A going-down principle for {é}tale groupoids and the Baum-Connes conjecture

Abstract: We study a going-down principle for {é}tale groupoids and its applications, extending the earlier results for locally compact groups by Chabert, Echterhoff and Oyono-Oyono, and for ample groupoids by B{ö}nicke and by B{ö}nicke-Dell'Aiera. The proof in the general {é}tale groupoid setting is based on a more detailed study of groupoid simplicial complexes. We also study a bicategorical functoriality involving the induction functors from {é}tale groupoid correspondences, which was introduced by Miller. This yields a bicategorical interpretation of the induction-restriction adjunction. As an application of the going-down principle, we provide a proof of the split injectivity of Baum-Connes assembly map for {é}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{ö}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {é}tale groupoids and the study of scope of validity of K{ü}nneth formulas.

• The paper establishes a going-down principle for all second countable locally compact Hausdorff étale groupoids by controlling measure-theoretic simplicial realizations through hypotheses (H1) and (H2).
• The paper develops a bicategorical induction framework that proves a compression isomorphism and reduces global KK-theoretic questions to proper open subgroupoids.
• The paper proves split injectivity of the Baum–Connes assembly map for strongly amenable groupoids at infinity, while also deriving continuity of topological K-theory and broader Künneth results.

Overview

This paper by Kai Mao establishes a going-down principle for second countable locally compact Hausdorff étale groupoids and applies it to the Baum–Connes conjecture. The result extends the going-down principle of Chabert–Echterhoff–Oyono-Oyono for locally compact groups [chabert2004going] and the versions for ample groupoids due to Bönicke [bonicke2020going] and Bönicke–Dell'Aiera [bonicke2019going]. The main technical obstacle in passing from ample to general étale groupoids is topological: the definition of a groupoid simplicial complex used for Rips complexes of ample groupoids does not behave well for arbitrary étale groupoids. The paper resolves this by introducing two hypotheses $(H_1)$ and $(H_2)$, which first appeared in Bessis's work [bessi2023], under which geometric realizations of groupoid simplicial complexes acquire the local compactness and properness properties needed for the descent argument.

The paper also develops a bicategorical framework for induction functors associated with étale groupoid correspondences in the sense of Miller [miller2022k], [miller2024functors], yielding a conceptual proof of the compression (induction-restriction) isomorphism. As applications, it proves split injectivity of the Baum–Connes assembly map for étale groupoids that are strongly amenable at infinity — recovering a result of Bönicke–Proietti [bonicke2024categorical] obtained via categorical methods — as well as continuity of topological K-theory and results on the scope of validity of Künneth formulas.

Groupoid simplicial complexes

The central technical contribution concerns the geometry of proper actions. For an étale groupoid $\mathcal G\rightrightarrows X$ acting properly on a locally compact Hausdorff space $Y$, the paper proves a local structure theorem: every point $z_0\in Y$ admits a neighborhood $V$ and a proper open subgroupoid $\mathcal H\subseteq\mathcal G$, isomorphic to $F\ltimes A$ where $F$ is the finite stabilizer of $z_0$ acting on a relatively compact open $(H_2)$0, such that $(H_2)$1 is $(H_2)$2-invariant and $(H_2)$3 for all $(H_2)$4. This is the direct analog of the classical statement for discrete groups, with finite subgroups replaced by proper open subgroupoids, and it underpins the descent technique used throughout.

A $(H_2)$5-simplicial complex is a pair $(H_2)$6 where $(H_2)$7 is a local homeomorphism, $(H_2)$8 is a family of finite subsets of fibers closed under taking nonempty subsets, and $(H_2)$9 is stable under the $\mathcal G\rightrightarrows X$0-action. The geometric realization $\mathcal G\rightrightarrows X$1 is defined as the subspace of the space $\mathcal G\rightrightarrows X$2 of probability Radon measures supported on single fibers, equipped with the weak-* topology; this measure-theoretic model replaces the usual simplex construction, which fails without ampleness. The hypotheses are:

• $\mathcal G\rightrightarrows X$3: for any compact $\mathcal G\rightrightarrows X$4, the set $\mathcal G\rightrightarrows X$5 is compact;
• $\mathcal G\rightrightarrows X$6: limits of convergent nets of simplices remain simplices.

Under these hypotheses the paper shows that $\mathcal G\rightrightarrows X$7 is locally compact Hausdorff, that compactly supported parts $\mathcal G\rightrightarrows X$8 are compact, and that if $\mathcal G\rightrightarrows X$9 is proper and $Y$0-compact then so is $Y$1. A key structural result identifies the centers of $Y$2-simplices, $Y$3, as a $Y$4-invariant closed subset of $Y$5, where $Y$6 is an étale $Y$7-space homeomorphic to a symmetric quotient of the fiber product $Y$8. This fact is essential because, unlike in the ample case, the anchor map on a zero-dimensional realization need not be a local homeomorphism but only a composition of a closed inclusion with one; the descent argument must therefore handle this extra exact sequence.

Rips complexes $Y$9 for compact $z_0\in Y$0 are shown to be proper, $z_0\in Y$1-compact $z_0\in Y$2-simplicial complexes satisfying both hypotheses, and the standard cutoff-function argument shows they form a cofinal family among proper cocompact $z_0\in Y$3-spaces up to $z_0\in Y$4-homotopy. Barycentric subdivision is constructed within this measure-theoretic framework and shown to preserve properness, $z_0\in Y$5-compactness, and both hypotheses, producing typed complexes to which the dimension-induction argument applies.

Induction functors and the bicategorical adjunction

The paper constructs a pseudofunctor

$z_0\in Y$6

from the bicategory $z_0\in Y$7 of second countable étale groupoids, correspondences, and bi-equivariant continuous maps to the strict 2-category whose 0-cells are Kasparov categories $z_0\in Y$8. On 1-cells, a correspondence $z_0\in Y$9 is sent to Miller's induction functor $V$0; the functoriality of $V$1 on KK-groups and the natural isomorphisms $V$2 were established by Miller. The new content lies in the treatment of 2-cells: for a bi-equivariant map $V$3, the paper defines $V$4 via the wrong-way class $V$5 of the induced local homeomorphism on orbit spaces, composed with the pullback identification $V$6. It is verified that $V$7 is functorial, that the lax associativity and unity constraints hold, and hence that internal adjunctions in $V$8 are transported to adjunctions of functors between Kasparov categories.

For a relatively clopen subgroupoid $V$9 (open subgroupoids that are proper are automatically relatively clopen), the correspondences $\mathcal H\subseteq\mathcal G$0 and $\mathcal H\subseteq\mathcal G$1 form an internal adjunction in $\mathcal H\subseteq\mathcal G$2, so $\mathcal H\subseteq\mathcal G$3. The compression isomorphism

$\mathcal H\subseteq\mathcal G$4

then follows abstractly from the adjunction, generalizing the result of Bönicke [bonicke2020going] beyond ample groupoids and beyond the properness assumption on $\mathcal H\subseteq\mathcal G$5 (properness of $\mathcal H\subseteq\mathcal G$6 suffices). The paper also sketches a contravariant variant giving a restriction-induction adjunction when the range map $\mathcal H\subseteq\mathcal G$7 is proper, making the adjunction ambidextrous in that case.

The going-down principle

A going-down functor for $\mathcal H\subseteq\mathcal G$8 is a collection of $\mathcal H\subseteq\mathcal G$9-graded cohomological functors $F\ltimes A$0 on separable commutative proper $F\ltimes A$1-$F\ltimes A$2-algebras, compatible with suspension and with subgroupoid induction, indexed by proper open subgroupoids $F\ltimes A$3 together with $F\ltimes A$4 itself; a going-down transformation $F\ltimes A$5 is a compatible family of natural transformations. The main theorem states:

If $F\ltimes A$6 is an isomorphism for every proper open subgroupoid $F\ltimes A$7 and every $F\ltimes A$8, then the induced map $F\ltimes A$9 on topological K-theory, defined as the colimit over $F$0-compact subsets of a classifying space $F$1, is an isomorphism for all $F$2.

The proof proceeds in three steps. First, the Rips complexes are shown to be homotopy-cofinal among $F$3-compact subsets of $F$4, reducing the problem to finite-dimensional proper $F$5-compact simplicial complexes satisfying $F$6 and $F$7. Second, after barycentric subdivision the complex may be assumed typed, and the stratum $F$8 is $F$9-homeomorphic to $z_0$0; since $z_0$1 is a $z_0$2-compact subset of an étale $z_0$3-space, the zero-dimensional case applies to it, and suspension compatibility transfers the conclusion to the stratum. Third, the long exact sequence attached to $z_0$4 and the five lemma complete the induction on dimension. The zero-dimensional case itself relies on the local structure theorem: a proper $z_0$5-compact set is a finite union of pieces of the form $z_0$6, handled by Mayer–Vietoris and induction on the number of pieces. As noted above, the handling of the zero-dimensional stratum requires one additional exact sequence relative to the ample-groupoid argument of [bonicke2020going], precisely because the anchor map there need not be a local homeomorphism.

Applying the principle to the canonical example $z_0$7 yields: if $z_0$8 restricts to a KK-equivalence over $z_0$9 for every proper open subgroupoid $(H_2)$00, then $(H_2)$01 induces an isomorphism $(H_2)$02. Notably, this version assumes invertibility only on coefficient algebras of the form $(H_2)$03, whereas the categorical proof of Bönicke–Proietti required invertibility against all separable $(H_2)$04-$(H_2)$05-algebras $(H_2)$06; the present theorem is therefore formally stronger in its hypothesis.

Split injectivity of Baum–Connes assembly

An étale groupoid is strongly amenable at infinity if it admits a $(H_2)$07-space $(H_2)$08 with proper surjective anchor admitting a continuous section, such that $(H_2)$09 is amenable. The paper first shows that the amenability can be transferred to the space of probability measures $(H_2)$10, which has convex fibers with affine action and inherits properness, surjectivity, and a continuous section of the anchor. Combining this with the affine-section homotopy result of Bönicke [bonicke2020going], the class $(H_2)$11 gives an element $(H_2)$12 that restricts to an invertible class over each proper open subgroupoid. The going-down theorem then makes $(H_2)$13 an isomorphism on $(H_2)$14, and naturality of the assembly map reduces $(H_2)$15 to the assembly map of the amenable transformation groupoid $(H_2)$16, which is an isomorphism by Tu's theorem [tu1999conjecture-bc]. Consequently:

If $(H_2)$17 is a second countable étale groupoid strongly amenable at infinity, then $(H_2)$18 is split injective for every separable $(H_2)$19-$(H_2)$20-algebra $(H_2)$21.

This recovers the result of Bönicke–Proietti [bonicke2024categorical] by analytic rather than categorical means. A remark contributed by Georges Skandalis records a more direct proof, bypassing the full going-down machinery, that follows Higson's argument for countable discrete groups [Higson99] using only the compression isomorphism and Mayer–Vietoris.

Two further applications follow the same template. First, continuity of topological K-theory: for an inductive system $(H_2)$22 of separable $(H_2)$23-$(H_2)$24-algebras with limit $(H_2)$25, the canonical map $(H_2)$26 is an isomorphism, since over proper open subgroupoids the claim reduces to continuity of K-theory for reduced crossed products. As a corollary, if $(H_2)$27 is exact or the connecting maps are injective, and $(H_2)$28 satisfies Baum–Connes with coefficients in each $(H_2)$29, then it satisfies Baum–Connes with coefficients in $(H_2)$30. Second, Künneth formulas: defining the class $(H_2)$31 of separable exact $(H_2)$32-$(H_2)$33-algebras satisfying the $(H_2)$34-Künneth formula, the paper proves that if $(H_2)$35 for every proper open subgroupoid $(H_2)$36, then $(H_2)$37. Combined with the relation between $(H_2)$38 and Baum–Connes validity, this extends results previously restricted to ample groupoids in [bonicke2019going]; in particular, if $(H_2)$39 satisfies Baum–Connes with coefficients, then the reduced twisted groupoid $(H_2)$40-algebra $(H_2)$41 of any twist belongs to $(H_2)$42.

Limitations and open questions

Several restrictions should be noted. The going-down principle and its corollaries require second countability of the groupoid and separability of the coefficient algebras; the bicategorical proof of the compression isomorphism is given only in this setting, although the analytic proof of [bonicke2020going] carries over verbatim without it. The split injectivity result covers strongly amenable groupoids at infinity, i.e., those admitting an amenable proper action with a continuous section of the anchor; whether plain amenability at infinity (without the section) suffices is not addressed here. The restriction-induction (ambidextrous) adjunction is stated only as a sketch and requires the properness of $(H_2)$43, a condition not automatic for relatively clopen subgroupoids. Finally, the Künneth results reduce the question to proper open subgroupoids but do not characterize $(H_2)$44 intrinsically; determining which exact $(H_2)$45-$(H_2)$46-algebras lie in $(H_2)$47 for a given $(H_2)$48 remains open, as does the corresponding question for groupoids satisfying Baum–Connes only partially.

Conclusion

The paper completes the program initiated by Chabert–Echterhoff–Oyono-Oyono and carried out for ample groupoids by Bönicke and Bönicke–Dell'Aiera, establishing the going-down principle for all second countable étale groupoids. The key innovations are the hypotheses $(H_2)$49 and $(H_2)$50 governing the topology of groupoid simplicial complexes realized inside spaces of probability measures, and the pseudofunctor $(H_2)$51 that interprets induction along correspondences and derives the compression isomorphism from an internal adjunction. The resulting split injectivity theorem for strongly amenable groupoids at infinity, together with the continuity and Künneth applications, demonstrates that the descent technique now applies across the full étale category rather than only to ample groupoids.