on the interval is C 1 smoothable. • The Grigorchuk-Machi group: GM = a, b, c, d h i
Some formulae: for li Z, ∈ a(l1, l2, l3,...) = (1+ l1, l2, l3,...), (l1, a(l2, l3,...)), l1 even, b(l1, l2, l3,...) = (l1, c(l2, l3,...)), l1 odd, (l1, a(l2, l3,...)), l1 even, c(l1, l2, l3,...) = (l1, d(l2, l3,...)), l1 odd, (l1, l2, l3,...), l1 even, d(l1, l2, l3,...) = (l1, b(l2, l3,...)), l1 odd. This is a torsion-free group of intermediate growth. It has a faithful• action on a rooted tree for which every vertex different from the root has one ancestor and infinitely many descendants. The natural action of GM on the interval is C 1 smoothable. • Nonexistence of embeddings group of intermediate growth (as for • 1+α example GM) in Diff+ ([0, 1]) is established by using (nontrivial extensions) of classical techniques in 1-dimensional dynamics. Theorem (Kim-Koberda) Given r > s 1 (not necessarily integers), there is a group of C s diffeomorphisms≥ of the interval that does not embed into the group of C r diffeomorphisms of the interval.
Critical regularity
Results that are sharp in what concerns (intermediate) regularity for• some nilpotent group actions (Tsuboi, Farb-Franks, Deroin-Kleptsyn-N, Castro-Jorquera-N, N, Parkhe). Critical regularity
Results that are sharp in what concerns (intermediate) regularity for• some nilpotent group actions (Tsuboi, Farb-Franks, Deroin-Kleptsyn-N, Castro-Jorquera-N, N, Parkhe).
Theorem (Kim-Koberda) Given r > s 1 (not necessarily integers), there is a group of C s diffeomorphisms≥ of the interval that does not embed into the group of C r diffeomorphisms of the interval. Theorem (Deroin, Kleptsyn, N) If Γ is a finitely-genetared group of C ω circle diffeomorphisms preserving a Cantor set on which the action is minimal, then this set has zero Lebesgue measure. Besides, there are only finitely many orbits of connected components of the complement.
Theorem (Deroin, Kleptsyn, N) If the action is minimal (plus extra technical hypothesis), then it is ergodic (w.r.t. the Lebesgue measure).
Derivatives are cocycles for group actions
This was first exploited by Sacksteder in the context of codimension-1 foliations. Theorem (Deroin, Kleptsyn, N) If the action is minimal (plus extra technical hypothesis), then it is ergodic (w.r.t. the Lebesgue measure).
Derivatives are cocycles for group actions
This was first exploited by Sacksteder in the context of codimension-1 foliations.
Theorem (Deroin, Kleptsyn, N) If Γ is a finitely-genetared group of C ω circle diffeomorphisms preserving a Cantor set on which the action is minimal, then this set has zero Lebesgue measure. Besides, there are only finitely many orbits of connected components of the complement. Derivatives are cocycles for group actions
This was first exploited by Sacksteder in the context of codimension-1 foliations.
Theorem (Deroin, Kleptsyn, N) If Γ is a finitely-genetared group of C ω circle diffeomorphisms preserving a Cantor set on which the action is minimal, then this set has zero Lebesgue measure. Besides, there are only finitely many orbits of connected components of the complement.
Theorem (Deroin, Kleptsyn, N) If the action is minimal (plus extra technical hypothesis), then it is ergodic (w.r.t. the Lebesgue measure). Rank-one phenomena (non-existence of Kazhdan subgroups in class• C 3/2).
Jorgensen inequality / Duminy’s theorem. • V (f ) := var(log(Df ); S1) = 4 dist(0, f (0)).
If a subgroup of PSL(2, R) is generated by elements f for which V (f ) 4 log(5/3), then either the group acts minimally or has a finite≤ orbit (Yamada, Guillot-N).
Groups of circle diffeomorphisms / M¨obiusgroups Jorgensen inequality / Duminy’s theorem. • V (f ) := var(log(Df ); S1) = 4 dist(0, f (0)).
If a subgroup of PSL(2, R) is generated by elements f for which V (f ) 4 log(5/3), then either the group acts minimally or has a finite≤ orbit (Yamada, Guillot-N).
Groups of circle diffeomorphisms / M¨obiusgroups
Rank-one phenomena (non-existence of Kazhdan subgroups in class• C 3/2). V (f ) := var(log(Df ); S1) = 4 dist(0, f (0)).
If a subgroup of PSL(2, R) is generated by elements f for which V (f ) 4 log(5/3), then either the group acts minimally or has a finite≤ orbit (Yamada, Guillot-N).
Groups of circle diffeomorphisms / M¨obiusgroups
Rank-one phenomena (non-existence of Kazhdan subgroups in class• C 3/2).
Jorgensen inequality / Duminy’s theorem. • If a subgroup of PSL(2, R) is generated by elements f for which V (f ) 4 log(5/3), then either the group acts minimally or has a finite≤ orbit (Yamada, Guillot-N).
Groups of circle diffeomorphisms / M¨obiusgroups
Rank-one phenomena (non-existence of Kazhdan subgroups in class• C 3/2).
Jorgensen inequality / Duminy’s theorem. • V (f ) := var(log(Df ); S1) = 4 dist(0, f (0)). Groups of circle diffeomorphisms / M¨obiusgroups
Rank-one phenomena (non-existence of Kazhdan subgroups in class• C 3/2).
Jorgensen inequality / Duminy’s theorem. • V (f ) := var(log(Df ); S1) = 4 dist(0, f (0)).
If a subgroup of PSL(2, R) is generated by elements f for which V (f ) 4 log(5/3), then either the group acts minimally or has a finite≤ orbit (Yamada, Guillot-N). Some references Some references Some references
Groups, Orders, and Dynamics (GOD). B. Deroin, A. Navas, C. Rivas MANY THANKS
MUITO OBRIGADO
MUCHAS GRACIAS