Dynamical characterizations of paradoxicality for groups Eduardo - PowerPoint PPT Presentation

Dynamical characterizations of paradoxicality for groups Eduardo Scarparo University of Copenhagen

Definition Let G be a group acting on a set X. Given A , B contained in X, we say that A is equidecomposable with B (A ∼ B) if there exist partitions with the same finite number of pieces of A = ⊔ n i =1 A i and B = ⊔ n i =1 B i , and elements s 1 , . . . , s n ∈ G such that B i = s i A i for 1 ≤ i ≤ n. A subset A of X is said to be paradoxical if there exist B , C ⊂ A such that B ∩ C = ∅ and B ∼ A ∼ C. Consider the left action of a group G on itself and the following conditions: 1 G is not paradoxical; (amenability) 2 G does not contain any paradoxical subset; (supramenability, Rosenblatt ’74) 3 G is not equidecomposable with a proper subset. Equivalently, no subset of G is equidecomposable with a proper subset of itself.

Abelian groups and, more generally, groups of subexponential growth are supramenable. It is not known if supramenability implies subexponential growth. Example Suppose a group G contains a free monoid SF 2 generated by two elements a and b . Then SF 2 ⊂ G is paradoxical. The lamplighter group ( � Z Z 2 ) ⋊ Z contains a free monoid generated by two elements. Hence, it is not supramenable. Theorem (Tarski ’29) A subset A of a group G is non-paradoxical if and only if there is a finitely additive, invariant measure µ : P ( G ) → [0 , + ∞ ] such that µ ( A ) = 1 .

Definition (Exel ’94 + McClanahan ’95) Let X be a topological space and { D g } g ∈ G be a family of open subsets of X. A partial action of a group G on X is a map θ : G → pHomeo ( X ) g �→ θ g : D g − 1 → D g such that: 1 θ e = Id X ; 2 For all g , h ∈ G, x ∈ D g − 1 , if θ g ( x ) ∈ D h − 1 , then x ∈ D ( hg ) − 1 and θ h ◦ θ g ( x ) = θ hg ( x ) . Example Let θ be a (global) action of a group G on a topological space X . Given a non-empty open subset D ⊂ X , define, for all g ∈ G , D g := D ∩ θ g ( D ). One can check that the restrictions of the maps θ g to the open subsets D g − 1 give rise to a partial action of G on D .

It is well-known that a group is amenable if and only if whenever it acts on a compact Hausdorff space X , there is an invariant probability measure on X . Definition Let ( { D g } g ∈ G , { θ g } g ∈ G ) be a partial action of a group G on a topological space X. We say a measure ν on X is invariant if, for all E ∈ B ( X ) and g ∈ G, we have that ν ( θ g ( E ∩ D g − 1 )) = ν ( E ∩ D g − 1 ) . Theorem A group G is supramenable if and only if whenever it partially acts on a compact Hausdorff space X, there is an invariant probability measure on X. Proof. If A ⊂ G is paradoxical, then the the restriction of the action of G on β G to β A ⊂ β G admits no invariant probability measure.

Proof. Conversely, assume G is supramenable. ˆ f ( g ) := f ( θ g ( x 0 )) . Since G is supramenable, there exists an invariant, finitely additive � ˆ measure µ on G such that µ ( A ) = 1. Then f �→ f d µ is an ”invariant” state on C ( X ). Using the Riesz representation theorem, this gives rise to an invariant probability measure on X . Given a partial action, one can associate to it a crossed product. Corollary A group is supramenable if and only if whenever it partially acts on a compact Hausdorff space, there is tracial state on C ( X ) ⋊ G.

Proposition If G is a countable, amenable, non-supramenable group, then there is a partial action of G on the Cantor set K such that C ( K ) ⋊ G is a simple and purely infinite algebra. This follows from a result of Kellerhals, Monod and Rørdam (’13) about actions of non-supramenable groups on K × N . Theorem (Xin Li ’15) If an exact group G contains a non-abelian free monoid, then, given any countable graph E, there is an action of G on the boundary-path space ∂ E of E such that C ∗ ( E ) ≃ C 0 ( ∂ E ) ⋊ r G.

Theorem Let G be a group acting on a set X. The following are equivalent: 1 For every finitely generated subgroup H of G and every x ∈ X, the H-orbit of x is finite; 2 ℓ ∞ ( X ) ⋊ r G is finite; 3 X is not equidecomposable with a proper subset of itself; 4 No subset of X is equidecomposable with a proper subset of itself. Corollary (Kellerhals-Monod-Rørdam ’13, S. ’15) Let G be a group. The following conditions are equivalent: 1 G is locally finite; 2 The uniform Roe algebra ℓ ∞ ( G ) ⋊ r G is finite; 3 G is not equidecomposable with a proper subset of itself; 4 No subset A ⊂ G is equidecomposable with a proper subset of itself.

Definition Let G be a group acting on a compact, totally disconnected metric space X. Given A , B clopen subsets of X, A is said to be equidecomposable with B (A ∼ B) if there exist partitions with the same finite number of clopen pieces of A = ⊔ n i =1 A i and B = ⊔ n i =1 B i , elements s 1 , . . . , s n ∈ G such that B i = s i A i for 1 ≤ i ≤ n. Example Let X := Z ∪ {±∞} and consider the homeomorphism T on X given by T ( x ) := x + 1 for x ∈ Z , T ( ±∞ ) := ±∞ , and the associated Z -action on X . Since T ([0 , + ∞ ]) = [1 , + ∞ ], it follows that [0 , + ∞ ] ∼ [1 , + ∞ ]. If some clopen subset of X is equidecomposable with a proper clopen subset of itself, then C ( X ) ⋊ r G is infinite. Does the converse hold?