Geodesic
On a matrix representation of a free group
Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 863-870

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An analog of the well-known Sanov representation of a free non-Abelian group by matrices of size $\ge3$ 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 $k\ge5$, these matrices generate a free group. 
@article{MZM_1998_64_6_a6,
     author = {A. N. Zubkov},
     title = {On a matrix representation of a free group},
     journal = {Matemati\v{c}eskie zametki},
     pages = {863--870},
     publisher = {mathdoc},
     volume = {64},
     number = {6},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a6/}
}
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A. N. Zubkov. On a matrix representation of a free group. Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 863-870. http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a6/