# 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)).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Contents

## Definition

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

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 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).