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.

Revised version. To appear in Ergodic Theory and Dynamical Systems

References in corpus (5)

• Some notes on property A
• Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms
• Almost elementary étale groupoids
• The groupoids of adaptable separated graphs and their type semigroup
• Strict comparison for $C^{\ast }$-algebras arising from Almost finite groupoids

Cited by in corpus (4)

• Strict comparison for $C^{\ast }$-algebras arising from Almost finite groupoids
• Purely infinite locally compact Hausdorff étale groupoids and their $C^{\ast }$-algebras
• Comparison and Simplicity of Commutator Subgroups of Full Groups
• Dynamic asymptotic dimension and Matui's HK conjecture