Dynamical comparison for local homeomorphisms

Abstract: We prove dynamical comparison for Deaconu--Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated $C<sup>*$-algebras are UCT Kirchberg algebras, recovering results by Carlsen--Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable ``large subgroupoids'' and establish comparison properties for these using the paradoxical towers technique developed by Gardella--Geffen--Kranz--Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.

• The paper proves dynamical comparison for Deaconu–Renault groupoids arising from minimal, surjective, non-injective local homeomorphisms on finite-dimensional compact metrizable spaces.
• The authors combine paradoxical towers for partial free-group actions with a new thin boundary property derived from finite covering dimension to handle purely infinite systems without invariant measures.
• The results imply the AH-conjecture for related Cantor groupoids, establish that the reduced groupoid C*-algebras are UCT Kirchberg algebras, and extend the small boundary property to minimal actions without freeness assumptions.

Overview

This paper, by Shirly Geffen, Shanshan Hua, and Julian Kranz, establishes dynamical comparison for a large class of purely infinite dynamical systems: Deaconu–Renault groupoids $G_T$ associated to minimal surjective non-injective local homeomorphisms $T\colon X \to X$ of compact metrizable spaces with finite Lebesgue covering dimension. The main theorem states that such $G_T$ always satisfies dynamical comparison. Since the injective case was already covered by prior results in the stably finite regime, the paper completes a dichotomy for these groupoids: under the stated hypotheses they are either stably finite (when $T$ is a homeomorphism) or purely infinite (when $T$ is not).

The proof strategy is notable for combining techniques from both regimes. In the purely infinite direction, the authors construct partial actions of non-abelian free groups as "large subgroupoids" and establish comparison via the paradoxical towers technique of Gardella–Geffen–Kranz–Naryshkin. The boundary of these subgroupoids is controlled by a groupoid version of the topological small boundary property, which the authors call the thin boundary property and derive from finite covering dimension of the unit space.

Main results

The central theorem asserts that for every minimal surjective local homeomorphism $T\colon X \to X$ of a compact metrizable space of finite covering dimension, the Deaconu–Renault groupoid $G_T$ satisfies dynamical comparison. Three corollaries follow:

• AH-conjecture: for minimal surjective local homeomorphisms of the Cantor set, $G_T$ satisfies Matui's AH-conjecture, via Xin Li's recent theorem relating groupoid homology to topological full group homology under dynamical comparison. This recovers the AH-conjecture for graph groupoids and extends Matui's classical result for global homeomorphisms to the non-injective case.
• UCT Kirchberg algebras: for minimal surjective non-injective local homeomorphisms on finite-dimensional spaces, $C^*_r(G_T)$ is a UCT Kirchberg algebra. This recovers a theorem of Carlsen–Thomsen by dynamical methods, using topological principality, amenability, nuclearity, and pure infiniteness (via Ma's criterion).
• Small boundary property: every minimal action of a countable discrete group on a compact metrizable space of finite covering dimension satisfies the classical small boundary property. This partially generalizes prior results of Szabó, Gardella et al., and Kerr, since it requires no freeness assumption — at the cost of assuming minimality.

Reduction to large fibers

A key structural reduction shows that the general theorem follows from the special case where every fiber has cardinality at least two, i.e. $|T^{-1}(x)| \geq 2$ for all $T\colon X \to X$0. The authors prove that for a minimal surjective non-injective local homeomorphism of a compact Hausdorff space, some finite iterate $T\colon X \to X$1 has uniformly large fibers — the sets $T\colon X \to X$2 are open, nested by surjectivity, and exhaust $T\colon X \to X$3 by minimality. A Zorn's lemma argument then produces a nonempty clopen set $T\colon X \to X$4, invariant under $T\colon X \to X$5, on which $T\colon X \to X$6 is a surjective minimal local homeomorphism. The subgroupoid $T\colon X \to X$7 is open, and an "open subgroupoid lemma" transfers dynamical comparison with no invariant measures from the subgroupoid to the ambient groupoid, provided the unit space has no isolated points.

In the large-fiber case, absence of invariant probability measures is shown directly: a covering argument using local invertibility on disjoint preimages of small sets yields $T\colon X \to X$8, a contradiction.

Paradoxical towers for partial actions

The core comparison argument embeds the Deaconu–Renault groupoid with a partial action of a free group. In the zero-dimensional case, the authors use de Castro–Steinberg's realization: $T\colon X \to X$9 for a semi-saturated orthogonal partial action $G_T$0 built from a clopen partition adapted to $G_T$1. The large-fiber condition guarantees that at least two generators act via surjections onto $G_T$2.

The main technical lemma generalizes the paradoxical towers construction of Gardella–Geffen–Kranz–Naryshkin to partial actions: for a non-elementary hyperbolic group $G_T$3 with trivial finite radical, independent loxodromic elements $G_T$4, and a finite set $G_T$5, there exist subsets $G_T$6 and elements $G_T$7 in the semigroup $G_T$8 such that the sets $G_T$9 ($T$0) are pairwise disjoint and the sets $T$1 are pairwise disjoint. The proof uses Gromov's ping-pong argument and the north–south dynamics of loxodromic elements on the Gromov boundary, choosing repelling fixed points in general position via topological freeness of the boundary action.

Combining this with an amenability-based Følner-type averaging argument (with the explicit constant $T$2), the authors show $T$3 for every nonempty open $T$4, establishing comparison with no invariant measures.

For higher-dimensional unit spaces, de Castro–Steinberg's realization is unavailable, so the authors instead build a "large open subgroupoid" of $T$5 from a partial action of $T$6 constructed on a partition of $T$7, where $T$8 is a closed $T$9-thin set arising from boundaries of a finite cover. They verify the three hypotheses of their abstract subgroupoid criterion: $T$0 embeds as an open subgroupoid, basic open sets have $T$1-thin range outside the subgroupoid, and the generators $T$2 act with $T$3-thin complements of their domains. The thin boundary property is precisely what makes the boundary error $T$4 negligible for comparison purposes.

Thin boundaries from finite covering dimension

The second major contribution is a purely group-theoretic-topological result: every minimal second countable Hausdorff étale groupoid with compact metrizable unit space of finite Lebesgue covering dimension satisfies the thin boundary property. The proof adapts the finite-dimensional general-position machinery of Lindenstrauss and Szabó to the groupoid setting.

A closed set $T$5 is assigned a rank via a recursive hierarchy $T$6: $T$7 if collisions of disjoint pieces of $T$8 under bisections always produce collision sets of rank $T$9. An induction on $T\colon X \to X$0 shows that sets in $T\colon X \to X$1 are $T\colon X \to X$2-thin: minimality provides covers by bisections landing in $T\colon X \to X$3 disjoint open sets, a dimension-theoretic coloring lemma refines these to $T\colon X \to X$4 disjoint families, and rank-lowering inductive hypotheses handle the collision sets.

The construction of sets with controlled boundaries proceeds via a "general position" condition for closed sets $T\colon X \to X$5 under finite families of controlled bisections: intersections $T\colon X \to X$6 must have dimension at most $T\colon X \to X$7 for separated collections $T\colon X \to X$8. A diagonal induction alternately refines the open set and its boundary neighborhoods to force the boundary into $T\colon X \to X$9.

Two features distinguish this from prior work. First, the notion of "separated" controlled bisections plays the role that freeness plays in Szabó's argument, so no freeness or principality assumption on the groupoid is needed — this is what yields the small boundary property corollary for possibly non-free minimal group actions. Second, the resulting thin boundary property is formally weaker than the topological small boundary property, but sufficient for dynamical comparison. The authors note that small-boundary methods, previously used primarily in the stably finite regime for nuclear dimension and $G_T$0-stability, here prove effective in the purely infinite regime — which they describe as somewhat unexpected, since prior comparison results for minimal amenable actions of nonamenable groups required no dimensionality assumptions.

Limitations and open questions

The main theorem requires finite Lebesgue covering dimension of the unit space; whether dynamical comparison holds for Deaconu–Renault groupoids over arbitrary compact metrizable spaces remains open. The small boundary property corollary assumes minimality, so it does not subsume all prior freeness-based results for non-minimal actions. In the zero-dimensional case, combinatorial graph-based models for surjective local homeomorphisms (due to Ara–Exel and Ara–Claramunt) suggest an alternative route to pure infiniteness via the extensive literature on purely infinite (generalized) graph $G_T$1-algebras; the extent to which the main result can be recovered from these models is left as a question. The paradoxical towers lemma is formulated for non-elementary hyperbolic groups with trivial finite radical, though only free groups are used in the application.

Conclusion

The paper establishes dynamical comparison for Deaconu–Renault groupoids of minimal surjective non-injective local homeomorphisms on finite-dimensional compact metrizable spaces, yielding Matui's AH-conjecture for these groupoids on the Cantor set and a dynamical proof that the reduced groupoid $G_T$2-algebras are UCT Kirchberg algebras. Methodologically, it demonstrates that small-boundary techniques transfer to the purely infinite regime and that paradoxical towers for partial free group actions, controlled by a new thin boundary property derived from covering dimension, suffice to prove comparison in a setting where no invariant measures exist. The work also delivers the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional spaces without any freeness hypothesis.