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The constructive membership problem for discrete free subgroups of rank 2 of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathrm{SL}_2(\mathbb{R})$

Part of: Other groups of matrices

Published online by Cambridge University Press:  01 August 2014

B. Eick ,
M. Kirschmer  and
C. Leedham-Green
B. Eick
Affiliation:
Institut Computational Mathematics, TU Braunschweig, Pockelsstrasse 14, 38106 Braunschweig, Germany email beick@tu-bs.de
M. Kirschmer
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany email markus.kirschmer@math.rwth-aachen.de
C. Leedham-Green
Affiliation:
School of Mathematical Sciences, Queen Mary College University of London, Mile End Road, London E1 4NS, United Kingdom email c.r.leedham-green@qmul.ac.uk

Abstract

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We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank $2$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$. This algorithm, together with methods for checking whether a two-generator subgroup of $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$ is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.

Supplementary materials are available with this article.

MSC classification

Secondary: 20H10: Fuchsian groups and their generalizations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 345 - 359
Copyright
© The Author(s) 2014 

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