On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras

Abstract: We construct the first explicit examples of locally compact Hausdorff étale groupoids that are not inner amenable and that do not arise as transformation groupoids associated to partial actions of discrete groups. This answers questions of Anantharaman--Delaroche and Exel. Our examples include all Higson--Lafforgue--Skandalis groupoids associated to non-amenable residually finite groups, as well as their principal variants constructed by Alekseev--Finn--Sell. These can be chosen to be second countable, ample, and in the latter case even principal. We also show that large classes of Deaconu--Renault groupoids with connected unit space do not arise from partial actions of discrete groups, including cases whose $C^*$-algebras are Kirchberg algebras in the UCT class. We contrast this with the totally disconnected case by giving ample transformation groupoid models for all unital Kirchberg algebras in the UCT class as well as many higher rank graph algebras. Finally, we characterize precisely when coarse groupoids arise from partial actions of discrete groups in terms of coarse embeddings into groups.

• The paper reveals that inner amenability is equivalent to amenability in certain exotic groupoids, providing the first known counterexamples with HLS and AFS constructions.
• It introduces property Δ as an obstruction to transformation groupoid modeling, delineating limitations for partial action models in specific étale groupoids.
• The study establishes that all unital UCT Kirchberg algebras admit ample transformation groupoid models, offering robust classification tools via K-theory and paradoxical comparisons.

Etale Groupoids, Partial Actions, Inner Amenability, and Models for Kirchberg Algebras

Overview and Context

The paper "On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras" (2604.17921) addresses several open structural problems in the interplay between locally compact Hausdorff étale groupoids and the theory of $C^*$-algebras, especially pertaining to Cartan subalgebras, amenability, and algebraic modeling of Kirchberg algebras. The authors construct explicit counterexamples to two longstanding questions: whether all such groupoids are inner amenable, and whether all arise as transformation groupoids associated to partial actions of discrete groups. They develop new obstruction mechanisms and provide sharp positive classification results for specific cases, notably for ample groupoids and models of unital UCT Kirchberg algebras.

Structural Negative Results

Inner Amenability

Inner amenability, a technical property required in recent generalizations of Ozawa’s exactness characterization for groupoids, had not been known to fail for any groupoid prior to this work. The authors prove that inner amenability is equivalent to amenability for certain exotic groupoids, specifically Higson–Lafforgue–Skandalis (HLS) and Alekseev–Finn–Sell (AFS) groupoids constructed from non-amenable residually finite groups. For such $\Gamma$, HLS and AFS groupoids are not inner amenable, providing the first explicit counterexamples and resolving a question posed by Anantharaman–Delaroche.

Partial Action Models

Transformation groupoids associated to partial actions of discrete groups offer rigid models for many étale groupoids. However, the paper constructs explicit groupoids (including all HLS and AFS groupoids for non-locally finite $\Gamma$, select Deaconu–Renault groupoids, and coarse groupoids for some spaces) that fail to admit any isomorphism to such a transformation structure. This answers a question asked by Exel, showing that the scope of partial action models is strictly limited.

Property $\Delta$ as an Obstruction

The authors introduce property $\Delta$ to formalize the obstruction: it captures the uniform finiteness of groupoid fibers approximated by compact pieces. Transformation groupoids always satisfy it, but the HLS and AFS constructions do not, due to their equicontinuous behavior at infinity. Moreover, property $\Delta$ does not coincide with inner amenability; Deaconu–Renault groupoids with connected unit spaces exhibit property $\bar{\Delta}$ (a stronger form), yet can lack partial action models when associated local homeomorphisms are non-injective on connected sets.

Positive Classification Results

Ample Groupoid Models for Kirchberg Algebras

A major outcome is the realization that all unital Kirchberg algebras in the UCT class can be modeled as crossed product $C^*$-algebras associated to ample transformation groupoids, i.e., partial actions of countable discrete groups on the Cantor set. The proof leverages Wu's global action models for stable UCT Kirchberg algebras [Wu2025], combined with paradoxical comparison properties and $K$-theory argumentation. The authors verify that any $K_0$-class can be represented via characteristic functions of compact open subsets, ensuring corner reductions correspond to the desired algebraic models.

Higher Rank Graph Groupoids

The paper also proves that path groupoids associated to higher rank graphs $\Gamma$0 which embed into their fundamental groupoids are transformation groupoids via partial actions of discrete groups, with locally proper pure cocycles yielding clopen domains. This result draws from the Castro–Kang cocycle characterization and functorial algebraic constructions, providing a sharp dichotomy: totally disconnected spaces support such models, whereas connected ones (e.g., unit circle doubling map) do not.

Coarse Groupoids

The authors completely characterize when a coarse groupoid $\Gamma$1 arises as a transformation groupoid, showing it is equivalent to $\Gamma$2 admitting an injective coarse map into a discrete group. Furthermore, clopen partial action domains correspond precisely to coarse embeddings. Explicit examples demonstrate spaces (with maximal locally finite coarse structure) failing to admit any injective coarse group embedding, so their groupoids fall outside partial action modeling.

Numerical and Contradictory Results

• First explicit examples of non-inner-amenable, non-transformation étale groupoids (HLS and AFS groupoids for non-amenable, residually finite $\Gamma$3).
• Kirchberg algebra models: All unital UCT Kirchberg algebras admit ample transformation groupoid models, with full $\Gamma$4-theory correspondence and action on the Cantor set.
• Graph dichotomy: Higher rank graphs with connected unit spaces (e.g., $\Gamma$5 via $\Gamma$6) admit no transformation groupoid structure; those embedding into their fundamental groupoid do.
• Coarse groupoid characterization: Transformation groupoid realization is equivalent to the existence of a coarse injection into a group; clopen partial action domains are equivalent to coarse embeddings.

Theoretical and Practical Implications

The results significantly constrain the landscape of $\Gamma$7-algebraic modeling via groupoids. The counterexamples show that inner amenability and transformation groupoid realization cannot be assumed even in ample, principal, or classifiable cases. The results impact attempts to resolve the UCT problem using groupoid techniques (as they show limitations of partial action models), while motivating further exploration of non-transformation groupoids with favorable analytic properties.

From a practical operator-algebraic perspective, the ability to model unital UCT Kirchberg algebras as ample transformation groupoids provides powerful classification tools, especially for constructions leveraging paradoxical comparison, $\Gamma$8-theory, and Cartan subalgebras.

Future Directions

The intricate relationship between inner amenability, groupoid structure, and $\Gamma$9-algebra classification suggests several avenues for future research:

• Further investigation into the analytic properties of non-inner-amenable groupoids, and their impact on exactness and nuclearity.
• Extensions of partial action models to broader classes via non-discrete groups or more general cocyclical structures.
• Systematic study of when a coarse space admits a coarse embedding or injection into a group.
• Deeper analysis of structural invariants that distinguish transformation groupoids among étale groupoids, especially in the context of classification programs.

Conclusion

This work supplies explicit structural counterexamples and sharp classification results regarding the realization of Hausdorff étale groupoids as partial transformation groupoids, their inner amenability, and their role as models for UCT Kirchberg algebras (2604.17921). While highlighting the rigidity of partial action models, it simultaneously demonstrates broad positive results for ample cases and higher rank graphs, offering a comprehensive framework for understanding the analytic and algebraic properties of groupoid $\Gamma$0-algebras.