Free Group Criterion

I've heard it said that the group of linear-
fractional transformations generated by tau
and sigma is freely generated by them - never
saw a proof. Here tau(z) = z + 2 and
sigma(z) = z/(2z + 1).

I found a keen proof, using the following lemma.
Not being JSH I'm not going to insist that I
be given a Field's Medal for this. It must be
well known, right?

Lemma: Suppose that G acts on X and G is generated
by a and b. Suppose that A and B are subsets of X,
such that the a^n A are disjoint (n in Z),
the b^n B are disjoint, and for every n <> 0
we have a^n A subset B and b^n B subset A.
Then G is freely generated by a and b.

************************

David C. Ullrich
.

Relevant Pages

  • Re: Free Group Criterion
    ... fractional transformations generated by tau ... and sigma is freely generated by them - never ... such that the a^n A are disjoint, ...
    (sci.math)
  • Re: Free Group Criterion
    ... David C. Ullrich wrote: ... I found a keen proof, ... such that the a^n A are disjoint, ...
    (sci.math)