arxiv: 2604.17921 · v2 · submitted 2026-04-20 · 🧮 math.OA · math.DS

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

Alcides Buss , Julian Kranz

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Pith reviewed 2026-05-10 03:56 UTC · model grok-4.3

classification 🧮 math.OA math.DS

keywords étale groupoidspartial actionsinner amenabilityKirchberg algebrasDeaconu-Renault groupoidsHigson-Lafforgue-Skandalis groupoidscoarse embeddingstransformation groupoids

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The pith

Higson-Lafforgue-Skandalis groupoids give explicit examples of étale groupoids that are neither inner amenable nor from partial actions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the first explicit examples of locally compact Hausdorff étale groupoids that fail to be inner amenable and fail to arise as transformation groupoids from partial actions of discrete groups. These examples consist of the Higson-Lafforgue-Skandalis groupoids for non-amenable residually finite groups together with their principal variants. This directly answers questions of Anantharaman-Delaroche and Exel. The work further shows that many Deaconu-Renault groupoids with connected unit space do not come from partial actions, including cases whose C*-algebras are Kirchberg algebras in the UCT class. In the totally disconnected case the paper supplies ample transformation groupoid models for all unital Kirchberg algebras in the UCT class and for many higher rank graph algebras, and it characterizes precisely when coarse groupoids arise from partial actions via coarse embeddings into groups.

Core claim

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. 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

What carries the argument

Higson-Lafforgue-Skandalis groupoids associated to non-amenable residually finite groups, which function as the explicit counterexamples to both inner amenability and representability by partial-action transformation groupoids.

If this is right

Large classes of Deaconu-Renault groupoids with connected unit space do not arise as transformation groupoids from partial actions of discrete groups.
Ample transformation groupoid models exist for every unital Kirchberg algebra in the UCT class and for many higher rank graph algebras when the unit space is totally disconnected.
Coarse groupoids arise from partial actions of discrete groups precisely when the underlying space admits a coarse embedding into a group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation between partial-action groupoids and the larger class of étale groupoids implies that some Kirchberg algebras in the UCT class admit groupoid models that lie outside the partial-action framework.
The contrast between connected and totally disconnected unit spaces suggests that connectedness may systematically obstruct representation by partial actions.
The characterization of coarse groupoids ties the existence of partial-action models to a geometric embedding property that can be checked independently.

Load-bearing premise

The Higson-Lafforgue-Skandalis groupoids associated to non-amenable residually finite groups (and their principal variants) are indeed not inner amenable and do not arise from partial actions of discrete groups.

What would settle it

A concrete partial action of a discrete group whose transformation groupoid is isomorphic to a Higson-Lafforgue-Skandalis groupoid for a non-amenable residually finite group, or an explicit verification that such a groupoid is inner amenable, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.17921 by Alcides Buss, Julian Kranz.

**Figure 1.** Figure 1: An illustration of the edges e appearing in (7.1). □ Theorem 7.6. Let A be a unital UCT Kirchberg algebra. Then there is a discrete group Γ and a partial action Γ ↷ X on the Cantor set with clopen domains such that A ∼= C ∗ (Γ ⋉ X). Proof. As pointed out above, we use the construction and notation from [Wu25]. Let B := A ⊗ K be the stabilization of A, which is a stable UCT Kirchberg algebra with K∗(B) ∼= K… view at source ↗

read the original abstract

We construct the first explicit examples of locally compact Hausdorff \'etale 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first explicit counterexamples to two open questions on étale groupoids via Higson-Lafforgue-Skandalis constructions from non-amenable groups, plus new models for Kirchberg algebras.

read the letter

The key takeaway is that this paper constructs the first explicit examples of locally compact Hausdorff étale groupoids that are not inner amenable and do not arise as transformation groupoids from partial actions of discrete groups. It settles questions from Anantharaman-Delaroche and Exel by taking Higson-Lafforgue-Skandalis groupoids associated to non-amenable residually finite groups, along with their principal variants, and verifying the properties through links to group non-amenability and incompatibility with transformation groupoid structures. These examples can be chosen second countable, ample, and in some cases principal. It also shows that many Deaconu-Renault groupoids with connected unit space fail to come from partial actions, including some whose C*-algebras are Kirchberg in the UCT class. In the totally disconnected setting it supplies ample transformation groupoid models for all unital Kirchberg algebras in the UCT class and many higher rank graph algebras, and it gives a characterization of coarse groupoids that arise from partial actions in terms of coarse embeddings into groups. What stands out is the move from existence arguments to concrete constructions that reuse known groupoids but add the needed checks. The strategy of relating non-inner amenability to the underlying group's properties looks direct and avoids circularity. The models for Kirchberg algebras give something usable for the classification program. The main soft spot is the technical verification that these specific groupoids really do not arise from partial actions. The paper claims to handle this via incompatibility with transformation groupoid properties, and the described arguments appear to cover the principal variants without obvious gaps, but the details would need checking in a review. No other load-bearing issues show up in the approach or citations. This is for people working on groupoid C*-algebras, amenability questions, and the classification of Kirchberg algebras. A reader who wants explicit counterexamples or concrete models will get direct value. It deserves peer review because it resolves open questions with verifiable constructions rather than abstract claims.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs the first explicit examples of locally compact Hausdorff étale groupoids that are not inner amenable and do not arise as transformation groupoids associated to partial actions of discrete groups. These include all Higson–Lafforgue–Skandalis groupoids associated to non-amenable residually finite groups, as well as their principal variants due to Alekseev–Finn–Sell; the examples can be chosen second countable, ample, and (in the latter case) principal. The paper further shows 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. In contrast, it supplies ample transformation groupoid models for all unital Kirchberg algebras in the UCT class and many higher-rank graph algebras in the totally disconnected setting. Finally, it characterizes precisely when coarse groupoids arise from partial actions of discrete groups in terms of coarse embeddings into groups.

Significance. If the central claims hold, the work resolves questions posed by Anantharaman–Delaroche and Exel by supplying the first explicit counterexamples to inner amenability and partial-action origins for étale groupoids. The explicit constructions based on Higson–Lafforgue–Skandalis groupoids and their principal variants, together with the Deaconu–Renault analysis and the coarse-groupoid characterization, provide concrete, falsifiable instances that advance the structural theory of groupoid C*-algebras. The supply of ample transformation-groupoid models for all unital Kirchberg algebras in the UCT class is a further concrete contribution that may facilitate computations and classification results.

minor comments (3)

The abstract and introduction should explicitly reference the precise statements of the questions by Anantharaman–Delaroche and Exel that are being answered, including section or theorem numbers from the cited works, to make the novelty immediately verifiable.
In the discussion of the Higson–Lafforgue–Skandalis groupoids (likely §3 or §4), the argument that non-amenability of the underlying group implies non-inner amenability of the groupoid should be cross-referenced to the exact proposition or lemma establishing the equivalence or implication used.
The characterization of coarse groupoids arising from partial actions (final section) would benefit from a short table or diagram summarizing the equivalence conditions (coarse embedding vs. partial action) for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and detailed summary of our work, as well as for the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs explicit examples using established families of groupoids (Higson-Lafforgue-Skandalis associated to non-amenable residually finite groups, their principal variants, and Deaconu-Renault groupoids) and supplies direct arguments relating non-inner amenability to the underlying group's non-amenability via groupoid structure, and incompatibility with partial-action transformation groupoids via property mismatch. The characterization of coarse groupoids arising from partial actions is derived from standard groupoid and coarse embedding properties without reduction to fitted inputs or self-referential definitions. No load-bearing self-citation chains, ansatzes smuggled via prior work, or renamings of known results as new derivations appear; the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of locally compact Hausdorff étale groupoids, C*-algebra theory, and coarse geometry; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard properties of locally compact Hausdorff étale groupoids and their C*-algebras
Invoked throughout to define the objects and state the claims.

pith-pipeline@v0.9.0 · 5499 in / 1213 out tokens · 59638 ms · 2026-05-10T03:56:12.658456+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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