On a matrix representation of a free group

A. N. Zubkov

doi:10.1007/bf02313032

On a matrix representation of a free group

A. N. Zubkov

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

1 Citation (Scopus)

Abstract

An analog of the well-known Sanov representation of a free non-Abelian group by matrices of size ≥ 3 is studied. Instead of transvections used in the Sanov representation, we use matrices with "filled" first (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The "filled" places are occupied by the same parameter k. It is proved that, for |k|≥ 5, these matrices generate a free group. However, for k = 2, this is not the case.

Original language	English
Pages (from-to)	745-752
Number of pages	8
Journal	Mathematical Notes
Volume	64
Issue number	6
DOIs	https://doi.org/10.1007/bf02313032
Publication status	Published - 1998
Externally published	Yes

Keywords

Congruence subgroup
Free group
Principal congruence subgroup
Sanov representation
Transvection

ASJC Scopus subject areas

General Mathematics

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10.1007/bf02313032

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title = "On a matrix representation of a free group",

abstract = "An analog of the well-known Sanov representation of a free non-Abelian group by matrices of size ≥ 3 is studied. Instead of transvections used in the Sanov representation, we use matrices with {"}filled{"} first (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The {"}filled{"} places are occupied by the same parameter k. It is proved that, for |k|≥ 5, these matrices generate a free group. However, for k = 2, this is not the case.",

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T1 - On a matrix representation of a free group

AU - Zubkov, A. N.

PY - 1998

Y1 - 1998

N2 - An analog of the well-known Sanov representation of a free non-Abelian group by matrices of size ≥ 3 is studied. Instead of transvections used in the Sanov representation, we use matrices with "filled" first (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The "filled" places are occupied by the same parameter k. It is proved that, for |k|≥ 5, these matrices generate a free group. However, for k = 2, this is not the case.

AB - An analog of the well-known Sanov representation of a free non-Abelian group by matrices of size ≥ 3 is studied. Instead of transvections used in the Sanov representation, we use matrices with "filled" first (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The "filled" places are occupied by the same parameter k. It is proved that, for |k|≥ 5, these matrices generate a free group. However, for k = 2, this is not the case.

KW - Congruence subgroup

KW - Free group

KW - Principal congruence subgroup

KW - Sanov representation

KW - Transvection

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SP - 745

EP - 752

JO - Mathematical Notes

JF - Mathematical Notes

IS - 6

ER -

On a matrix representation of a free group

Abstract

Keywords

ASJC Scopus subject areas

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