Geodesic
Some diophantine problems arising from the theory of cyclically-presented groups
Glasgow mathematical journal, Tome 41 (1999) no. 2, pp. 157-165

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Let n∈N and let Fn be the free group on n generators. Let w be an arbitrary word in Fn, and let σ be an n-cycle in Sn. We consider groups of the type Γ(n,w)=Fn/N, where N is the normal closure in Fn of the “cycled words’’ w, σ(w), σ2(w),...,σn−1(w), and solve, by means of classical algebraic number theory, the following problems.A. When is Γ(n,w)ab infinite?B. When is Γ(n,w) a perfect group? 
Odoni, R. W. K. Some diophantine problems arising from the theory of cyclically-presented groups. Glasgow mathematical journal, Tome 41 (1999) no. 2, pp. 157-165. doi: 10.1017/S0017089599950383
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