Effective virtual and residual properties of some arithmetic hyperbolic $3$–manifolds
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- by Jason DeBlois, Nicholas Miller and Priyam Patel PDF
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Abstract:
We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesically immerse in non-compact right-angled reflection orbifolds, extending work of the third author from the compact case. The first result gives examples to which the second applies, and for these we give explicit bounds on geodesic residual finiteness growth.References
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Additional Information
- Jason DeBlois
- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- MR Author ID: 785234
- Email: jdeblois@pitt.edu
- Nicholas Miller
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- Address at time of publication: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
- MR Author ID: 1222359
- Email: nickmbmiller@berkeley.edu
- Priyam Patel
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84111
- Email: patelp@math.utah.edu
- Received by editor(s): November 11, 2019
- Received by editor(s) in revised form: April 27, 2020
- Published electronically: September 9, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8219-8257
- MSC (2010): Primary 20E26, 57M10
- DOI: https://doi.org/10.1090/tran/8190
- MathSciNet review: 4169687