Finitely generated subgroups of lattices in $\mathrm {PSL}_2\mathbb {C}$
HTML articles powered by AMS MathViewer
- by Yair Glasner, Juan Souto and Peter Storm PDF
- Proc. Amer. Math. Soc. 138 (2010), 2667-2676 Request permission
Abstract:
Let $\Gamma$ be a lattice in $\mathrm {PSL}_2 (\mathbb {C})$. The pro-normal topology on $\Gamma$ is defined by taking all cosets of nontrivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup $\Delta < \Gamma$ is closed in the pro-normal topology. As a corollary we deduce that if $H$ is a maximal subgroup of a lattice in $\mathrm {PSL}_2( \mathbb {C})$, then either $H$ is of finite index or $H$ is not finitely generated.References
- I. Agol. Tameness of hyperbolic 3-manifolds. Preprint available at http://front.math.ucdavis.edu/math.GT/0405568.
- I. Agol, D. D. Long, and A. W. Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. (2) 153 (2001), no. 3, 599–621. MR 1836283, DOI 10.2307/2661363
- Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385–446. MR 2188131, DOI 10.1090/S0894-0347-05-00513-8
- Richard D. Canary, A covering theorem for hyperbolic $3$-manifolds and its applications, Topology 35 (1996), no. 3, 751–778. MR 1396777, DOI 10.1016/0040-9383(94)00055-7
- Tsachik Gelander and Yair Glasner, Countable primitive groups, Geom. Funct. Anal. 17 (2008), no. 5, 1479–1523. MR 2377495, DOI 10.1007/s00039-007-0630-y
- Rita Gitik, Doubles of groups and hyperbolic LERF 3-manifolds, Ann. of Math. (2) 150 (1999), no. 3, 775–806. MR 1740992, DOI 10.2307/121056
- R. I. Grigorchuk and J. S. Wilson, A structural property concerning abstract commensurability of subgroups, J. London Math. Soc. (2) 68 (2003), no. 3, 671–682. MR 2009443, DOI 10.1112/S0024610703004745
- M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
- G. A. Margulis and G. A. Soĭfer, Maximal subgroups of infinite index in finitely generated linear groups, J. Algebra 69 (1981), no. 1, 1–23. MR 613853, DOI 10.1016/0021-8693(81)90123-X
- Ashot Minasyan, On residual properties of word hyperbolic groups, J. Group Theory 9 (2006), no. 5, 695–714. MR 2253961, DOI 10.1515/JGT.2006.045
- Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
- W. Thurston. The topology and geometry of 3-manifolds. Princeton Univ. Lecture Notes, 1976-1979. Available from the MSRI website www.msri.org.
- William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277
Additional Information
- Yair Glasner
- Affiliation: Department of Mathematics, Ben Gurion University of the Negev, 84105 Beer Sheva, Israel
- MR Author ID: 673281
- ORCID: 0000-0002-6231-3817
- Juan Souto
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-2026
- Peter Storm
- Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
- Received by editor(s): October 26, 2009
- Published electronically: March 16, 2010
- Additional Notes: The first author was partially supported by ISF grant 888/07
The third author was partially supported by a National Science Foundation Postdoctoral Fellowship. - Communicated by: Alexander N. Dranishnikov
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2667-2676
- MSC (2010): Primary 20B15, 20E26, 57N10
- DOI: https://doi.org/10.1090/S0002-9939-10-10310-4
- MathSciNet review: 2644883