Finitely generated subgroups of lattices in

Authors:
Yair Glasner, Juan Souto and Peter Storm

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2667-2676

MSC (2010):
Primary 20B15, 20E26, 57N10

Published electronically:
March 16, 2010

MathSciNet review:
2644883

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a lattice in . The pro-normal topology on 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 is closed in the pro-normal topology. As a corollary we deduce that if is a maximal subgroup of a lattice in , then either is of finite index or is not finitely generated.

**1.**I. Agol.

Tameness of hyperbolic 3-manifolds.

Preprint available at`http://front. math.ucdavis.edu/math.GT/0405568`.**2.**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**, 10.2307/2661363**3.**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**, 10.1090/S0894-0347-05-00513-8**4.**Richard D. Canary,*A covering theorem for hyperbolic 3-manifolds and its applications*, Topology**35**(1996), no. 3, 751–778. MR**1396777**, 10.1016/0040-9383(94)00055-7**5.**Tsachik Gelander and Yair Glasner,*Countable primitive groups*, Geom. Funct. Anal.**17**(2008), no. 5, 1479–1523. MR**2377495**, 10.1007/s00039-007-0630-y**6.**Rita Gitik,*Doubles of groups and hyperbolic LERF 3-manifolds*, Ann. of Math. (2)**150**(1999), no. 3, 775–806. MR**1740992**, 10.2307/121056**7.**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**, 10.1112/S0024610703004745**8.**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****9.**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**, 10.1016/0021-8693(81)90123-X**10.**Ashot Minasyan,*On residual properties of word hyperbolic groups*, J. Group Theory**9**(2006), no. 5, 695–714. MR**2253961**, 10.1515/JGT.2006.045**11.**Peter Scott,*Subgroups of surface groups are almost geometric*, J. London Math. Soc. (2)**17**(1978), no. 3, 555–565. MR**0494062****12.**W. Thurston.

The topology and geometry of 3-manifolds. Princeton Univ. Lecture Notes, 1976-1979.

Available from the MSRI website`www.msri.org`.

**13.**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**, 10.2307/1971277

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Additional Information

**Yair Glasner**

Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, 84105 Beer Sheva, Israel

**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

DOI:
https://doi.org/10.1090/S0002-9939-10-10310-4

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

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.