Sanov subgroup in SL(2,Z)

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This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) free group:F2 and the group is (up to isomorphism) special linear group:SL(2,Z) (see subgroup structure of special linear group:SL(2,Z)).
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Definition

This is the subgroup of special linear group:SL(2,Z) generated by the matrices:

\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 \\ 2 & 1 \\\end{pmatrix}

It is a free group of rank two with the above two elements as a freely generating set for it. Further information: Sanov subgroup in SL(2,Z) is free of rank two

Arithmetic functions

The subgroup has index 12 in the whole group. In fact, any finite index free subgroup of rank two in the special linear group of degree two must have index 12.

Image in projective special linear group

Consider the quotient map SL(2,\mathbb{Z}) \to PSL(2,\mathbb{Z}). The kernel of this map is of order two. The Sanov subgroup, being free, does not contain any non-identity element of order two, hence it intersects the kernel trivially, so its image in PSL(2,\mathbb{Z}) is isomorphic to it. By the index considerations, this image is a subgroup isomorphic to free group:F2 of index six inside projective special linear group:PSL(2,Z). For more, see Sanov subgroup in PSL(2,Z).

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