Topological full groups of ample groupoids with applications to graph algebras

Nyland, Petter; Ortega, Eduard

doi:10.1142/s0129167x19500186

Cited by 24 publications

(28 citation statements)

References 67 publications

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“…This means that G E∞ either is simple itself, or contains precisely one nontrivial normal subgroup, namely D( G E∞ ) (of index 2). The group G E∞ is nonamenable [Mat15b], but does have the Haagerup property [NO19]. One can also deduce that G E∞ is C * -simple by the results in [BS19].…”

Section: Establishing Property Trmentioning

confidence: 75%

Matui's AH conjecture for Graph Groupoids

Nyland¹,

Ortega²

2020

Preprint

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We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the K-theory of its groupoid C * -algebra.

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Section: Establishing Property Trmentioning

confidence: 75%

Matui's AH conjecture for Graph Groupoids

Nyland¹,

Ortega²

2020

Preprint

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“…They studied the corresponding Higman-Thompson group and proved that it is isomorphic to the topological full group of the groupoid associated to pX A , σ A q. Their results were generalized for graph groupoids in [17]. Proof.…”

Section: Tables and The Higman-thompson Groupsmentioning

confidence: 99%

Higman-Thompson groups from self-similar groupoid actions

Deaconu¹

2021

Preprint

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Given a self-similar groupoid action pG, Eq on a finite directed graph, we prove some properties of the corresponding ample groupoid of germs GpG, Eq. We study the analogue of the Higman-Thompson group associated to pG, Eq using G-tables and relate it to the topological full group of GpG, Eq, which is isomorphic to a subgroup of unitaries in the algebra C ˚pG, Eq. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for GpG, Eq in some particular cases. introductionSelf-similar group actions are defined using subgroups of the automorphism group of a rooted tree which is viewed as the path space of a graph with one vertex and n edges. Given a finite directed graph E with no sources, the corresponding path space gives rise to a union of trees (or forest) T E , and it is natural to consider self-similar actions of subgroupoids G of PIsopT E q, the set of partial isomorphsims of T E .Given such a self-similar action pG, Eq, we study the properties of the generalized Exel-Pardo ample groupoid GpG, Eq. Since the source map of the graph is in general not G-equivariant, this determines certain modifications in the proofs of similar results from [5], where G is a group acting on E by graph automorphisms. We study the topological full group vGpG, Eqw and we relate it to the Higman-Thompson type group defined by G-tables for the action pG, Eq. After recalling some results about groupoid homology, we discuss the Matui's AH-conjecture for GpG, Eq and related issues.We begin by recalling the definition of a self-similar groupoid action pG, Eq from [7] and give several examples, illustrating some differences with the case when G is a group acting by automorphisms of the graph E. We recall some facts about the structure of the associated C ˚algebra C ˚pG, Eq defined using a C ˚-correspondence over C ˚pGq and 2020 Mathematics Subject Classification. Primary 46L05.

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“…In fact, when adding sinks to ultragraphs, or to graphs, many unforeseen difficulties arise. For example, the results describing topological full groups as invariant for continuous orbit equivalence of shift spaces (or groupoids) associated to graphs (or ultragraphs) are only valid for graphs (or ultragraphs) without sinks (see [32] and [11]), and there is no clear way to extend these results to include graphs and ultragraphs with sinks. Furthermore, when studying KMS states associated to graphs the existence of sinks induce new KMS states (see [12]).…”

Section: Introductionmentioning

confidence: 99%

KMS states and continuous orbit equivalence for ultragraph shift spaces with sinks

Tasca¹

2020

Preprint

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We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).

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Topological full groups of ample groupoids with applications to graph algebras

Cited by 24 publications

References 67 publications

Matui's AH conjecture for Graph Groupoids

Matui's AH conjecture for Graph Groupoids

Higman-Thompson groups from self-similar groupoid actions

KMS states and continuous orbit equivalence for ultragraph shift spaces with sinks

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