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).
In this paper we deal with congruence equations arising from suborbital graphs of the normalizer of Γ0(m) in P SL(2, R) . We also propose a conjecture concerning the suborbital graphs of the normalizer and the related congruence equations. In order to prove the existence of solution of an equation over prime finite field, this paper utilizes the Fuchsian group action on the upper half plane and Farey graphs properties.
In this paper, we investigate suborbital graphs formed by the action of Γ 2 which is the group generated by the second powers of the elements of the modular group Γ onQ. Firstly, conditions for being an edge, self-paired and paired graphs are provided, then we give necessary and sufficient conditions for the suborbital graphs to contain a circuit and to be a forest. Finally, we examine the connectivity of the subgraph Fu,N and show that it is connected if and only if N ≤ 2.
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