Homology and Matui’s Hk Conjecture for Groupoids on One-Dimensional Solenoids

Yi, Inhyeop

doi:10.1017/s0004972719000522

Cited by 11 publications

(13 citation statements)

References 14 publications

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“…This result includes most of the previously known examples (see e.g. [50]) that were based on separate computations of the K-theory and homology, and hence provides a more conceptual explanation. 5.3.…”

Section: 2supporting

confidence: 65%

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Dynamic asymptotic dimension and Matui's HK conjecture

Bönicke¹,

Dell'Aiera²,

Gabe³

et al. 2021

Preprint

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We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid. As a consequence, the K-theory of the C ˚-algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two satisfy Matui's HK-conjecture.We also construct explicit maps from the groupoid homology groups to the K-theory groups of their C ˚-algebras in degrees zero and one, and investigate their properties. Contents 1. Introduction Outline of the paper 2. Models for groupoid homology 2.1. The category of G-modules and the coinvariant functor 2.2. Semi-simplicial G-spaces and homology 2.3. Projective resolutions and Tor 3. Colourings and homology 3.1. Colourings and nerves 3.2. Homology vanishing 3.3. Maps between nerves and G 3.4. Anti-Čech homology 3.5. Dynamic asymptotic dimension 4. The HK conjecture 4.1. The one-dimensional comparison map 4.2. Dynamic asymptotic dimension one 4.3. The spectral sequence of Proietti and Yamashita 5. Examples and applications 5.1. Free actions on totally disconnected spaces 5.2. Smale spaces with totally disconnected (un)stable sets 5.3. Bounded geometry metric spaces 5.4. Examples with topological property (T) References

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Section: 2supporting

confidence: 65%

“…The main reason Scarparo's example fails to satisfy the conjecture is the existence of torsion in its isotropy groups. For ample groupoids with exclusively torsion free isotropy groups the conjecture has been confirmed in several interesting cases [15,31,33,50] even beyond the originally postulated minimal setting.…”

Section: Introductionmentioning

confidence: 89%

Dynamic asymptotic dimension and Matui's HK conjecture

Bönicke¹,

Dell'Aiera²,

Gabe³

et al. 2021

Preprint

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“…Subsequently, other authors have expanded upon this. The HK conjecture has been shown to hold for Katsura-Exel-Pardo groupoids [Ort18], Deaconu-Renault groupoids of rank 1 and 2 [FKPS18] and groupoids of unstable equivalence relations on one-dimensional solenoids [Yi20].…”

mentioning

confidence: 99%

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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“…The first counterexample is due to Scarparo [19] and a stronger counterexample (it does not even satisfy the rational version of the conjecture) can be found in [14]. On the other hand, there have been a number of positive results, starting with Matui's original work [12], also see [1,8,13,15,22]. In particular, there has been quite a bit of success verifying the conjecture for particular classes of principal groupoids, see in particular [1, Corollary C] and [15,Remark 3.5].…”

Section: Introductionmentioning

confidence: 99%

A counterexample to the HK-conjecture that is principal

Deeley¹

2021

Preprint

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Scarparo has constructed counterexamples to Matui's HK-conjecture and Ortega and Scarparo have constructed counterexamples to the rational HK-conjecture. These counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo's original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.

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Homology and Matui’s Hk Conjecture for Groupoids on One-Dimensional Solenoids

Abstract: We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$-algebras on one-dimensional solenoids.

Cited by 11 publications

References 14 publications

Dynamic asymptotic dimension and Matui's HK conjecture

Dynamic asymptotic dimension and Matui's HK conjecture

Matui's AH conjecture for Graph Groupoids

A counterexample to the HK-conjecture that is principal

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