[PPT] - Hlder regularity of certain non-solvable groups Sang-hyun Kim PowerPoint Presentation

SLIDE 1

Hölder regularity of certain non-solvable groups

Sang-hyun Kim (KIAS, Korea) Thomas Koberda (UVa, US) and Cristóbal Rivas (USACH, Chile)

Teichmüller Theory: Classical, Higher, Super and Quantum CIRM — Luminy, October 8, 2020

SLIDE 2

Critical regularity of a group

: compact interval is —diffeo, . is —Hölder con is —Hölder continuous if . cf. .

I Diffk

+(I) = {f : I → I | f

Ck f′ > 0} Diffk+τ

+ (I) = {f ∈ Diffk +(I) ∣ f (k)

τ } g : I → ℝ τ [g]τ := sup

x≠y

|gx − gy| |x − y|τ < ∞ Diffk+1

+ (I) ≠ Diffk+Lip +

(I)

(K.—Koberda 2020) f.g. such that for all . Rigidity Study representations of for a “lattice” and a topological group .

1 ≤ r < s ⟹ Diffr

+(I) ≥ Diffs +(I)

1 ≤ r ⟹ ∃ Gr ≤ Diffr

+(I)

Gr / ↪ Diffs

+(I)

s > r Γ → G Γ G

[g]1 := [g]Lip

Theme Which f.g. groups arise as subgps of ?

Diffr

+(I)

Definition (critical regularity) CritReg(G) := sup{r ≥ 0 | G ↪ Diffr

+(I)}

(Deroin—Kleptsyn—Navas 2007) If is countable, then biLip. i.e. CritReg

r

.

G ≤ Homeo+(I) G ∼ (G) ≥ 1 = − ∞

group theory analysis

⟹

SLIDE 3

Definition (critical regularity) CritReg(G) := sup{r ≥ 0 | G ↪ Diffr

+(I)}

Motivation from the foliation theory

{π1(B) → Diffr(F)}/conj . ↭ {Cr foliated B-bundles over F}/isom . ρ ↭ F → ( ˜

B × F)/(x, t) ∼ (g . x, ρ(g) . t) ∀g ∈ π1(B)

↓

B

Thurston Stability (1974) cpt cnt transver. orientable codimension-one foliation w/ a cpt leaf s.t.

r

.

(Mn, ℱ)

C1 L H1(L; ℝ) = 0 ⟹ M ≅ L × I L → M → S1

W. Thurston

(1946-2012)

SLIDE 4

e.g. is trivial. e.g. Not locally indicable: . f , but not , foliation on . (Plante—Thurston 1976) ∀ nilpotent subgroup of is abelian. (Farb–Franks 2003) ∀ f.g. (res.) tor-free nilpotent gp embeds into .

H1(L) = 0 ⇒ ∀ρ : π1(L) → Diff1[0,1)

π1(MSFS) = ⟨a, b, c ∣ a2 = b3 = c7 = abc⟩ ≤ ˜ PSL(2,ℝ) ≤ Homeo+(I) π1(MSFS) / ↪ Diff1

+(I)

C0 C1 M × S1 Diff2

+(I)

Diff1

+(I)

Thurston Stability Lemma (1974) is locally indicable. i.e. surjects onto

Diff1[0,1) ∀1 ≠ Hf.g. ≤ Diff1[0,1) ℤ .

Motivation from the foliation theory

Thurston Stability (1974) cpt cnt transver. orientable codimension-one foliation w/ a cpt leaf s.t.

r

.

(Mn, ℱ)

C1 L H1(L; ℝ) = 0 ⟹ M ≅ L × I L → M → S1

W. Thurston

(1946-2012)

P . Kropholler

W. Thurston

SLIDE 5

(Plante—Thurston 1976) ∀ nilpotent subgroup of is abelian. (Farb–Franks 2003) ∀ f.g. (res.) tor-free nilpotent gp embeds into .

Diff2

+(I)

Diff1

+(I)

CritReg: non-exponential growth groups

Heisenberg group (Castro—Jorquera—Navas 2014) , i.e. CritReg (Jorquera—Navas—Rivas 2017) , but , i.e. CritReg (Navas 2008) : intermediate growth for Moreover, Grigorchuk—Machi group

Heis = N3 ≤ Diff1

+(I)

Heis ↪ Diff2−ϵ

+ (I)

(Heis) = 2 N4 ↪ Diff1.5−ϵ

+

(I) / ↪ Diff1.5+ϵ

+

(I) (N4) = 1.5 G ⟹ G / ↪ Diff1+τ

+ (I)

τ > 0. ↪ Diff1

+(I)

Definition (critical regularity) CritReg(G) := sup{r ≥ 0 | G ↪ Diffr

+(I)}

SLIDE 6

Heisenberg group (Castro—Jorquera—Navas 2014) , i.e. CritReg (Jorquera—Navas—Rivas 2017) , but , i.e. CritReg (Navas 2008) : intermediate growth for Moreover, Grigorchuk—Machi group

Heis = N3 ≤ Diff1

+(I)

Heis ↪ Diff2−ϵ

+ (I)

(Heis) = 2 N4 ↪ Diff1.5−ϵ

+

(I) / ↪ Diff1.5+ϵ

+

(I) (N4) = 1.5 G ⟹ G / ↪ Diff1+τ

+ (I)

τ > 0. ↪ Diff1

+(I)

CritReg: more examples

(Witte ’94; Burger–Monod ’99, Ghys ’99) higher-rank lattice (Navas ’02 / Bader–Furman–Gelander–Monod ’07) Property T group (Brown–Fisher–Hurtado) d > n rank-d lattice (Farb–Franks ’01) (Baik–K–Koberda ’16) , virtually.

/ ↪ Diff1

+(S1)

/ ↪ Diff1.5

+ (S1)

/ ↪ Diff1

+ (Xn−manifold)

Mod(Sg≥3,p≤1) / ↪ Diff2

+(M)

3g − 3 + p ≤ 1 ⟺ Mod(Sg,p) ↪ Diff2

+(M)

SLIDE 7

CritReg: exponential growth groups?

Thompson’s group : PL homeo of [0,1] w/ dyadic breakpts & slopes (Thurston, using

representation)

Thompson’s group . (Ghys—Sergiescu 1987) is conjugate into In particular, CritReg .

F F 2ℤ PPSL(2,ℤ) F ↪ Diff1

+[0,1]

F ↪ PL[0,1] Diff∞

+ [0,1]

(F) = ∞ f g

“two-chain” (K—K—Lodha 2019) , .

∀N ≫ 0 < fN, gN > ≅ F

Question Are there exponential growth examples? (for

smooth groups)

C1

SLIDE 8

: finite simplicial graph. The RAAG (right-angled Artin group) on is: e.g. (Agol, Wise 2012)

fin. vol. hyp. 3-mfd gp virtually

(M. Kapovich) (Baik—K—Koberda 2019) . Cor (BKK) virtually

Γ Γ A(Γ) := < V(Γ) | [a, b] = 1∀{a, b} ∈ E(Γ) > A( △ ) ≅ ℤ3 A( ∴ ) ≅ F3 A( ∙ − ∙ − ∙ ) ≅ F2 × ℤ ∀ ↪ ∃A(Γ) ∀A(Γ) ↪ Symp(S2) A( ∙ − ∙ − ∙ − ∙ ) / ↪ Diff2

+(I or S1)

Mod(Σg,p) ↪ Diff2

+(I or S1)

⟺ 3g − 3 + p ≤ 1.

S2 → E Mhyp,3 ↓

Question Are there exponential growth examples? (for

smooth groups)

C1

Right-angled Artin groups

SLIDE 9

Right-angled Artin groups

(FF2003) . (K—Koberda 2018) CritReg

r

⟹ A(Γ) ≤ Diff1

+(I)

A(Γ) ↪ Diff2

+(I) ⟺ (F2 × ℤ) * ℤ /

↪ A(Γ) ⟺ A(Γ) ↪ Diff∞

+ (I)

(A(Γ)) ∈ [1,2] = ∞

S2 → E Mhyp,3 ↓

: finite simplicial graph. The RAAG (right-angled Artin group) on is: e.g. (Agol, Wise 2012)

fin. vol. hyp. 3-mfd gp virtually

(M. Kapovich) (Baik—K—Koberda 2019) . Cor (BKK) virtually

Γ Γ A(Γ) := < V(Γ) | [a, b] = 1∀{a, b} ∈ E(Γ) > A( △ ) ≅ ℤ3 A( ∴ ) ≅ F3 A( ∙ − ∙ − ∙ ) ≅ F2 × ℤ ∀ ↪ ∃A(Γ) ∀A(Γ) ↪ Symp(S2) A( ∙ − ∙ − ∙ − ∙ ) / ↪ Diff2

+(I or S1)

Mod(Σg,p) ↪ Diff2

+(I or S1)

⟺ 3g − 3 + p ≤ 1.

A( ∙ − ∙ − ∙ ∙ )

Question (1) CritReg (2) CritReg

(A(Γ)) = ? ((F2 × ℤ) * ℤ) = ?

SLIDE 10

Right-angled Artin groups

Theorem A (K—Koberda—Rivas) If and are non-solvable, then for all

G H (G × H) * ℤ / ↪ Diff1+τ

+ (I)

τ > 0.

Cor CritReg CritReg

((F2 × F2) * ℤ) = 1. (F * ℤ) = 1.

F × F ↪ F

A( ∙ − ∙ − ∙ ∙ )

e x p

n

e n t i a l g r

w

t h Theorem B (K—Koberda—Rivas) .

F * ℤ / ↪ Diff1

+(I)

(Navas)

(BS(1,2) × ℤ) * ℤ / ↪ Diff1

+(I)

Question (1) CritReg (2) CritReg

(A(Γ)) = ? ((F2 × ℤ) * ℤ) = ?

SLIDE 11

Theorem A (K—Koberda—Rivas) If and are non-solvable, then for all Theorem B (K—Koberda—Rivas) .

G H (G × H) * ℤ / ↪ Diff1+τ

+ (I)

τ > 0. F * ℤ / ↪ Diff1

+(I)

Overlapping actions

f ∈ Homeo(X) ⇝ supp f := X∖Fix f G ≤ Homeo(X) ⇝ supp G := ∪g∈G supp g

The —Lemma (K—Koberda 2018) If satisfy , then .

abt a, b, t ∈ Diff1

+(I or S1)

supp a ∩ supp b = Ø ⟨a, b, t⟩ ≇ (ℤ × ℤ) * ℤ

, for all

K * ℤ ↪ Diff1

+(I)

⟹ supp a ∩ supp b ≠ Ø a, b ∈ K

Theorem C (K—Koberda—Rivas) If and then .

G × H ≤ Diff1+τ

+ (I)

k ≫ 0, supp G(k) ∩ supp H(k) = Ø

: overlapping action

K

hidden relations!

∃

SLIDE 12

Spse for some . Then , overlapping. Theorem C implies

r

.

(G × H) * ℤ ↪ Diff1+τ

+ (I)

τ > 0 G × H ≤ Diff1+τ

+ (I)

G(k) = 1 H(k) = 1 □

Overlapping actions

Theorem C Theorem A

⟹

Proof of Theorem B Show that a —blow-up of the standard action

f is non-overlapping.

C1 F □

Theorem (Brum—Matte Bon—Rivas—Triestino) Every faithful —action of on is semiconjugate to the standard action.

C1 F I

Theorem A (K—Koberda—Rivas) If and are non-solvable, then for all Theorem B (K—Koberda—Rivas) . Theorem C (K—Koberda—Rivas) If and then .

G H (G × H) * ℤ / ↪ Diff1+τ

+ (I)

τ > 0. F * ℤ / ↪ Diff1

+(I)

G × H ≤ Diff1+τ

+ (I)

k ≫ 0, supp G(k) ∩ supp H(k) = Ø

, for all

K * ℤ ↪ Diff1

+(I)

⟹ supp a ∩ supp b ≠ Ø a, b ∈ K

SLIDE 13

C

n

r a d i a n a c t i

n

Conradian action and (k,1)—nesting

Theorem C (K—Koberda—Rivas) If and , then .

G × H ≤ Diff1+τ

+ (I)

k ≫ 0 supp G(k) ∩ supp H(k) = Ø

(Navas 2008) : intermediate growth No rank-two free semigroup in No two-chain of supporting intervals no faithful —action (cf. Navas 2008, Deroin—Kleptsyn—Navas 2007 Castro—Jorquera—Navas 2014) (1) , Conradian, , —nesting.

G ⟹ G ⟹ ⟹ C1+τ G ≤ Homeo+(ℝ) G(k) ≠ 1 c ∈ Z(G) Fix c = Ø ⟹∃(k,1)

“two-chain” ≠ , “ L e v e l s t r u c t u r e ”

≈

∃gi giJi

For all i = 2,..,k

Ji−1 Ji

—nesting

∃(k,1)

SLIDE 14

Smoother diffeomorphisms are slower!

Conradian action and (k,1)—nesting

(cf. Navas 2008, Deroin—Kleptsyn—Navas 2007 Castro—Jorquera—Navas 2014) (1) , Conradian, , —nesting, i.e.

G ≤ Homeo+(ℝ) G(k) ≠ 1 c ∈ Z(G) Fix c = Ø ⟹∃(k,1)

J1 J2 ⋮ Jk cJ1 cJ2 ⋮ cJk Ji−1 ∃gi Ji giJi gi

For all i = 2,..,k ≠ , “ L e v e l s t r u c t u r e ”

≈

(2) , —nesting .

G ≤ Diff1+τ

+ (I) ∃(k,1)

⟹ τ(1 + τ)k−2 ≤ 1

J f x f(x)

| f(x) − x| ≤ [f (k)]τ|J|k+τ

SLIDE 15

Generalization of Bonatti—Crovisier—Wilkinson

Centralizer—Conradian Lemma (KKR) , centralizes is Conradian.

G ≤ Diff1+τ

+ (I) τ > 0

G c ∈ Diff1

+(I)

⟹ G ↾supp c

c g1 g2 g1 g2 c

Theorem C (K—Koberda—Rivas) If and , then .

G × H ≤ Diff1+τ

+ (I)

k ≫ 0 supp G(k) ∩ supp H(k) = Ø

find Conradian restrictions! Proof of Theorem C

gk ∈ G(k), hk ∈ H(k)

SLIDE 16

Theorem A (K—Koberda—Rivas) If and are non-solvable, then for all Corollary for . Question (1) ? (2) Are there overlapping ? is overlapping for all . Remark (Bonatti—Farinelli 2015) Overlapping

G H (G × H) * ℤ / ↪ Diff1+τ

+ (I)

τ > 0. (F2 × F2) * ℤ = A( □ ∙ ) / ↪ Diff1+τ

+ (I)

τ > 0 (F2 × ℤ) * ℤ ↪ Diff1+τ

+ (I)

F2 × ℤ ↪ Diff1+τ

+ (I)

G ≤ Homeo+(I) ⟺ supp a ∩ supp b ≠ Ø a, b ∈ G ∃ F2 × ℤ ≤ Diff1

+(I) .

Summary and further questions

F2 F2 F2 F2 F2 ℤ

Centralizer—Conradian Lemma (KKR) , centralizes is Conradian.

G ≤ Diff1+τ

+ (I) τ > 0

G c ∈ Diff1

+(I)

⟹ G ↾supp c

c g1 g2 g1 g2 Generalization of Bonatti—Crovisier—Wilkinson c

Theorem C (K—Koberda—Rivas) If and , then .

G × H ≤ Diff1+τ

+ (I)

k ≫ 0 supp G(k) ∩ supp H(k) = Ø

find Conradian restrictions!

SLIDE 17

Summary and further questions

Theorem A (K—Koberda—Rivas) If and are non-solvable, then for all Corollary for . Question (1) ? (2) Are there overlapping ? is overlapping for all . Remark (Bonatti—Farinelli 2015) Overlapping

G H (G × H) * ℤ / ↪ Diff1+τ

+ (I)

τ > 0. (F2 × F2) * ℤ = A( □ ∙ ) / ↪ Diff1+τ

+ (I)

τ > 0 (F2 × ℤ) * ℤ ↪ Diff1+τ

+ (I)

F2 × ℤ ↪ Diff1+τ

+ (I)

G ≤ Homeo+(I) ⟺ supp a ∩ supp b ≠ Ø a, b ∈ G ∃ F2 × ℤ ≤ Diff1

+(I) .