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.