Given a word $w \in X^{\pm 1}$ representing an element of the free group $F(X)$ there is a (usually non-unique) sequence $w=w\_0 \to w\_1 \to \cdots \to w\_r$ with $|w\_i|>|w\_{i+1}|$ where $w\_r$ is the unique reduced form of $w$, i.e. there are no subwords of the form $xx^{-1}, x \in X^{\pm 1}$. I know of this fact, even have a proof of it in some notes I wrote. **Does anybody know a reference for this?**


Similarly let $\mathbb X$ be a graph of groups and let $w$ be some word in some symbols representing an element of the fundamental group $\pi\_1(\mathbb X,v)$. Britton's lemma famously, and conveniently, describes what normal forms are in the case of HNN extensions. **Does anybody know of a reference** for the fact that given $w$ as above, there is a sequence $w=w\_0 \to w\_1 \to \cdots \to w\_r$ of decreasing *syllable length* such that $w\_r$ is reduced (i.e. has minimal syllable length among all words representing the same group element)?


Again, I can prove this myself, but a reference would be convenient.



Passing to normal forms in graphs of groups
In comments, the OP indicates that what they really want is a *uniqueness* result for reduced words in arbitrary graphs of groups. (Indeed, what the question actually asks for, that any word can be transformed into a reduced word by successively cancelling, is obvious by induction on length.)


The desired uniqueness result exists, but people don't usually write it down in full generality, because it's a bit of a mess to state. Instead, they usually write down the following morally equivalent result, which can be thought of as uniqueness for the trivial element.


**Theorem:** Any word in the fundamental group of a graph of groups that represents the trivial element is not reduced.


For instance, this is Theorem 11 of Serre's classic *Trees*.


The appropriate uniqueness result follows immediately: given two reduced words $u,v$ both representing the same element $g$, we have that $uv^{-1}$ represents the trivial element, and so is not reduced. Since $u$ and $v$ are themselves reduced, the only possible cancellation occurs at the concatenation point of the words. Now apply this cancellation and induct on length. Following convention, I will leave it as an exercise to state the uniqueness result that this proves. ;)


Finally, I'll add an important philosophical remark, of which I'm sure the OP is well aware. The existence and uniqueness of normal forms for graphs of groups is equivalent to the statement that the Bass--Serre tree is a tree -- existence shows that the graph is connected, and uniqueness shows that it has no loops. So if you're content to prove that it's a tree via another technique (e.g. the topological approach of Scott--Wall) then you can also deduce existence and uniqueness.


