SLIDE 1
Dynamical characterizations of paradoxicality for groups
Eduardo Scarparo
University of Copenhagen
SLIDE 2 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=1Ai and
B = ⊔n
i=1Bi, and elements s1, . . . , sn ∈ G such that Bi = siAi 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.
SLIDE 3
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 SF2 generated by two elements a and b. Then SF2 ⊂ G is paradoxical. The lamplighter group (
Z Z2) ⋊ 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.
SLIDE 4 Definition (Exel ’94 + McClanahan ’95) Let X be a topological space and {Dg}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 → θg : Dg−1 → Dg such that:
1 θe = IdX; 2 For all g, h ∈ G, x ∈ Dg−1, if θg(x) ∈ Dh−1, then x ∈ D(hg)−1
and θh ◦ θg(x) = θhg(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, Dg := D ∩ θg(D). One can check that the restrictions of the maps θg to the open subsets Dg−1 give rise to a partial action of G on D.
SLIDE 5 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 ({Dg}g∈G, {θg}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(θg(E ∩ Dg−1)) = ν(E ∩ Dg−1). Theorem A group G is supramenable if and only if whenever it partially acts
- n 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
- n βG to βA ⊂ βG admits no invariant probability measure.
SLIDE 6
Proof. Conversely, assume G is supramenable. ˆ f (g) := f (θg(x0)). Since G is supramenable, there exists an invariant, finitely additive measure µ on G such that µ(A) = 1. Then f → ˆ f dµ 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.
SLIDE 7
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) ≃ C0(∂E) ⋊r G.
SLIDE 8 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.
SLIDE 9
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=1Ai and
B = ⊔n
i=1Bi, elements s1, . . . , sn ∈ G such that Bi = siAi 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?
