Ergodicity and type of nonsingular Bernoulli actions Richard Kadison and his mathematical legacy – A memorial conference University of Copenhagen 29 - 30 November 2019 Stefaan Vaes 1/17
Bernoulli actions Bernoulli actions of a countable group G For any standard probability space ( X 0 , µ 0 ), consider G � ( X 0 , µ 0 ) G = � ( X 0 , µ 0 ) given by ( g · x ) h = x g − 1 h . g ∈ G ◮ ( G = Z ) Kolmogorov-Sinai : entropy of µ 0 is a conjugacy invariant. ◮ ( G = Z ) Ornstein : entropy is a complete invariant. ◮ Bowen : beyond amenable groups, sofic groups. ◮ Popa : orbit equivalence rigidity, von Neumann algebra rigidity. � What about G � ( X 0 , µ g ) given by ( g · x ) h = x g − 1 h ? g ∈ G Main motivation: produce interesting families of type III group actions. 2/17
Group actions of type III ◮ The classical Bernoulli action G � ( X , µ ) = ( X 0 , µ 0 ) G • is ergodic, • preserves the probability measure µ . ◮ An action G � ( X , µ ) is called non-singular if µ ( g · U ) = 0 whenever µ ( U ) = 0 and g ∈ G . ◮ Write U ∼ V if there exists a measurable bijection ∆ : U → V with ∆( x ) ∈ G · x for a.e. x ∈ U . ◮ A nonsingular ergodic G � ( X , µ ) is of type III if U ∼ V for all non-negligible U , V ⊂ X . • There is no G -invariant measure in the measure class of µ . • The Radon-Nikodym derivative d ( g · µ ) / d µ must be sufficiently wild. 3/17
Group actions of type III 1 Let G � ( X , µ ) be a nonsingular group action. ◮ Write ω ( g , x ) = d ( g − 1 · µ ) ( x ), the Radon-Nikodym 1-cocycle. d µ ◮ The action G � X × R given by g · ( x , s ) = ( g · x , s + log( ω ( g , x ))) preserves the (infinite) measure µ × e − s ds . ◮ This is called the Maharam extension. It is the ergodic analogue of the Connes-Takesaki continuous core for von Neumann algebras. An ergodic nonsingular action G � ( X , µ ) is of type III 1 if its Maharam extension remains ergodic. Associated ergodic flow R � L ∞ ( X × R ) G . G � ( X , µ ) is of type III iff this flow is not just R � R . G � ( X , µ ) is of type III λ iff this flow is R � R / Z log λ . 4/17
Bernoulli actions of type III � Consider G � ( X , µ ) = ( X 0 , µ g ) given by ( g · x ) h = x g − 1 h . g ∈ G 1 All µ g are equal : type II 1 , ergodic, probability measure preserving. 2 Interesting gray zone : when is G � ( X , µ ) of type III, or type III 1 ? 3 The µ g are quite different : type I, the action is dissipative , meaning � that X = g · U up to measure zero. g ∈ G 4 The µ g are very different : the action is singular. 5/17
Kakutani’s criterion � ◮ The action G � ( X 0 , µ g ) is nonsingular if and only if g ∈ G d ( µ gh , µ h ) 2 < ∞ . � for every g ∈ G , we have h ∈ G ◮ Take X 0 = { 0 , 1 } with 0 < µ g (0) < 1. Assume that δ ≤ µ g (0) ≤ 1 − δ for all g ∈ G . Then, the action is nonsingular if and only if | µ gh (0) − µ h (0) | 2 < ∞ for all g ∈ G . � h ∈ G Then c : G → ℓ 2 ( G ) : c g ( h ) = µ h (0) − µ g − 1 h (0) is a 1-cocycle for the left regular representation, meaning that c gh = c g + λ g c h . 6/17
An easy no-go theorem Theorem (V-Wahl, 2017) If H 1 ( G , ℓ 2 ( G )) = { 0 } , there are no nonsingular Bernoulli actions of type III. More precisely, every nonsingular Bernoulli action of G is the sum of a classical, probability measure preserving Bernoulli action and a dissipative Bernoulli action. ◮ The groups with H 1 ( G , ℓ 2 ( G )) = { 0 } are precisely the nonamenable groups with β (2) 1 ( G ) = 0. ◮ Large classes of nonamenable groups have β (2) 1 ( G ) = 0 : • property (T) groups, • groups that admit an infinite, amenable, normal subgroup, • direct products of infinite groups. 7/17
What if H 1 ( G , ℓ 2 ( G )) � = { 0 } ? This is very delicate ! Even for the case G = Z . ◮ (Krengel, 1970) The group G = Z admits a nonsingular Bernoulli action without invariant probability measure. ◮ (Hamachi, 1981) The group G = Z admits a nonsingular Bernoulli action of type III . ◮ (Kosloff, 2009) The group G = Z admits a nonsingular Bernoulli action of type III 1 . In all cases: no explicit constructions. 8/17
Dissipative versus conservative Recall: G � ( X , µ ) is dissipative iff X = � g ∈ G g · U up to measure zero. G � ( X , µ ) is conservative iff we return to every U ⊂ X with µ ( U ) > 0. Theorem (V-Wahl, 2017) Let G � � g ∈ G ( { 0 , 1 } , µ g ) be nonsingular. Let c g ( h ) = µ h (0) − µ g − 1 h (0). − 1 � 2 � c g � 2 ◮ If � � exp < ∞ , the action is dissipative. 2 g ∈ G ◮ If µ g (0) ∈ [ δ, 1 − δ ] for all g ∈ G − 3 δ − 2 � c g � 2 � � � and if exp = + ∞ , the action is conservative. 2 g ∈ G The growth of g �→ � c g � 2 should be sufficiently slow. 9/17
A naive example Take Z � � n ∈ Z ( { 0 , 1 } , µ n ) where ◮ µ n (0) = p if n < 0, ◮ µ n (0) = q if n ≥ 0. One might expect: if p � = q , then the action is of type III λ . But (Krengel 1970 and Hamachi 1981): if p � = q , the action is dissipative. Indeed: � c n � 2 2 ∼ | n | and � n ∈ Z exp( − ε | n | ) < + ∞ for every ε > 0. 10/17
Ergodicity of nonsingular Bernoulli actions Let G � ( X , µ ) = � g ∈ G ( { 0 , 1 } , µ g ) be any nonsingular Bernoulli action. Assume that µ g (0) ∈ [ δ, 1 − δ ] for all g ∈ G . ◮ (Kosloff, 2018) When G = Z and G � ( X , µ ) is conservative, then G � ( X , µ ) is ergodic. ◮ (Danilenko, 2018) When G is amenable and G � ( X , µ ) is conservative, then G � ( X , µ ) is ergodic. Tool: let R be the tail equivalence relation on ( X , µ ) given by x ∼ y iff x g � = y g for at most finitely many g ∈ G . ◮ They prove that any G -invariant function is R -invariant. ◮ Key role: Hurewicz ratio ergodic theorem (K) / a new pointwise ergodic theorem (D). 11/17
Ergodicity of nonsingular Bernoulli actions Let G � ( X , µ ) = � g ∈ G ( { 0 , 1 } , µ g ) be any nonsingular Bernoulli action. Theorem (Bj¨ orklund-Kosloff-V, 2019) ◮ If G is abelian and G � ( X , µ ) is conservative, then G � ( X , µ ) is ergodic. So, no assumption that µ g (0) ∈ [ δ, 1 − δ ]. ◮ If G is arbitrary and G � ( X , µ ) is strongly conservative, then G � ( X , µ ) is ergodic. So, no amenability assumption. Assume that µ g (0) ∈ [ δ, 1 − δ ]. Write c g ( h ) = µ h (0) − µ g − 1 h (0). g ∈ G exp( − 8 δ − 1 � c g � 2 If � 2 ) = + ∞ , then G � ( X , µ ) is strongly conservative and thus ergodic. 12/17
Type of nonsingular Bernoulli actions Let G � ( X , µ ) = � g ∈ G ( { 0 , 1 } , µ g ) be a conservative Bernoulli action. ◮ Basically no systematic results on the type of G � ( X , µ ). ◮ (Bj¨ orklund-Kosloff, 2018) If G is amenable and lim g →∞ µ g (0) exists, then G � ( X , µ ) is either II 1 or III 1 . Theorem (Bj¨ orklund-Kosloff-V, 2019) Let G be abelian and not locally finite. ◮ If lim g →∞ µ g (0) does not exist: type III 1 . ◮ If lim g →∞ µ g (0) = λ and 0 < λ < 1, then type II 1 or type III 1 , g ∈ G ( µ g (0) − λ ) 2 being finite or not. depending on � ◮ If lim g →∞ µ g (0) = λ and λ ∈ { 0 , 1 } , then type III. Answering Krengel: a Bernoulli action of Z is never of type II ∞ . 13/17
Type of nonsingular Bernoulli actions Let G � ( X , µ ) = � g ∈ G ( { 0 , 1 } , µ g ) be nonsingular and µ g (0) ∈ [ δ, 1 − δ ]. Write c g ( h ) = µ h (0) − µ g − 1 h (0). Theorem (Bj¨ orklund-Kosloff-V, 2019) Assume that G has only one end. g ∈ G exp( − 8 δ − 1 � c g � 2 Assume that � 2 ) = + ∞ . Then, G � ( X , µ ) is of type III 1 , unless g ∈ G ( µ g (0) − λ ) 2 < + ∞ . Then type II 1 . for some 0 < λ < 1, we have � Corollary (answering conjecture of V-Wahl): a group G admits a type III 1 Bernoulli action iff H 1 ( G , ℓ 2 ( G )) � = { 0 } . Recall: the growth condition on the cocycle implies that G � ( X , µ ) is strongly conservative. 14/17
Ends of groups Recall. A finitely generated group G has more than one end if its Cayley graph has more than one end: there exists a finite subset F ⊂ G with disconnected complement. Proposition. A finitely generated group G has more than one end iff there exists a subset W ⊂ G such that ◮ W is almost invariant: | gW △ W | < ∞ for all g ∈ G , ◮ both W and G \ W are infinite. Use this as definition of “having more than one end” for arbitrary countable groups. 15/17
Ends of groups Stallings’ Theorem A countable group G has more than one end if and only if G is in one of the following families. ◮ Nontrivial amalgamated free products and HNN extensions over finite subgroups. ◮ Virtually cyclic groups. ◮ Locally finite groups. Due to Stallings for finitely generated groups. Due to Dicks & Dunwoody for arbitrary groups. 16/17
Ends of groups and nonsingular Bernoulli actions Let W ⊂ G be almost invariant. Define ◮ µ g (0) = p if g ∈ W , ◮ µ g (0) = q if g �∈ W . Then: G � ( X , µ ) = � g ∈ G ( { 0 , 1 } , µ g ) is a nonsingular Bernoulli action. But (remember G = Z and W = N ) : the action could be dissipative. Theorem (Bj¨ orklund-Kosloff-V, 2019) ◮ Infinite, locally finite groups admit Bernoulli actions of each possible type: II 1 , II ∞ , III 0 , III λ and III 1 . ◮ Every nonamenable group with more than one end admits nonsingular Bernoulli actions of type III λ for each λ close enough to 1. 17/17
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