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:

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

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 $12$.

External links

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