Gabriel Favre: Amenable-like properties of étale groupoids

Abstract.

This thesis consists of three papers related to analytic and representation theoretic properties of étale groupoids. In the first paper, we characterize algebraically the type I and CCR property for ample groupoids and their non-commutative duals: Boolean inverse semigroups. Our results use and generalize Thoma’s work on discrete groups. Algebraic characterizations in the more general context of non-Hausdorff groupoids have been obtained in the author’s licentiate thesis. They use a non-Hausdorff version of the Clark-van Wyk topological characterization. We also characterize type I inverse semigroups using the Booleanization of inverse semigroups introduced by Lawson. The inverse semigroups of type I are characterized by excluding specific subquotients of their Booleanization.

In the second paper, we show that any free action of a connected Lie group of polynomial growth on a finite dimensional locally compact space has a finite tubular dimension by constructing a tubular cover of appropriate multiplicity. As a consequence, the C∗-algebras associated to the corresponding transformation groupoids all have finite nuclear dimension. The proof strategy is adapted from the strategy for R-actions of Hirshberg-Wu to the polynomial growth setting. As a corollary, we obtain that the groupoids associated to model sets in connected simply connected nilpotent Lie groups admit a classifiable C∗-algebra. In the third paper, we study inner amenability for groupoids attached to irregular point sets in general second countable locally compact groups. Upon imposing a regularity condition on the point set–finite local complexity–we are able to show inner amenability of the corresponding ample groupoid. The motivation for this work is the question of Anantharaman-Delaroche asking whether all étale groupoids are inner amenable. As a motivating example, model sets arising from arithmetic lattices give inner amenable groupoids, even in non-amenable groups.

Link to the thesis on DiVA