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CONTINUED FRACTIONS FOR A CLASS OF TRIANGLE GROUPS

Part of: Other groups of matrices Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  07 February 2013

KARIANE CALTA  and
THOMAS A. SCHMIDT
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
*
✉toms@math.orst.edu
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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.

Keywords

continued fractionsFuchsian groupsDiophantine approximation

MSC classification

Secondary: 11J70: Continued fractions and generalizations 11K50: Metric theory of continued fractions 11J17: Approximation by numbers from a fixed field 11J81: Transcendence (general theory) 20H10: Fuchsian groups and their generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 1-2 , October 2012 , pp. 21 - 42
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

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