Groupoids Research Articles

The AH conjecture for Cantor minimal dihedral systems

The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani–Rokhlin partitions adapted to such systems.

On the Hochschild homology of proper Lie groupoids

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and we prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski’s ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski’s conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space.

Almost finiteness and homology of certain non-free actions

We show that Cantor minimal \mathbb{Z}\rtimes\mathbb{Z}_2 -systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal \mathbb{Z}\rtimes\mathbb{Z}_2 -systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.

An equivariant PPV theorem and Paschke–Higson duality

We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.

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

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).

A counterexample to the HK-conjecture that is principal

AbstractScarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known 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.

The type semigroup, comparison, and almost finiteness for ample groupoids

AbstractWe prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

(Non)exotic Completions of the Group Algebras of Isotropy Groups

Abstract Motivated by the problem of characterizing KMS states on the reduced C$^*$-algebras of étale groupoids, we show that the reduced norm on these algebras induces a C$^*$-norm on the group algebras of the isotropy groups. This C$^*$-norm coincides with the reduced norm for the transformation groupoids, but, as follows from examples of Higson–Lafforgue–Skandalis, it can be exotic already for groupoids of germs associated with group actions. We show that the norm is still the reduced one for some classes of graded groupoids, in particular, for the groupoids associated with partial actions of groups and the semidirect products of exact groups and groupoids with amenable isotropy groups.

The Higson–Roe sequence for étale groupoids. II. The universal sequence for equivariant families

This is the second part of our series about the Higson–Roe sequence for étale groupoids. We devote this part to the proof of the universal K-theory surgery exact sequence which extends the seminal results of N. Higson and J. Roe to the case of transformation groupoids. In the process, we prove the expected functoriality properties as well as the Paschke–Higson duality theorem.

Asymptotic continuous orbit equivalence of expansive systems

We introduce notions of asymptotic continuous orbit equivalence and (strongly) asymptotic conjugacy for expansive systems, and characterize them in terms of the transformation groupoids, the principal groupoids coming from the local conjugacy relations an

Nonstandard Quantum Complex Projective Lin

In our attempt to explore how the quantum nonstandard complex projective spaces $\mathbb{C}P_{q,c}^{n}$ studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of $C\big(\mathbb{C}P_{q,c}^{1}\big) $ with the pull-back of two copies of the Toeplitz $C^*$-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting non-obvious details, such as a prominent generator of $C\big(\mathbb{C}P_{q,c}^{1}\big) $ being a concrete weighted double shift.

Inverse ratio ventilation combined with PEEP in infants undergoing thoracoscopic surgery with one lung ventilation for lung cystadenomas: a randomized control trial of 63 cases

To investigate the effect of inverse ratio ventilation (IRV) combined with positive end-expiratory pressure (PEEP) in infants undergoing thoracoscopic surgery with single lung ventilation (OLV) for lung cystadenomas. A total of 66 infants undergoing thoracoscopic surgery with OLV for lung cystadenomas in our hospital from February, 2018 to February, 2019 were randomized into conventional ventilation groups (group N, n=33) and inverse ventilation group (group R, n=33). Hemodynamics and respiratory parameters of the infants were recorded and arterial blood gas analysis was performed at 15 min after two lung ventilation (TLV) (T1), OLV30 min (T2), OLV60 min (T3), and 15 min after recovery of TLV (T4). Bronchoalveolar lavage fluid was collected before and after surgery to detect the expression level of advanced glycation end product receptor (RAGE). Sixty-three infants were finally included in this study. At T2 and T3, Cdyn, PaO2 and OI in group R were significantly higher (P < 0.05) and Ppeak, PaCO2 and PA-aO2 were significantly lower than those in group N (P < 0.05). There was no significant difference in HR or MAP between the two groups at T2 and T3 (P > 0.05). The level of RAGE significantly increased after the surgery in both groups (P < 0.05), and was significantly lower in R group than in N group (P < 0.05). In infants undergoing thoracoscopic surgery with OLV for pulmonary cystadenoma, appropriate IRV combined with PEEP does not affect hemodynamic stability and can increases pulmonary compliance, reduce the peak pressure, and improve oxygenation to provide pulmonary protection.

Applications of Neutrosophic N -Structures in n-Ary Groupoids

This paper includes the notions of neutrosophic n-ary N-subgroupoids of n-ary groupoids and some properties.

Groupoid description of modular structures.

Modular structures are crystal structures built by subperiodic (zero-, mono- or diperiodic) substructures, called modules. The whole set of partial operations relating substructures in a modular structure build up a groupoid; modular structures composed of identical substructures are described by connected groupoids, or groupoids in the sense of Brandt. A general approach is presented to describe modular structures by Brandt's groupoids and how to obtain the corresponding space groups, in which only the partial operations that have an extension to the whole crystal space appear.

Graph varieties axiomatized by semimedial, medial, and some other groupoid identities

Graph varieties axiomatized by semimedial, medial, and some other groupoid identities

Certain Types of Groupoids

In this work, we try to construct anew types of groupoids and disucss their properties

On the Center of a Hypergroupoid

In this paper, we define the center of a hypergroupoid and investigate its properties. We also define isomorphism action on groupoids and show that for some particular groupoid $$\mathbb {G} $$ , the center of hypergroupoid $$\mathbb {G}^\mathbb {G}$$ consists of all $$ x^\mathbb {G} $$ , where x is in the center of $$\mathbb {G}$$ .

Homology of odometers

We compute the homology groups of transformation groupoids associated with odometers and show that certain $(\mathbb{Z}\rtimes \mathbb{Z}_{2})$-odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_{r}^{\ast }(G)$. We also show that transformation groupoids of odometers satisfy the AH conjecture.

Some properties of AG*-groupoids and AG-bands under SI-product operation

Some properties of AG*-groupoids and AG-bands under SI-product operation