The type semigroup, comparison, and almost finiteness for ample groupoids

Abstract

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

Dynamic asymptotic dimension and Matui's HK conjecture

“…The following result adds to this list of connections. It might be known to experts but does not seem to appear in the literature so far (except for degree zero, which has been treated in [2]).…”

“…Let 𝐾 be as in the definition of closeness for 𝛼 and 𝛽, and assume also that 𝐾 is so large that the colouring underlying  (𝑚) is 𝐾-bounded. Let 𝑙 ⩾ 𝑚 be large enough so that if 𝜄 ∶  (𝑚) →  (𝑙) is the composition of the morphisms in the definition of the anti-Čech sequence, then for all 𝑈 ∈  (𝑚) , we have that 𝜄(𝑈) ⊇ 𝑈𝐾 2 (such an 𝑙 exists by definition of an anti-Čech sequence). It will suffice to show that 𝜄•𝛼 and 𝜄•𝛽 induce the same map 𝐻 * (𝐶) → 𝐻 * ( (𝑙) ).…”

“…As 𝛼 and 𝛽 are close with respect to 𝐾, we have that 𝛼(𝑥 0 ) and 𝛽(𝑥 0 ) are both subsets of g𝐾 for some g ∈ 𝐺, whence there are 𝑘 𝛼 and 𝑘 𝛽 in 𝐾 such that g 𝛼 = g𝑘 𝛼 and g 𝛽 = g𝑘 𝛽 . Hence, g 𝛼 = g 𝛽 𝑘 𝛽 𝑘 −1 𝛼 , so in particular g 𝛼 is in g 𝛽 𝐾 2 . Now, by choice of 𝜄, 𝜄(𝛽(𝑥 𝑗 )) ⊇ 𝛽(𝑥 𝑗 )𝐾 2 for all 𝑗, whence g 𝛼 is in 𝜄(𝛽(𝑥 𝑗 )) for all 𝑗.…”

“…Then the colouring underlying  (1) * is 𝐾-bounded for some compact open subset 𝐾 of 𝐺, which we may assume contains 𝐺 0 , and that satisfies 𝐾 = 𝐾 −1 . Set 𝐾 1 ∶= 𝐾 2 . Then Lemma 3.20 gives a morphism Ψ (1) ∶  anti-Čech sequence there is 𝑚 2 > 𝑚 1 such that  (𝑚 2 ) is 𝐾 1 -Lebesgue, whence Lemma 3.19 gives a morphism Φ (1) ∶ 𝐺 𝐾 1 * →  (𝑚 2 ) .…”

“…Lemma 4.11. Let 𝜓 1 ∈ 𝐾𝐾 𝐺 (𝐶 0 (𝐺 (2) ), 𝑁 1 ) = 𝐾𝐾 𝐺 (𝑃 1 , 𝑁 1 ) be the morphism from the following part of a phantom tower for 𝐴 = 𝐶 0 (𝐺 0 )…”