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

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

Critical regularity of a group 1 ≤ r < s ⟹ Di ff r + ( I ) ≥ Di ff s + ( I ) : compact interval I (K.—Koberda 2020) G r ≤ Di ff r 1 ≤ r ⟹ ∃ f.g. such that + ( I ) Di ff k C k + ( I ) = { f : I → I | f is —diffeo, f ′ > 0} . ↪ Di ff s for all . G r / + ( I ) s > r analysis ⟹ group theory Di ff k + τ + ( I ) ∣ f ( k ) + ( I ) = { f ∈ Di ff k is —Hölder con } τ Rigidity g : I → ℝ Study representations of Γ → G is —Hölder continuous if τ | gx − gy | Γ for a “lattice” and a topological group . G < ∞ . [ g ] τ := sup [ g ] 1 := [ g ] Lip | x − y | τ Definition (critical regularity) x ≠ y CritReg ( G ) := sup{ r ≥ 0 | G ↪ Di ff r + ( I )} + ( I ) ≠ Di ff k +Lip Di ff k +1 cf. . ( I ) + (Deroin—Kleptsyn—Navas 2007) Theme G ≤ Homeo + ( I ) G ∼ If is countable, then biLip. Di ff r Which f.g. groups arise as subgps of ? + ( I ) ( G ) ≥ 1 = − ∞ i.e. CritReg or .

Motivation from the foliation theory Definition (critical regularity) { π 1 ( B ) → Di ff r ( F ) }/ conj . CritReg ( G ) := sup{ r ≥ 0 | G ↪ Di ff r + ( I )} { C r foliated B -bundles over F }/ isom . ↭ B × F ) / ( x , t ) ∼ ( g . x , ρ ( g ) . t ) ∀ g ∈ π 1 ( B ) ρ ↭ F → ( ˜ ↓ B Thurston Stability (1974) cpt cnt transver. orientable ( M n , ℱ ) C 1 codimension-one foliation W. Thurston H 1 ( L ; ℝ ) = 0 w/ a cpt leaf s.t. L (1946-2012) ⟹ L → M → S 1 M ≅ L × I or .

Motivation from the foliation theory Thurston Stability Lemma (1974) Thurston Stability (1974) Di ff 1 [0,1) is locally indicable. cpt cnt transver. orientable ∀ 1 ≠ H f.g. ≤ Di ff 1 [0,1) ( M n , ℱ ) ℤ . i.e. surjects onto C 1 codimension-one foliation H 1 ( L ) = 0 ⇒ ∀ ρ : π 1 ( L ) → Di ff 1 [0,1) e.g. is trivial. W. Thurston H 1 ( L ; ℝ ) = 0 w/ a cpt leaf s.t. L (1946-2012) π 1 ( M SFS ) = ⟨ a , b , c ∣ a 2 = b 3 = c 7 = abc ⟩ e.g. ⟹ P . Kropholler ≤ ˜ PSL(2, ℝ ) ≤ Homeo + ( I ) W. Thurston L → M → S 1 M ≅ L × I or . ↪ Di ff 1 π 1 ( M SFS ) / Not locally indicable: . + ( I ) C 0 C 1 M × S 1 f , but not , foliation on . (Plante—Thurston 1976) Di ff 2 ∀ nilpotent subgroup of is abelian. + ( I ) (Farb–Franks 2003) Di ff 1 ∀ f.g. (res.) tor-free nilpotent gp embeds into . + ( I )

CritReg: non-exponential growth groups Definition (critical regularity) Heisenberg group CritReg ( G ) := sup{ r ≥ 0 | G ↪ Di ff r + ( I )} Heis = N 3 ≤ Di ff 1 + ( I ) (Plante—Thurston 1976) (Castro—Jorquera—Navas 2014) Di ff 2 Heis ↪ Di ff 2 − ϵ ∀ nilpotent subgroup of is abelian. + ( I ) , i.e. CritReg + ( I ) ( Heis ) = 2 (Farb–Franks 2003) (Jorquera—Navas—Rivas 2017) Di ff 1 N 4 ↪ Di ff 1.5 − ϵ ↪ Di ff 1.5+ ϵ ∀ f.g. (res.) tor-free nilpotent gp embeds into . + ( I ) , but , ( I ) / ( I ) + + i.e. CritReg ( N 4 ) = 1.5 (Navas 2008) : intermediate growth G ↪ Di ff 1+ τ ⟹ G / for + ( I ) τ > 0. ↪ Di ff 1 Moreover, Grigorchuk—Machi group + ( I )

CritReg: more examples (Witte ’94; Burger–Monod ’99, Ghys ’99) Heisenberg group ↪ Di ff 1 + ( S 1 ) higher-rank lattice / Heis = N 3 ≤ Di ff 1 + ( I ) (Navas ’02 / Bader–Furman–Gelander–Monod ’07) (Castro—Jorquera—Navas 2014) Heis ↪ Di ff 2 − ϵ , i.e. CritReg + ( I ) ( Heis ) = 2 ↪ Di ff 1.5 + ( S 1 ) Property T group / (Jorquera—Navas—Rivas 2017) (Brown–Fisher–Hurtado) d > n N 4 ↪ Di ff 1.5 − ϵ ↪ Di ff 1.5+ ϵ , but , ( I ) / ( I ) + + + ( X n − manifold ) ↪ Di ff 1 rank-d lattice / i.e. CritReg ( N 4 ) = 1.5 (Navas 2008) : intermediate growth G ↪ Di ff 2 (Farb–Franks ’01) Mod(S g ≥ 3,p ≤ 1 ) / + ( M ) ↪ Di ff 1+ τ ⟹ G / for + ( I ) τ > 0. 3 g − 3 + p ≤ 1 ⟺ (Baik–K–Koberda ’16) ↪ Di ff 1 Moreover, Grigorchuk—Machi group + ( I ) Mod( S g , p ) ↪ Di ff 2 , virtually. + ( M )

CritReg: exponential growth groups? Thompson’s group F Question 2 ℤ : PL homeo of [0,1] w/ dyadic breakpts & slopes F Are there exponential growth examples? PPSL(2, ℤ ) (Thurston, using -representation) C 1 (for -smooth groups) F ↪ Di ff 1 Thompson’s group . + [0,1] (Ghys—Sergiescu 1987) Di ff ∞ F ↪ PL[0,1] is conjugate into + [0,1] ( F ) = ∞ In particular, CritReg . (K—K—Lodha 2019) ∀ N ≫ 0 < f N , g N > ≅ F , . f “two-chain” g

Right-angled Artin groups Γ : finite simplicial graph. Question Are there exponential growth examples? Γ The RAAG ( right-angled Artin group ) on is: C 1 (for -smooth groups) A ( Γ ) := < V ( Γ ) | [ a , b ] = 1 ∀ { a , b } ∈ E ( Γ ) > A ( △ ) ≅ ℤ 3 e.g. A ( ∴ ) ≅ F 3 A ( ∙ − ∙ − ∙ ) ≅ F 2 × ℤ (Agol, Wise 2012) ∀ ↪ ∃ A ( Γ ) fin. vol. hyp. 3-mfd gp virtually S 2 → E ∀ A ( Γ ) ↪ Symp( S 2 ) (M. Kapovich) ↓ M hyp,3 (Baik—K—Koberda 2019) ↪ Di ff 2 + ( I or S 1 ) A ( ∙ − ∙ − ∙ − ∙ ) / . ↪ Di ff 2 + ( I or S 1 ) Cor (BKK) Mod( Σ g , p ) virtually ⟺ 3 g − 3 + p ≤ 1.

Right-angled Artin groups ⟹ A ( Γ ) ≤ Di ff 1 (FF2003) . + ( I ) (K—Koberda 2018) Γ : finite simplicial graph. A ( ∙ − ∙ − ∙ ∙ ) A ( Γ ) ↪ Di ff 2 + ( I ) ⟺ ( F 2 × ℤ ) * ℤ / ↪ A ( Γ ) Γ The RAAG ( right-angled Artin group ) on is: ⟺ A ( Γ ) ↪ Di ff ∞ A ( Γ ) := < V ( Γ ) | [ a , b ] = 1 ∀ { a , b } ∈ E ( Γ ) > + ( I ) A ( △ ) ≅ ℤ 3 e.g. A ( ∴ ) ≅ F 3 ( A ( Γ )) ∈ [1,2] = ∞ CritReg or A ( ∙ − ∙ − ∙ ) ≅ F 2 × ℤ Question (Agol, Wise 2012) ( A ( Γ )) = ? (1) CritReg ∀ ↪ ∃ A ( Γ ) fin. vol. hyp. 3-mfd gp virtually (( F 2 × ℤ ) * ℤ ) = ? (2) CritReg S 2 → E ∀ A ( Γ ) ↪ Symp( S 2 ) (M. Kapovich) ↓ M hyp,3 (Baik—K—Koberda 2019) ↪ Di ff 2 + ( I or S 1 ) A ( ∙ − ∙ − ∙ − ∙ ) / . ↪ Di ff 2 + ( I or S 1 ) Cor (BKK) Mod( Σ g , p ) virtually ⟺ 3 g − 3 + p ≤ 1.

Right-angled Artin groups Theorem A (K—Koberda—Rivas) Question If and are non-solvable, G H ( A ( Γ )) = ? (1) CritReg ↪ Di ff 1+ τ ( G × H ) * ℤ / then for all + ( I ) τ > 0. (( F 2 × ℤ ) * ℤ ) = ? (2) CritReg h t w o r g l a i Cor CritReg (( F 2 × F 2 ) * ℤ ) = 1. t n e n A ( ∙ − ∙ − ∙ ∙ ) o p x e ( F * ℤ ) = 1. CritReg F × F ↪ F Theorem B (K—Koberda—Rivas) ↪ Di ff 1 F * ℤ / . + ( I ) (Navas) ↪ Di ff 1 ( BS (1,2) × ℤ ) * ℤ / + ( I )

Overlapping actions Theorem A (K—Koberda—Rivas) f ∈ Homeo( X ) ⇝ supp f := X ∖ Fix f If and are non-solvable, G H G ≤ Homeo( X ) ⇝ supp G := ∪ g ∈ G supp g ↪ Di ff 1+ τ ( G × H ) * ℤ / then for all + ( I ) τ > 0. Theorem B (K—Koberda—Rivas) The —Lemma (K—Koberda 2018) abt ↪ Di ff 1 F * ℤ / . + ( I ) a , b , t ∈ Di ff 1 + ( I or S 1 ) If satisfy hidden relations! ∃ supp a ∩ supp b = Ø , ⟨ a , b , t ⟩ ≇ ( ℤ × ℤ ) * ℤ then . : overlapping action K * ℤ ↪ Di ff 1 , K + ( I ) ⟹ supp a ∩ supp b ≠ Ø for all a , b ∈ K Theorem C (K—Koberda—Rivas) G × H ≤ Di ff 1+ τ k ≫ 0, If and + ( I ) supp G ( k ) ∩ supp H ( k ) = Ø then .

Overlapping actions Theorem C ⟹ Theorem A ( G × H ) * ℤ ↪ Di ff 1+ τ Spse for some . + ( I ) τ > 0 K * ℤ ↪ Di ff 1 , + ( I ) G × H ≤ Di ff 1+ τ Then , overlapping. + ( I ) ⟹ supp a ∩ supp b ≠ Ø for all a , b ∈ K G ( k ) = 1 H ( k ) = 1 □ Theorem C implies or . Theorem A (K—Koberda—Rivas) If and are non-solvable, Theorem (Brum—Matte Bon—Rivas—Triestino) G H ↪ Di ff 1+ τ C 1 then ( G × H ) * ℤ / for all Every faithful —action of on is + ( I ) τ > 0. F I semiconjugate to the standard action. Theorem B (K—Koberda—Rivas) ↪ Di ff 1 F * ℤ / . + ( I ) Proof of Theorem B Theorem C (K—Koberda—Rivas) C 1 Show that a —blow-up of the standard action G × H ≤ Di ff 1+ τ k ≫ 0, If and + ( I ) □ of is non-overlapping. F supp G ( k ) ∩ supp H ( k ) = Ø then .

Conradian action and (k,1)—nesting (Navas 2008) : intermediate growth G Theorem C (K—Koberda—Rivas) ⟹ No rank-two free semigroup in G G × H ≤ Di ff 1+ τ k ≫ 0 If and , n + ( I ) ⟹ No two-chain of supporting intervals o i t c a n a i d a r n supp G ( k ) ∩ supp H ( k ) = Ø o C C 1+ τ then . ⟹ no faithful —action (cf. Navas 2008, Deroin—Kleptsyn—Navas 2007 Castro—Jorquera—Navas 2014) “two-chain” G ( k ) ≠ 1 G ≤ Homeo + ( ℝ ) (1) , Conradian, ” e r u t c u c ∈ Z ( G ) Fix c = Ø , r t s l e v e L “ ≈ , ≠ ∃ ( k ,1) —nesting ⟹∃ ( k ,1) —nesting. ∃ g i J i − 1 J i g i J i For all i = 2,.., k

Conradian action and (k,1)—nesting G ≤ Di ff 1+ τ (2) + ( I ) ∃ ( k ,1) , —nesting ⟹ τ (1 + τ ) k − 2 ≤ 1 (cf. Navas 2008, Deroin—Kleptsyn—Navas 2007 . Castro—Jorquera—Navas 2014) Smoother diffeomorphisms are slower! G ( k ) ≠ 1 G ≤ Homeo + ( ℝ ) (1) , Conradian, ” e f r u J t c u r c ∈ Z ( G ) Fix c = Ø , t s l e v f ( x ) e x L “ ≈ , ≠ ⟹∃ ( k ,1) —nesting, i.e. | f ( x ) − x | ≤ [ f ( k ) ] τ | J | k + τ J 1 cJ 1 J 2 cJ 2 ⋮ ⋮ J k cJ k ∃ g i J i − 1 J i g i J i For all i = 2,.., k g i