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CONTINUED FRACTIONS FOR A CLASS OF TRIANGLE GROUPS

Published online by Cambridge University Press:  07 February 2013

KARIANE CALTA
Affiliation:
Vassar College, Poughkeepsie, NY 12604-0257, USA email kacalta@vassar.edu
THOMAS A. SCHMIDT*
Affiliation:
Oregon State University, Corvallis, OR 97331, USA
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Abstract

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We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

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