Proving that a group generated by x,y and z and a given relation is actually free I'm trying to show that a group generated by elements $x,y,z$ with a given relation $xyxz^{-2}=1$ (where $1$ is the identity) is in fact a free group.
What are some usual ways of going about this kind of problem? Just hints please!
Regards
 A: You have $y=x^{-1}z^2x^{-1}$ from your relation, so $G=\langle x,z\rangle$ without any defining relation.
A: (This is essentially just an expansion of Boris Novikov's solution.)
There is an important technique known as Tietze transformations for manipulating presentations of groups.  The two Tietze transformations are:


*

*Add a new relation which can be derived from the existing relations.  Inversely, remove a relation that can be derived from the others.

*Add a new generator, together with a relation that expresses it in terms of the existing generators.  Inversely, you can remove a generator if this generator only appears in one relation, and this relation defines the generator in terms of other generators.
A special case of (1) is to replace a relation with an equivalent relation.  Technically, this involves two applications of (1): you must first add the new relation, and then remove the original.
Tietze's theorem is that these moves do not change the isomorphism type of a group defined by a presentation.  Moreover, if two finitely presented groups are isomorphic, it is possible to get from the first presentation to the second using a finite sequence of Tietze transformations.
Now, the given group has the following presentation:
$$
\langle x,y,z \mid xyxz^{-2} = 1\rangle.
$$
The given relation is equivalent to the relation $y = x^{-1}z^2 x$, so we can replace it using two Tietze transformations of type (1):
$$
\langle x,y,z \mid y=x^{-1}z^2 x^{-1}\rangle.
$$
We can now use a Tietze transformation of type (2) to remove the generator $y$.  This leaves
$$
\langle x,z \mid -\rangle
$$
which is a presentation for the free group of rank two.
A: Are you aware of the relations of your problem to the subject of presentation of groups, in general, and in particular to the Nielsen-Schreier therorem and Nielsen transformations? This should qualify as one of the "usual ways" of going about this kind of problem at least.
