# Sanov subgroup in SL(2,Z) is free of rank two

From Groupprops

(Redirected from Upper-triangular and lower-triangular unipotent matrices generate free non-abelian subgroup in special linear group over integers)

## Contents

## Statement

Consider , the Special linear group:SL(2,Z) (?) (this is the subgroup of the general linear group:GL(2,Z) comprising the matrices of determinant one). Then, the subgroup of generated by the elements:

is a free group with these two elements forming a freely generating set.

## Related facts

## Facts used

- Ping-pong lemma (the form involving two elements): If acts on a set , and and are such that neither nor is contained in the other. Further, suppose that and for all nonzero integers . Then, the subgroup of generated by and is a free group with as a freely generating set.

## Proof

We construct an action that satisfies the hypothesis for fact (1) (the ping-pong lemma).

Let with acting on it the usual way (by matrix multiplication with column vectors). Consider the subsets:

.

We now prove the conditions for the ping-pong lemma:

- Neither of and is contained in the other: In fact, it is cler from the definition that they are disjoint non-empty sets.
- for : Suppose such that . Then . Since , by assumption.
- for : This is similar to the previous part.