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

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 inside the space of filters in Λ. When Λ is finitely aligned, the path space coincides with the space of filters in Λ. We construct a semigroup action of P on the path space whose semidirect product groupoid is a Hausdorff ample groupoid. We call the semidirect product groupoid 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 the path groupoid coincides with the Toeplitz groupoid of Renault--Williams and with the path groupoid of Yeend under each of their hypotheses. If Q is countable and amenable, then the path groupoid is amenable by a theorem of Renault--Williams. We also define a boundary-path space that is a closed invariant subset of the unit space of the path groupoid. The reduction of the path groupoid to the boundary-path space is the boundary-path groupoid of Λ, which too we reconcile with the relevant groupoids of Renault--Williams and Yeend. For a particular nonfinitely aligned P-graph, we show that our path groupoid and Spielberg's groupoid have C*-algebras with different ideal structures.