Path and boundary-path groupoids of nonfinitely aligned higher-rank graphs over weakly quasi-lattice ordered groups

<p dir="ltr">Given a weakly quasi-lattice ordered group (Q, P ) and a P-graph Λ (not necessarily finitely aligned), we construct a locally compact Hausdorff path space F_c(Λ) inside the space F(Λ) of filters in Λ. When Λ is finitely aligned, F_c(Λ) coincides with F(Λ). We construct a semigroup action T of P on F_c(Λ) whose semidirect product groupoid GΛ is a Hausdorff ample groupoid. We call GΛ the path groupoid of Λ, which is in general distinct from the one that Spielberg associates to nonfinitely aligned left cancellative small categories. We show GΛ coincides with the Toeplitz groupoid of Renault and Williams and with the path groupoid of Yeend under each of their hypotheses. If Q is countable and amenable, then GΛ is amenable by a theorem of Renault and Williams. We also define a boundary-path space ∂Λ that is a closed invariant subset of the unit space of GΛ. The reduction G_∂Λ of GΛ to ∂Λ is the boundary-path groupoid of Λ, which too we reconcile with the relevant groupoids of Renault-Williams and Yeend. For a particular non-finitely aligned P-graph Λ₀, we show that our path groupoid GΛ₀ and Spielberg’s groupoid of Λ₀ have C*-algebras with different ideal structures.</p>