HK and GL

Abstract: We study the HK conjecture and the gap-labelling problem for transformation groupoids associated with free actions of poly-$\Z$ groups on Cantor sets. The main tool is a comparison of the long exact sequences in groupoid homology and cohomology with the Pimsner--Voiculescu exact sequence for crossed products by $\Z$. In addition to the canonical homology comparison maps $μ_0$ and $μ_1$, we introduce cohomology comparison maps associated with suitable $K$-theory classes of the acting group. Together with Poincaré duality, these maps detect the higher homology terms occurring in the HK conjecture. We apply this method to free actions of poly-$\Z$ groups of small Hirsch length. For actions of $\Z$, $\Z<sup>2$, and the Klein bottle group, we recover HK and gap-labelling. For several classes of groups of Hirsch length three and four, we either prove HK or obtain explicit exact sequences describing the $K$-groups in terms of groupoid homology and cohomology. For gap-labelling, we combine the de la Harpe--Skandalis determinant, the trace formula for the Pimsner--Voiculescu boundary map, and transposition decompositions in topological full groups. This gives gap-labelling up to a factor of two for all free actions of poly-$\Z$ groups of Hirsch length three and for certain groups of Hirsch length four, including $\Z<sup>4$. We also recover gap-labelling for $\Z<sup>3$-actions and prove gap-labelling up to a factor of two for $\Z<sup>5$-actions by using cohomology comparison maps for mapping tori.

• The paper presents explicit comparison maps linking groupoid homology and K-theory to verify the HK and gap‐labelling conjectures for poly-Z actions.
• It employs long exact sequences and Pimsner–Voiculescu techniques to compute K-groups and elucidate torsion obstructions in these dynamical systems.
• The analysis extends known low-dimensional results and establishes a framework for addressing higher-dimensional poly-Z group actions on Cantor sets.

Homology–$K$-Theory Conjecture and Gap-Labelling for Polycyclic $\mathbf{Z}$-Group Actions on Cantor Sets

Introduction and Problem Context

This paper presents a thorough and systematic investigation into two central conjectures at the intersection of topological dynamics, operator algebras, and noncommutative geometry: the HK (homology–$K$-theory) conjecture and the gap-labelling (GL) conjecture for transformation groupoids arising from free actions of poly-$\mathbf{Z}$ groups on Cantor sets. The author develops novel tools for the comparison of groupoid (co)homology and $K$-theory, substantially expanding known cases where the conjectures can be verified (both integrally and up to explicit torsion obstructions), and elucidating the structural reasons for existing gaps and failures in the conjectures.

The HK and GL Conjectures: Formulation and Prior Knowledge

The HK conjecture posits a canonical, parity-preserving isomorphism between the $K$-theory of the reduced groupoid $C^*$-algebra and the direct sum of the groupoid's homology groups, mirroring the classical Chern character in the context of ample groupoids:

$$K_i(C^*_r(G)) \cong \bigoplus_{j\geq 0} H_{2j+i}(G), \quad i=0,1 \,.$$

Torsion phenomena in the homology or $K$-theory pose delicate complications, as exemplified by several counterexamples.

The gap-labelling conjecture, originating in the theory of aperiodic solids and the study of spectral gaps of Schrödinger operators, connects the ordered $K_0$-group of a crossed product $\mathbf{Z}$0-algebra and the topological invariants of the underlying dynamics (via invariant measures and traces). In the groupoid formalism, GL asks whether for every invariant probability measure $\mathbf{Z}$1 on the unit space,

$\mathbf{Z}$2

For classical $\mathbf{Z}$3-actions on Cantor sets, this recovers the original Cantor-dynamical formulation.

Earlier results established the conjectures for low-dimensional cases (e.g., $\mathbf{Z}$4 and $\mathbf{Z}$5 actions), shifts of finite type, and certain products. However, counterexamples, especially for odometric systems and higher cohomological dimensions, indicate that the full integral form of HK and GL fails in general [Sc20ETDS, De23ETDS].

Key Contributions: Comparison Maps, Exact Sequences, and Low-Dimensional Computations

1. Comparison of Homological Long Exact Sequences and $\mathbf{Z}$6-Theory Pimsner–Voiculescu Sequences

The central technical innovation is the construction (and proof of compatibility) of explicit comparison maps between groupoid homology and $\mathbf{Z}$7-theory ($\mathbf{Z}$8, $\mathbf{Z}$9), as well as between groupoid cohomology and $K$0-theory ($K$1, $K$2) mediated by specific $K$3-theory classes of the acting group. The interplay between the long exact sequences for (co)homology and the Pimsner–Voiculescu six-term exact sequence is developed in depth, allowing for computations that track individual comparison maps rather than just isomorphism classes.

An explicit demonstration is given that the above comparison maps are compatible with the exact sequences, yielding diagrammatic commutativity results that enable inductive computations for poly-$K$4 group actions.

2. Extension of HK and GL Verifications to Poly-$K$5 Actions

The method is applied systematically to actions of various poly-$K$6 groups, stratified by Hirsch length. For $K$7 and $K$8 actions, the known HK and GL results are recovered through these new frameworks, with an explicit isomorphism between the relevant homology/cohomology groups and $K$9-groups.

The case of the Klein bottle group (a semidirect product of $\mathbf{Z}$0 by $\mathbf{Z}$1 with an orientation-reversing automorphism) is resolved in detail: the $\mathbf{Z}$2-theory fits into a split extension involving $\mathbf{Z}$3 and $\mathbf{Z}$4; the homology comparison map $\mathbf{Z}$5 is an isomorphism, and the image of $\mathbf{Z}$6 is shown to be injective.

For actions of poly-$\mathbf{Z}$7 groups of Hirsch length three and four, explicit exact sequences are computed that describe the structure of the $\mathbf{Z}$8-groups in terms of groupoid homology and cohomology, as well as their relation to the comparison maps. In particular, for several classes of Hirsch length four groups including $\mathbf{Z}$9, the $K$0-groups are assembled from $K$1, $K$2, and $K$3, with $K$4 appearing as a direct summand.

For Hirsch length five (notably, actions by $K$5), the method yields injectivity of $K$6 under suitable orientation assumptions and identifies the limits of current understanding in these higher dimensions.

3. Refined Groupoid-Theoretic Gap-Labelling and Torison Obstructions

The groupoid-theoretic formulation of GL in terms of the dimension map $K$7 and traces of $K$8-theory classes is pursued, and the precise obstruction to surjectivity is analyzed. It is shown that in many cases, even when full GL cannot be established, gap-labelling holds up to a factor of two, that is,

$K$9

This factor arises sharply from torsion effects detectable via the structure of topological full groups and the de la Harpe–Skandalis determinant, with explicit calculations tied to transposition decompositions.

Strong results are obtained for groups of Hirsch length three and certain groups of length four (including $K$0). For $K$1 actions, the original gap-labelling theorem is recovered, while for $K$2 actions, the method, combined with mapping torus computations, yields gap-labelling up to a factor of two. The obstruction is precisely localized in the analysis, paralleling but in explicit detail the denominators that arise in higher-dimensional Chern character theories.

Cohomology Comparison Maps and Poincaré Duality

A significant novelty is the introduction of cohomology comparison maps associated to classes in $K$3-theory, allowing detection of otherwise undetectable summands in both homology and $K$4-theory. The use of Poincaré duality is essential for relating these comparisons and interpreting higher homology terms in the context of the HK conjecture.

Mapping Tori, Connes Pairing, and Higher-Dimensional Gap-Labelling

For free actions of $K$5, mapping torus techniques and the Connes pairing are invoked. For odd $K$6, antisymmetry in the pairing ensures certain cohomological contributions have vanishing trace, which is decisive for gap-labelling in three (and further, up to a factor in five) dimensions. These arguments are shown to be compatible with the exact sequences, yielding explicit computations of the relevant groups and maps.

Implications and Future Directions

This work clarifies, in an explicit and constructive manner, the structural underpinnings of known successes and failures of both the HK and GL conjectures for transformation groupoids of totally disconnected dynamical systems. The explicit comparison procedures developed furnish a template for future investigations, particularly for actions of more general groups or in higher-dimensional settings.

The methods elucidate where torsion and divisibility obstructions arise, providing a toolkit for both further positive results and the identification of new counterexamples. The role of groupoid (co)homology and topological full group structure is likely to persist as central in any further breakthroughs. For practical applications, especially in the study of aperiodic order and solid state physics, the analysis here gives a concrete route toward computing and understanding the invariants controlling the spectrum of associated Schrödinger operators.

Conclusion

This paper establishes a new standard for the systematic study of the HK and GL conjectures for groupoids arising from poly-$K$7 actions on Cantor sets. Through the development of explicit comparison maps, the careful alignment of homological and $K$8-theoretic exact sequences, and the computation of obstruction factors, the work delivers both structural insight and substantial progress on long-standing conjectures. It sets the stage for further advances in the homological analysis of transformation groupoids, $K$9-algebraic dynamics, and the mathematical foundations of aperiodic order.