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In this paper, we provide a model for cross sections to the geodesic andhorocycle flows on $\operatorname{SL}(2, \mathbb{R})/G_q$ using an extension ofa heuristic of P. Arnoux and A. Nogueira. Our starting point is a continuedfraction algorithm related to the group $G_q$, and a cross section to thehorocycle flow on $\operatorname{SL}(2, \mathbb{R})/G_q$ from a previous paper.As an application, we get the natural extension and invariant measure for asymmetric $G_q$-Farey interval map resulting from projectivizing theaforementioned continued fraction algorithm.

On Cross Sections to the Geodesic and Horocycle Flows on Quotients of $\operatorname{SL}(2, \mathbb{R})$ by Hecke Triangle Groups $G_q$