Generalized classes of suborbital graphs for the congruence subgroups of the modular group

Pradthana Jaipong ; Wanchai Tapanyo

Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 20-36.

Résumé

Let $\mathrm{\Gamma }$ be the modular group. We extend a nontrivial $\mathrm{\Gamma }$-invariant equivalence relation on $\hat{\mathbb{Q}}$ to a general relation by replacing the group ${\mathrm{\Gamma }}_0(n)$ by ${\mathrm{\Gamma }}_K(n)$, and determine the suborbital graph ${\mathcal{F}}_{u,n}^K$, an extended concept of the graph ${\mathcal{F}}_{u,n}$. We investigate several properties of the graph, such as, connectivity, forest conditions, and the relation between circuits of the graph and elliptic elements of the group ${\mathrm{\Gamma }}_K(n)$. We also provide the discussion on suborbital graphs for conjugate subgroups of $\mathrm{\Gamma }$.

Détail

Mots-clés : modular group, congruence subgroups, suborbital graphs.

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