Abstract
We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual residual finiteness growth defined in terms of word length. This equivalence implies that our results recover the quantification of residual finiteness from Bou-Rabee et al. (Math Z, arXiv:1402.6974 [math.GR]) for hyperbolic manifolds that virtually immerse into a compact reflection orbifold.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Agol, I.: Geometrically Finite Subgroup Separability for the Figure-8 Knot Group. Preliminary Report/ Private Communication
Agol, I.: The virtual Haken conjecture, with an appendix by I. Agol, D. Groves and J. Manning. Doc. Math. 18, 1045–1087 (2013)
Agol, I., Long, D.D., Reid, A.W.: The Bianchi groups are separable on geometrically finite subgroups. Ann. Math. (2) 152(3), 599–621 (2001)
Basmajian, A.: Universal length bounds for non-simple closed geodesics on hyperbolic surfaces. J. Topol. 6(2), 513–534 (2013)
Beardon, A.F.: The Geometry of Discrete Groups. Springer, Berlin (1983)
Bou-Rabee, K.: Quantifying residual finiteness. J. Algebra 323, 729–737 (2010)
Bou-Rabee, K., Hagen, M., Patel, P.: Quantifying residual finiteness of virtually special groups. Math. Z. (to appear). arXiv:1402.6974 [math.GR]
Bou-Rabee, K., McReynolds, D.B.: Asymptotic growth and least common multiples in groups. Bull. Lond. Math. Soc. 43(6), 1059–1068 (2011)
Bou-Rabee, K., McReynolds, D.B.: Characterizing linear groups in terms of growth properties. arXiv:1403.0983
Bou-Rabee, K., McReynolds, D.B.: Extremal behavior of divisibility functions. Geom. Dedicata (to appear). arXiv:1211.4727 [math.GR]
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)
Buskin, N.V.: Efficient separability in free groups. Sib. Math. J. 50(4), 603–608 (2009)
Haglund, F.: Finite index subgroups of graph products. Geom. Dedicata 135, 167–209 (2008)
Haglund, F., Wise, D.T.: Special cube complexes. Geom. Funct. Anal. 17(5), 1551–1620 (2008)
Haglund, F., Wise, D.T.: Coxeter groups are virtually special. Adv. Math. 224, 1890–1903 (2010)
Kassabov, M., Matucci, F.: Bounding the residual finiteness of free groups. Proc. Am. Math. Soc. 139(7), 2281–2286 (2011)
Kozma, G., Thom, A.: Divisibility and Laws in Finite Simple Groups (preprint). arXiv:1403.2324 [math.GR]
Mal’cev, A.I.: On the faithful representation of infinite groups by matrices. Mat. Sb. 8(50), 405–422 (1940)
Meyerhoff, R.: A lower bound for the volume of hyperbolic 3-manifolds. Can. J. Math. 39(5), 1038–1056 (1987)
Patel, P.: On a theorem of Peter Scott. Proc. Am. Math. Soc. 142(8), 2891–2906 (2014)
Potyagailo, L., Vinberg, É.B.: On right-angled re ection groups in hyperbolic spaces. Comment. Math. Helv. 80(1), 63–73 (2005)
Rivin, I.: Geodesics with one self-intersection, and other stories. Adv. Math. 231(5), 2391–2412 (2012)
Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. (2) 17, 555–565 (1978); Correction: ibid. (2) 32, 217–220 (1985)
Todhunter: Spherical Trigonometry, Project Gutenberg eBook (2006)
Vinberg, É.B.: Absence of crystallographic groups of re ections in Lobachevskiĭ spaces of large dimension. Trudy Moskov. Mat. Obshch. 47, 68–102 (1984)
Wise, D.T.: The structure of groups with a quasiconvex hierarchy, 205 pp (2011) (preprint)
Acknowledgments
Much of the work in this paper originally appeared in the author’s thesis, written under the supervision of Feng Luo, whom the author thanks for his help, support, and encouragement. The author would also like to sincerely thank Ian Agol for sharing his work and ideas, as well as for insightful conversations. Many thanks are also due to Tian Yang for his suggestions regarding the proofs of Lemmas 3.1 and 4.1, to Alan Reid and Nicholas Miller for helpful conversations, to Ben McReynolds for his encouragement to write this paper, and to David Duncan for comments on an early draft. The author would especially like to thank an anonymous referee whose extensive and thorough suggestions have greatly increased the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Patel, P. On the residual finiteness growths of particular hyperbolic manifold groups. Geom Dedicata 185, 87–103 (2016). https://doi.org/10.1007/s10711-016-0169-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-016-0169-x