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Finitely generated subgroups of lattices in $ \mathrm{PSL}_2\mathbb{C}$


Authors: Yair Glasner, Juan Souto and Peter Storm
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
Published electronically: March 16, 2010
MathSciNet review: 2644883
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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.


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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.

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