Groupoid Actions Research Articles

Imprimitivity theorems and self-similar actions on Fell bundles

We introduce the notion of self-similar actions of groupoids on other groupoids and Fell bundles. This leads to a new imprimitivity theorem arising from such dynamics, generalizing many earlier imprimitivity theorems involving group and groupoid actions.

RECURRENCE SETS FOR PARTIAL INVERSE

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. Some applications to the basic dynamical properties of the action are indicated, covering topics as topological transitivity, limit points and periodic points. We then explore the connections of these recurrence sets with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions).

𝐾𝐾-duality for self-similar groupoid actions on graphs

We extend Nekrashevych’s K K KK -duality for C ∗ C^* -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid ( G , E ) (G,E) acting faithfully on a finite directed graph E E , we associate two C ∗ C^* -algebras, O ( G , E ) \mathcal {O}(G,E) and O ^ ( G , E ) \widehat {\mathcal {O}}(G,E) , to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in K K KK -theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

Stratification of the transverse momentum map

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions

Abstract We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures. Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

Zappa–Szép products for partial actions of groupoids on left cancellative small categories

We study groupoid actions on left cancellative small categories and their associated Zappa–Szép products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa–Szép products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective, and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.

Integration of generalized complex structures

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.

Amenability for actions of étale groupoids on 𝐶*-algebras and Fell bundles

We generalize Renault’s notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable C ∗ C^* -algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the C ∗ C^* -algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel’s approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid actions, we show that the approximation property implies measurewise amenability of the underlying action.

Canonical stratification of definable Lie groupoids

Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over R, or more generally, Shiota's X-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J.N. Mather and V.A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.

Topological Group-Groupoids and Equivalent Categories

The concept of groupoid was offered by Brandt (1926). The structure of the topological groupoid was given by Ehresmann (1958). A groupoid action is a significant appliance in algebraic topology offered by Ehresmann. Another algebraic notion is a covering given by Brown (1988). The topological group-groupoids (Γ-groupoid) were first put forward by Icen & Ozcan (2001). The definition of coverings of topological Γ groupoid and actions of topological Γ-groupoid were also presented by Icen et al. (2005). In this paper, we are going to create a category TΓGpdCov(Γ) of covering morphisms of TΓ-groupoid and a category TΓGpdOp(Γ) of actions of TΓ-groupoid. We will then prove that these categories are equivalent.

Uniform enveloping semigroupoids for groupoid actions

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems.We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter—Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.

Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

Let [Formula: see text] be a group-type partial action of a connected groupoid [Formula: see text] on a ring [Formula: see text] and [Formula: see text] be the corresponding partial skew groupoid ring. In the first part of this paper, we investigate the relation of several ring theoretic properties between [Formula: see text] and [Formula: see text]. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action [Formula: see text], we characterize when [Formula: see text] is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented.

Formal groupoids

We provide foundations for a characteristic free study of foliated varieties in terms of infinitesimal actions of formal groupoids. The ultimate goal is the bi-rational geometry of the same, and to this end we prove a cone theorem for foliations in curves, together with structure theorems for extremal rays, and, of course, a minimal model theorem for surfaces. All possible wild ramification effects of Deligne–Mumford champ are built in, along with the occasional use of Artin champ to address the mathbb Q-Gorenstein condition.

Topological dynamics of groupoid actions

Some basic notions and results in topological dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.

Galois correspondence for group-type partial actions of groupoids

Let $G$ be a finite groupoid and $\alpha=(S_g,\alpha_g)_{g\in G}$ a unital partial action of group-type of $G$ on a commutative ring $S=\oplus_{y\in G_0}S_y$. We prove a Galois correspondence between a class of wide subgroupoids of $G$ and a class of subrings of $S$. We recover known results for global groupoid actions and we give several examples to illustrate the correspondence.

Classification of outer actions of discrete amenable groupoids on injective factors

We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama–Takesaki type resolution group approach.

Groupoid actions on sets, duality and a morita context

Let and be groupoids. We present the notion of a -set and we prove a duality theorem in this context, which extends the duality theorem for graded algebras by groups. For A a unital -graded algebra and X a finite split -set, we show that there is an isomorphism between the category of the X-graded left A-modules and the category of the left -modules. As an application of this isomorphism, we construct a Morita context.

The Koopman representation for self-similar groupoid actions

We introduce the [Formula: see text]-algebra [Formula: see text] generated by the Koopman representation [Formula: see text] of an étale groupoid [Formula: see text] acting on a measure space [Formula: see text]. We prove that for a level transitive self-similar action [Formula: see text] with [Formula: see text] finite and [Formula: see text] constant, there is an invariant measure [Formula: see text] on [Formula: see text] and that [Formula: see text] is residually finite dimensional with a normalized self-similar trace. We also discuss [Formula: see text]-fold similarities of Hilbert spaces in connection to representations of the graph algebra [Formula: see text] and self-similar representations of [Formula: see text] in connection to the Cuntz–Pimsner algebra [Formula: see text].

Morphisms of groupoid actions and recurrence

Topological groupoids admit various types of morphisms. We push these notions to the level of continuous groupoid actions to obtain various types of groupoid action morphisms. Some dynamical properties and their relation to these morphisms are studied. Among them are recurrence, various forms of transitivity, minimality, limit, recurrent, periodic and almost periodic points.

Partial groupoid actions on sets and topological spaces

Partial groupoid actions on sets and topological spaces