Geodesic
On a Result of A. M. Macbeath on Normal Subgroups of a Fuchsian Group
Canadian mathematical bulletin, Tome 13 (1970) no. 1, pp. 15-16

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A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.This theorem will follow from a consideration of the exact sequence * 
Jonsson, W. On a Result of A. M. Macbeath on Normal Subgroups of a Fuchsian Group. Canadian mathematical bulletin, Tome 13 (1970) no. 1, pp. 15-16. doi: 10.4153/CMB-1970-003-0
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     title = {On a {Result} of {A.} {M.} {Macbeath} on {Normal} {Subgroups} of a {Fuchsian} {Group}},
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