Geodesic
Subgroups of infinite index in the modular group II
Glasgow mathematical journal, Tome 22 (1981) no. 1, pp. 101-118

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Let H be a subgroup of Γ, the modular group. Let h be the number of orbits of under the action of H. In each orbit, the stabilizers are H-conjugate. Let U be the mapping z↦z + 1. Each stabilizer is Γ-conjugate to 〈Uc〉 for some non-negative integer c. The integer c is the cusp-width of the orbit. Let h0 be the number of orbits with non-trivial stabilizer, i.e. with c>0. The sequence (c(1), ..., c(h0)) of non-zero cuspwidths is the cusp-split of H. Clearly, h0<h, and h∞ = h−h0 is the number of orbits with trivial stabilizer. 
Stothers, W. W. Subgroups of infinite index in the modular group II. Glasgow mathematical journal, Tome 22 (1981) no. 1, pp. 101-118. doi: 10.1017/S0017089500004523
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