Sanov subgroup in SL(2,Z)

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).

External links

• Math Overflow discussion of finite index free subgroups in the special linear group of degree two