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- Title
Bounds on Entries in Bianchi Group Generators.
- Authors
Martin, Daniel E
- Abstract
Upper and lower bounds are given for the maximum Euclidean curvature among faces in the Ford domain for |$\text {PSL}_2({\mathcal{O}})$| in the upper-half space model of hyperbolic space, where |${\mathcal{O}}$| is an imaginary quadratic ring of integers with discriminant |$\Delta $|. We prove these bounds are asymptotically within |$(\log |\Delta |)^{8.54}$| of one another. This improves on the previous best upper bound, which is roughly off by a factor between |$\Delta ^2$| and |$|\Delta |^{5/2}$| depending on the smallest prime dividing |$\Delta $|. The gap between our upper and lower bounds is determined by an analog of Jacobsthal's function, introduced here for imaginary quadratic fields.
- Subjects
BIANCHI groups; GENERATORS of groups; RINGS of integers; HYPERBOLIC spaces; CURVATURE
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 11, p9155
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnac092