## Improved computation of fundamental domains for arithmetic Fuchsian groups

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James Rickards
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## Abstract:

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.## References

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## Additional Information

**James Rickards**- Affiliation: Department of Mathematics, University of Colorado Boulder, 2300 Colorado Ave, Boulder, Colorado, USA 80309
- MR Author ID: 939651
- ORCID: 0000-0003-3387-9368
- Email: james.rickards@colorado.edu
- Received by editor(s): November 12, 2021
- Received by editor(s) in revised form: June 15, 2022
- Published electronically: August 3, 2022
- Additional Notes: This research was supported by an NSERC Vanier Scholarship at McGill University. The author was partially supported by NSF-CAREER CNS-1652238 (PI Katherine E. Stange).
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2929-2954 - MSC (2020): Primary 11Y40; Secondary 11F06, 20H10, 11R52
- DOI: https://doi.org/10.1090/mcom/3777
- MathSciNet review: 4473108