Connectedness of a suborbital graph for congruence subgroups. (Q2636691)

For a permutation group \(G\) acting on a set \(S\), Sims introduced and developed a theory of so-called suborbital graphs. If \(G\) acts transitively on \(S\), define a suborbital graph as follows. Choose a pair \((s,t)\in S\times S\) and look at the graph with vertex set \(S\) and an edge from \(u\) to \(v\) if \(u=g.s\), \(v=g.t\) for some \(g\in G\). For instance, the modular group \(\Gamma\) has an action on \(\widehat{\mathbb Q}=\mathbb Q\cup\{\infty\}\). For the pair \((\infty,1)\), the resultant graph is the well-studied Farey graph. In the article [in Groups, Vol. 2, Proc. Int. Conf., St. Andrews/UK 1989, Lond. Math. Soc. Lect. Note Ser. 160, 316-338 (1991; Zbl 0728.20040)] \textit{G. A. Jones} et al. studied the suborbital graphs \(G_{u,n}\) corresponding to a pair \((\infty,u/n)\) with \(n\geq 0\) and \((u,n)=1\). They examined properties like being connected and being a forest. The equivalence relation on \(\widehat{\mathbb Q}\) studied by them is induced by the congruence subgroup \(\Gamma_0(n)\). In the present paper, the authors study an analogous suborbital graph where the equivalence on \(\widehat{\mathbb Q}\) is induced by the smaller congruence subgroup \(\Gamma_1(n)\). The resulting graph \(H_{u,n}\) is the full subgraph of \(G_{u,n}\) whose vertex set is a \(\Gamma_1(n)\)-orbit. The authors prove the following results: \(\bullet\) \(H_{1,n}\) is connected if and only if \(n\leq 2\); \(\bullet\) \(H_{u,n}\) is disconnected if \(n>2\); \(\bullet\) \(H_{1,n}\) and \(H_{n-1,n}\) are forests if \(n>1\); \(\bullet\) If \(H_{u,n}\) contains a triangle, then \(\Gamma_1(n)\) contains an elliptic element of order \(3\) but the converse is not true in general; \(\bullet\) \(H_{u,n}\) contains a \(2\)-gon if and only if \(\Gamma_1(n)\) contains an elliptic element of order \(2\).