The Modular group ? acts on the set of extended rational numbers ?Q
transitively. Here, our main purpose is to examine some properties of
hyperbolic paths of minimal lengths in the suborbital graphs for ?. We
characterize all vertices of these hyperbolic paths in the suborbital graphs
which are trees.
In this paper, we examine some properties of suborbital graphs for the Picard group. We obtain edge and circuit conditions, then propose a conjecture for the graph to be forest. This paper is an extension of some results in (Jones et al. in The Modular Group and Generalized Farey Graphs, pp. 316-338, 1991).
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