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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The subgroup has index 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 .
Image in projective special linear group
Consider the quotient map . 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 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).