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Continued fraction representations of units associated with certain Hecke groups

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Abstract.

We consider the Hecke groups generated by \( S(z)=z+\lambda \) and \( T(z)=-1/z \) with \( \lambda=2\cos(\pi/q) \) for \( q\geqq 3 \). We show that when \( q=4 \) or 6, the units in \( \mathbb{Z}[\lambda] \) are infinite pure periodic \( \lambda \)-fractions, and hence cannot be cusp points (images of \( \infty \) by a member of the group.) The case when \( q=7 \) is quite different; examples of units that are finite \( \lambda \)-fractions and units that are infinite \( \lambda \)-fractions are given. We conclude with a conjecture on the structure of these infinite repeating units.

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Eingegangen am 3.2.2000

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Rosen, D., Towse, C. Continued fraction representations of units associated with certain Hecke groups. Arch. Math. 77, 294–302 (2001). https://doi.org/10.1007/PL00000494

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