Sanov subgroup in SL(2,Z) is free of rank two
From Groupprops
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.