A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs

We give a local treatment of finite alignment by identifying the finitely aligned part of any (not necessarily finitely aligned) higher-rank graph. We show the finitely aligned part is itself a constellation and forms a finitely aligned relative category of paths together with the original higher-rank graph. We show that the elements of the finitely aligned part are precisely those whose cylinder sets are compact, which allows us to give novel definitions of locally compact path and boundary-path spaces for nonfinitely aligned higher-rank graphs. We extend a semigroup action and the associated semidirect product groupoid developed by Renault and Williams to define ample Hausdorff path and boundary-path groupoids. The groupoids are amenable for nonfinitely aligned k-graphs by a result of Renault and Williams. In the finitely aligned case, the path groupoids coincide with Spielberg’s groupoids, and the boundary-path groupoid has an inverse semigroup model via a result of Ortega and Pardo.

💡 Research Summary

This paper develops a systematic local treatment of finite alignment for higher‑rank graphs (k‑graphs) that are not necessarily finitely aligned. The authors introduce a new notion of “finite alignment at a vertex (or at a path) λ” for a P‑graph Λ (Definition 3.1). A path λ belongs to the set FA(Λ) precisely when, for every pair of extensions μ∈λΛ and ν∈Λ, the intersection μΛ∩νΛ can be expressed as a finite union of principal right ideals λ_iΛ. This local definition allows the extraction of a distinguished sub‑structure FA(Λ) from any Λ, regardless of global finite alignment.

The first major result (Lemma 3.4, Proposition 3.5) shows that FA(Λ) is a right ideal and, although it may fail to be a full higher‑rank graph, it forms a “constellation” in the sense of Gould–Higgins – a one‑sided categorical object. By adjoining the range vertices r(FA(Λ)), the authors obtain a subcategory FA_r(Λ) of Λ such that (FA_r(Λ), Λ) is a finitely aligned relative category of paths (Spielberg’s terminology). Thus the finitely aligned part of any Λ can be treated as a well‑behaved relative path category even when Λ itself is not finitely aligned.

The second contribution concerns path spaces. The classical filter space F(Λ) (BSV13) is locally compact when Λ is finitely aligned, but may lose this property for non‑finitely aligned graphs (Yee07). The authors prove a key equivalence (Lemma 4.8): λ∈FA(Λ) if and only if the cylinder set Z(λ)⊂F(Λ) is compact. Using this, they define a new locally compact path space FF_A(Λ) as the open subspace of F(Λ) consisting of filters that contain only finitely aligned elements, and a boundary‑path space ∂Λ as the closed subspace of FF_A(Λ) consisting of infinite filters (Theorem 4.9). Both spaces are Hausdorff, locally compact, and admit a natural action of the degree semigroup P.

Building on Renault–Williams’ semigroup‑action framework (RW17), the paper constructs a directed, locally compact semigroup action (X,P,T) where X is either FF_A(Λ) or ∂Λ (Theorem 5.14). For each m∈P, the domain and image are open and the map T_m is a local homeomorphism. The associated semidirect product groupoid G(X,P,T) (Section 5.2) is shown to be étale, Hausdorff, and ample, with a basis of compact open bisections Z(U,m,n,V). The resulting “path groupoid” G_path = G(FF_A(Λ),P,T) and “boundary‑path groupoid” G_boundary = G(∂Λ,P,T) are therefore ample Hausdorff groupoids. By invoking the amenability theorem of Renault–Williams, both groupoids are amenable for any k‑graph, finitely aligned or not (Corollary 5.20).

In the finitely aligned case, the authors verify that their constructions recover known objects: G_path coincides topologically with Spielberg’s path groupoid, and G_boundary is topologically isomorphic to the tight groupoid of an inverse semigroup model due to Ortega–Pardo (Theorem 6.6, Corollary 6.8). Hence the new framework genuinely extends the classical theory without losing any of its desirable properties.

The paper’s significance lies in providing a robust, locally compact path‑space and groupoid machinery for non‑finitely aligned higher‑rank graphs. This resolves a longstanding obstacle: the lack of local compactness prevented the direct application of groupoid C*‑algebra techniques to many interesting examples. With FF_A(Λ) and ∂Λ, one can now form reduced and full C*‑algebras of the associated ample groupoids, guarantee amenability, and thus obtain faithful representations, K‑theoretic computations, and connections to Leavitt path algebras in a broader setting. The work also suggests further extensions to more general categories of paths or inverse semigroup actions, opening avenues for future research in non‑commutative geometry and operator algebras.