Homogeneous number of free generators

Menny Aka; Tsachik Gelander; Gregory A. Soǐfer

doi:10.1515/jgt-2014-0001

Homogeneous number of free generators

Menny Aka, Tsachik Gelander, Gregory A. Soǐfer

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We address two questions of Simon Thomas. First, we show that for any n ≥ 3 one can find a four-generated free subgroup of SL_n(ℤ) which is profinitely dense. More generally, we show that an arithmetic group Γ that admits the congruence subgroup property has a profinitely-dense free subgroup with an explicit bound on its rank. Next, we show that the set of profinitely-dense, locally-free subgroups of such an arithmetic group Γ is uncountable.

Original language	English
Pages (from-to)	525-539
Number of pages	15
Journal	Journal of Group Theory
Volume	17
Issue number	4
DOIs	https://doi.org/10.1515/jgt-2014-0001
State	Published - 1 Jul 2014

Bibliographical note

Funding Information:
We acknowledge the support of the ERC grant 226135, the ERC grant 203418, the ISEF foundation, the Ilan and Asaf Ramon memorial foundation, the “Hoffman Leadership and Responsibility” fellowship program, the ISF grant 1003/11, the BSF grant 2010295, the SFB grant 701 “Spektrale Strukturen und TopologischeMethoden in der Mathematik”, the Emmy Noether Research Institute for Mathematics Bar-Ilan University, the Israel Science Foundation under ISF grant 657/09 and the Max Planck Institute for Mathematics, Bonn.

Funding

We acknowledge the support of the ERC grant 226135, the ERC grant 203418, the ISEF foundation, the Ilan and Asaf Ramon memorial foundation, the “Hoffman Leadership and Responsibility” fellowship program, the ISF grant 1003/11, the BSF grant 2010295, the SFB grant 701 “Spektrale Strukturen und TopologischeMethoden in der Mathematik”, the Emmy Noether Research Institute for Mathematics Bar-Ilan University, the Israel Science Foundation under ISF grant 657/09 and the Max Planck Institute for Mathematics, Bonn.

Funders	Funder number
Emmy Noether Research Institute for Mathematics Bar-Ilan University
Hoffman Leadership and Responsibility
Ilan and Asaf Ramon memorial foundation
Max Planck Institute for Mathematics
Bloom's Syndrome Foundation	2010295
ISEF Foundation
Seventh Framework Programme	226135, 203418
Iowa Science Foundation	1003/11
European Commission
Israel Science Foundation	657/09

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abstract = "We address two questions of Simon Thomas. First, we show that for any n ≥ 3 one can find a four-generated free subgroup of SLn(ℤ) which is profinitely dense. More generally, we show that an arithmetic group Γ that admits the congruence subgroup property has a profinitely-dense free subgroup with an explicit bound on its rank. Next, we show that the set of profinitely-dense, locally-free subgroups of such an arithmetic group Γ is uncountable.",

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Homogeneous number of free generators

Abstract

Bibliographical note

Funding

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