Katsura--Exel--Pardo Groupoids and the AH~Conjecture

Nyland, Petter; Ortega, Eduard

doi:10.48550/arxiv.2007.06638

Cited by 2 publications

(7 citation statements)

References 6 publications

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“…The AH conjecture was verified for the Katsura-Exel-Pardo groupoids by P. Nyland and E. Ortega in [18]. The groupoid GpG, Eq obtained from Example 2.8 is one the these.…”

Section: Topological Full Groups and The Ah Conjecturementioning

confidence: 82%

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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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“…The AH conjecture was verified for the Katsura-Exel-Pardo groupoids by P. Nyland and E. Ortega in [18]. The groupoid GpG, Eq obtained from Example 2.8 is one the these.…”

Section: Topological Full Groups and The Ah Conjecturementioning

confidence: 82%

“…The groupoid GpG, Eq obtained from the self-similar action in Example 2.6 satisfies the HK conjecture, see [2]. The AHconjecture may well be true for this groupoid, but the technique of proof will be different from the one in [18]. In particular, we would obtain that vGpG, Eqw ab " 0 and that vGpG, Eqw is perfect.…”

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confidence: 99%

Higman-Thompson groups from self-similar groupoid actions

Deaconu¹

2021

Preprint

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“…This conjecture has been confirmed for several cases, and so far no counter-examples have been found. Among the cases the conjecture has been confirmed there are the Katsura-Exel-Pardo groupoids, or what is the same, the groupoids of a special selfsimilar action of Z over a finite graph [10]. With the techniques developed in [10], that are inspired from the ones in [7], one can see that an analysis of the properties of the kernel of the canonical cocycle is crucial.…”

Section: The Ah-conjecturementioning

confidence: 99%

“…We compute the low homology groups of this groupoid using similar techniques that the ones used in [11]. Finally in Section 4 we verify, using techniques developed in [10], the AH-conjecture for the groupoid G (Γ,X) , that is, there exists an exact sequence…”

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confidence: 95%

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The homology of the groupoid of the self-similar dihedral group

Ortega,

Sanchez

2020

Preprint

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We compute the K-theory of the C * -algebra associated to the self-similar infinite dihedral group, and the homology of its associated étale groupoid. We see that the rational homology differs from the K-theory, strongly contradicting a conjecture posted by Matui. Moreover, we compute the abelianization of the topological full group of the groupoid associated to the self-similar infinite dihedral group.In [8] Matui conjectured that the homology groups of a minimal effective, ample groupoid G totally captures the K-theory of their associated reduced groupoid C *algebra, and called it the HK conjecture, that is,This conjecture has been proved for important classes of groupoids, like the transformation groupoids of Cantor minimal systems, products of Cuntz-Krieger groupoids andmany others (see [2,6,8,11]). More generally, it was proved in [12] that the conjecture is true whenever H n (G, Z) = 0 for n ≥ 3. However, the conjecture was shown to be false by Scarparo in [13] for a class of Odomoters of the infinite dihedral group. Then one can ask if a weakening of the conjecture could be true, the rational HK-conjecture, that isThe rational HK-conjecture it is satisfied by the transformations groupoids of Z nminimal actions on the Cantor space [8, Section 3]. Scarparo counter-example is still valid for i = 0 but not for i = 1. In here we present a complete counter-example for the rational HK-conjecture. Our example arises from a self-similar action of the infinite dihedral group over a rooted tree and it associated groupoid, that not only encodes the global action of the group but also the local self-similarities of this action [9, Section 5]. This groupoid is going to be minimal and locally contracting, and its associated (reduced) groupoid C * -algebra is going to be simple and purely infinite. This contrast with the counter-examples of Scarparo, whose associated C * -algebras were AF-algebras. Therefore, there is not obvious relationship between the example presented here and the

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Katsura--Exel--Pardo Groupoids and the AH~Conjecture

Cited by 2 publications

References 6 publications

Higman-Thompson groups from self-similar groupoid actions

Higman-Thompson groups from self-similar groupoid actions

The homology of the groupoid of the self-similar dihedral group

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