Generating pairs for finite index subgroups of SL(n,Z)

Chen Meiri

doi:10.1016/j.jalgebra.2016.09.026

Generating pairs for finite index subgroups of SL(n,Z)

Chen Meiri

Mathematics

Technion - Israel Institute of Technology

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

Abstract

Let n≥3. Lubotzky [2] asked if every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by two elements. Venkataramana [11] proved that every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by three elements. Since then it was widely believed that the answer to Lubotzky's question is positive and we show that this is indeed the case. In fact, we prove a stronger statement: for “almost every” element g∈SL(n,Z) there exists h∈SL(n,Z) such that [SL(n,Z):〈g,h^m〉]<∞ for every m≥1.

Original language	English
Pages (from-to)	420-424
Number of pages	5
Journal	Journal of Algebra
Volume	470
DOIs	https://doi.org/10.1016/j.jalgebra.2016.09.026
State	Published - 15 Jan 2017

Keywords

Arithmetic groups
Generators
SL(n,Z)
Small subgroups

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2016.09.026

Cite this

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title = "Generating pairs for finite index subgroups of SL(n,Z)",

abstract = "Let n≥3. Lubotzky [2] asked if every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by two elements. Venkataramana [11] proved that every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by three elements. Since then it was widely believed that the answer to Lubotzky's question is positive and we show that this is indeed the case. In fact, we prove a stronger statement: for {\textquotedblleft}almost every{\textquotedblright} element g∈SL(n,Z) there exists h∈SL(n,Z) such that [SL(n,Z):〈g,hm〉]<∞ for every m≥1.",

keywords = "Arithmetic groups, Generators, SL(n,Z), Small subgroups",

author = "Chen Meiri",

note = "Funding Information: Acknowledgments. The author was partially supported by ISF grant 662/15 and is thankful to Nir Avni, Tsachik Gelander and Alex Lubotzky for several fruitful talks. Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

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T1 - Generating pairs for finite index subgroups of SL(n,Z)

AU - Meiri, Chen

N1 - Funding Information: Acknowledgments. The author was partially supported by ISF grant 662/15 and is thankful to Nir Avni, Tsachik Gelander and Alex Lubotzky for several fruitful talks. Publisher Copyright: © 2016 Elsevier Inc.

PY - 2017/1/15

Y1 - 2017/1/15

N2 - Let n≥3. Lubotzky [2] asked if every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by two elements. Venkataramana [11] proved that every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by three elements. Since then it was widely believed that the answer to Lubotzky's question is positive and we show that this is indeed the case. In fact, we prove a stronger statement: for “almost every” element g∈SL(n,Z) there exists h∈SL(n,Z) such that [SL(n,Z):〈g,hm〉]<∞ for every m≥1.

AB - Let n≥3. Lubotzky [2] asked if every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by two elements. Venkataramana [11] proved that every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by three elements. Since then it was widely believed that the answer to Lubotzky's question is positive and we show that this is indeed the case. In fact, we prove a stronger statement: for “almost every” element g∈SL(n,Z) there exists h∈SL(n,Z) such that [SL(n,Z):〈g,hm〉]<∞ for every m≥1.

KW - Arithmetic groups

KW - Generators

KW - SL(n,Z)

KW - Small subgroups

UR - https://www.scopus.com/pages/publications/85007591751

U2 - 10.1016/j.jalgebra.2016.09.026

DO - 10.1016/j.jalgebra.2016.09.026

M3 - Article

SN - 0021-8693

VL - 470

SP - 420

EP - 424

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Generating pairs for finite index subgroups of SL(n,Z)

Abstract

Keywords

ASJC Scopus subject areas

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