Topological Full Groups of Étale Groupoids

Abstract

This is a survey of the recent development of the study of topological full groups ofétale groupoids on the Cantor set.Étale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations. Minimal Z-actions, minimal Z N -actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so on. Homology groups ofétale groupoids, groupoid C * -algebras and their Kgroups are also investigated.

Topological full groups of ample groupoids with applications to graph algebras

“…And the AH-conjecture relates the topological full group to the groupoid homology. These conjectures have been verified in several cases [Mat16b], in particular for (products of) graph groupoids arising from strongly connected finite graphs. For the more general graph groupoids studied in the present paper, the second named author will, together with Toke Meier Carlsen, attack these conjectures in a forthcoming paper.…”

“…Remark 11.18. The terminology AF-groupoid is due to Renault [Ren80], and is also used by Matui in [Mat12] and [Mat16b]. Note however, that Matui only considered the case of a compact unit space therein.…”

“…For descriptions of the topological full group in certain classes of examples, see Proposition 9.4, Remark 11.22 and Remark 11.28. See also [Mat16b] for a survey of about topological full groups of étale groupoids with compact unit space.…”

Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

“…Then our groups F A N , T A N , V A N for the matrix A N are nothing but the Brown's triple F N,1 , T N,1 , V N,1 for r = 1, respectively. Let A N,r be the r × r block matrix whose entries are N × N matrices such that such that D A N,r = D A N ⊗ D r , where D r is the commutative C * -algebra of the diagonal elements of the r × r full matrix algebra M r (C), our groups F A N,r , T A N,r , V A N,r for the matrix A N,r are nothing but the Brown's triple F N,r , T N,r , V N,r (see [18], [19]). Since det(id−A N,r ) = 1−N , the classification of the Higman-Thompson groups V N,r corresponds to that of the C * -algebras O N ⊗ M r (C) through Theorem 1.1 (see [24,Corollary 6.6], [21]).…”

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

“…Moreover, if we further assume G being also second countable and non-elementary, then its Cuntz semigroup Cu(C * r (G)) is almost divisible and Cu(C * r (G)) and Cu(C * r (G) ⊗ Z) are canonically order-isomorphic, where Z denotes the Jiang-Su algebra.Almost finiteness for an ample groupoid was introduced by Matui in [20]. He studied their topological full groups as well as the applications of almost finiteness to the homology of étale groupoids (see [21] for a survey of results in this direction). In [12], David Kerr specialised to almost finite group actions and treated them as a topological analogue of probability measure preserving hyperfinite equivalence relations, with the ultimate goal of transferring ideas from the classification of equivalence relations and von Neumann algebras to the world of (amenable) topological dynamics and C * -algebras.Recently, the classification program for C * -algebras has culminated in the outstanding theorem that all separable, simple, unital, nuclear, Z-stable C * -algebras satisfying the universal coefficient theorem (UCT) are classified by their Elliott-invariant (see [30, Corollary D] and [8, Corollary D]).…”