Finitely generated subgroups of lattices in
Authors:
Yair Glasner, Juan Souto and Peter Storm
Journal:
Proc. Amer. Math. Soc. 138 (2010), 26672676
MSC (2010):
Primary 20B15, 20E26, 57N10
Published electronically:
March 16, 2010
MathSciNet review:
2644883
Fulltext PDF
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Abstract: Let be a lattice in . The pronormal topology on is defined by taking all cosets of nontrivial normal subgroups as a basis. This topology is finer than the profinite topology, but it is not discrete. We prove that every finitely generated subgroup is closed in the pronormal 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.
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 I. Agol.
Tameness of hyperbolic 3manifolds. Preprint available at http://front. math.ucdavis.edu/math.GT/0405568.
 2.
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The Bianchi groups are separable on geometrically finite subgroups. Ann. of Math. (2), 153(3):599621, 2001. MR 1836283 (2002e:20099)
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Shrinkwrapping and the taming of hyperbolic 3manifolds. J. Amer. Math. Soc., 19(2):385446 (electronic), 2006. MR 2188131 (2006g:57030)
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 R. Canary.
A covering theorem for hyperbolic manifolds and its applications. Topology, 35(3):751778, 1996. MR 1396777 (97e:57012)
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Countable primitive groups. Geom. Funct. Anal., 17(5):14791523, 2008. MR 2377495 (2008m:20003)
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Doubles of groups and hyperbolic LERF 3manifolds. Ann. of Math. (2), 150(3):775806, 1999. MR 1740992 (2001a:20044)
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A structural property concerning abstract commensurability of subgroups. J. London Math. Soc. (2), 68(3):671682, 2003. MR 2009443 (2004i:20056)
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Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 1993. MR 1253544 (95m:20041)
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Maximal subgroups of infinite index in finitely generated linear groups. J. of Algebra, 69(1):123, 1981. MR 613853 (83a:20056)
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 Ashot Minasyan.
On residual properties of word hyperbolic groups. J. Group Theory, 9(5):695714, 2006. MR 2253961 (2007e:20091)
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 P. Scott.
Subgroups of surface groups are almost geometric. J. London Math. Soc. (2), 17(3):555565, 1978. MR 0494062 (58:12996)
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 W. Thurston.
The topology and geometry of 3manifolds. Princeton Univ. Lecture Notes, 19761979. Available from the MSRI website www.msri.org.
 13.
 W. Thurston.
Hyperbolic structures on 3manifolds. I: Deformation of acylindrical manifolds. Ann. of Math. (2), 124(2):203246, 1986. MR 855294 (88g:57014)
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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 481092026
Peter Storm
Affiliation:
Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, Pennsylvania 191046395
DOI:
http://dx.doi.org/10.1090/S0002993910103104
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.
