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
Abstract 
References 
Similar Articles 
Additional Information
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.
 1.
I. Agol.
Tameness of hyperbolic 3manifolds. 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
(2002e:20099), http://dx.doi.org/10.2307/2661363
 3.
Danny
Calegari and David
Gabai, Shrinkwrapping and the taming of
hyperbolic 3manifolds, J. Amer. Math. Soc.
19 (2006), no. 2,
385–446. MR 2188131
(2006g:57030), http://dx.doi.org/10.1090/S0894034705005138
 4.
Richard
D. Canary, A covering theorem for hyperbolic 3manifolds and its
applications, Topology 35 (1996), no. 3,
751–778. MR 1396777
(97e:57012), http://dx.doi.org/10.1016/00409383(94)000557
 5.
Tsachik
Gelander and Yair
Glasner, Countable primitive groups, Geom. Funct. Anal.
17 (2008), no. 5, 1479–1523. MR 2377495
(2008m:20003), http://dx.doi.org/10.1007/s000390070630y
 6.
Rita
Gitik, Doubles of groups and hyperbolic LERF 3manifolds, Ann.
of Math. (2) 150 (1999), no. 3, 775–806. MR 1740992
(2001a:20044), http://dx.doi.org/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
(2004i:20056), http://dx.doi.org/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
(95m:20041)
 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
(83a:20056), http://dx.doi.org/10.1016/00218693(81)90123X
 10.
Ashot
Minasyan, On residual properties of word hyperbolic groups, J.
Group Theory 9 (2006), no. 5, 695–714. MR 2253961
(2007e:20091), http://dx.doi.org/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
(58 #12996)
 12.
W. Thurston.
The topology and geometry of 3manifolds. Princeton Univ. Lecture Notes, 19761979. Available from the MSRI website www.msri.org.
 13.
William
P. Thurston, Hyperbolic structures on 3manifolds. I. Deformation
of acylindrical manifolds, Ann. of Math. (2) 124
(1986), no. 2, 203–246. MR 855294
(88g:57014), http://dx.doi.org/10.2307/1971277
 1.
 I. Agol.
Tameness of hyperbolic 3manifolds. 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(3):599621, 2001. MR 1836283 (2002e:20099)
 3.
 D. Calegari and D. Gabai.
Shrinkwrapping and the taming of hyperbolic 3manifolds. J. Amer. Math. Soc., 19(2):385446 (electronic), 2006. MR 2188131 (2006g:57030)
 4.
 R. Canary.
A covering theorem for hyperbolic manifolds and its applications. Topology, 35(3):751778, 1996. MR 1396777 (97e:57012)
 5.
 T. Gelander and Y. Glasner.
Countable primitive groups. Geom. Funct. Anal., 17(5):14791523, 2008. MR 2377495 (2008m:20003)
 6.
 R. Gitik.
Doubles of groups and hyperbolic LERF 3manifolds. Ann. of Math. (2), 150(3):775806, 1999. MR 1740992 (2001a:20044)
 7.
 R. I. Grigorchuk and J. S. Wilson.
A structural property concerning abstract commensurability of subgroups. J. London Math. Soc. (2), 68(3):671682, 2003. MR 2009443 (2004i:20056)
 8.
 M. Gromov.
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)
 9.
 G. Margulis and G. Soĭfer.
Maximal subgroups of infinite index in finitely generated linear groups. J. of Algebra, 69(1):123, 1981. MR 613853 (83a:20056)
 10.
 Ashot Minasyan.
On residual properties of word hyperbolic groups. J. Group Theory, 9(5):695714, 2006. MR 2253961 (2007e:20091)
 11.
 P. Scott.
Subgroups of surface groups are almost geometric. J. London Math. Soc. (2), 17(3):555565, 1978. MR 0494062 (58:12996)
 12.
 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
20B15,
20E26,
57N10
Retrieve articles in all journals
with MSC (2010):
20B15,
20E26,
57N10
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
PII:
S 00029939(10)103104
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.
