Following section 1.4, which is entitled "Constructions Using Amalgams" is the innocent-enough looking exercise:


Show that the group defined by the presentation (presumably on the generators $x\_1, x\_2, x\_3$)


$x\_2x\_1x\_2^{-1}=x\_1^2 $


$ x\_3x\_2x\_3^{-1} = x\_2^2$


$ x\_1x\_3x\_1^{-1} = x\_3^2$


is trivial.


Now, if one simply (well, not simply in practice but simply as in I could write code to do it much more easily than I can actually do it myself...one quickly loses track of inverses etc. and discovers new magical relations that are, in fact, too good to be true) manipulates relations, writes certain words and then tries to move generators past each other, one can obtain this result. Which leaves me to wonder: why precisely is this exercise here?


Guess 1: This exercise stands in contrast to an example in this section where we have the same set of relations except on 4 generators, so 4 relations, and so the same kind of proof does not work out at all (you can no longer move all the generators past one-another); in fact Serre uses amalgams to demonstrate that the aforementioned group contains the free group on two generators. One may perhaps be tempted (as I was, indeed, for way too long) to attempt to use amalgams to solve the exercise which is the topic of this post, to no avail. Maybe the point here is that sometimes amalgams get you nothing; that some types of questions lend themselves to proof by generator and relation manipulation but others to more "abstract" diagram chasing? 


Guess 2: There is an elegant proof somehow using amalgams and it escaped me.


Guess 3: Some other reason.


Anyway, perhaps from my years of secondary, college, and finally graduate math textbook experience, I feel somehow uneasy solving an exercise at the end of a section without employing any tools at all from that section. I worry I've missed the point.


Any ideas? (Excluding perhaps that I am overthinking this and need to let it go?)



About an exercise in Serre's "Trees"
I don't have the book next to me but I'm quite sure I remember the exercise and, as far as I was able to determine, your first guess is the correct one. This isn't an amalgam exercise in any natural way (or any way at all), but rather an example to put the case of 4 generators in perspective. 


