For a group $G$ and a tuple $J = (g\_1,g\_2 ... g\_n) \in G^k$ for $k$ some constant, define a parametrized word $w : G^k \rightarrow G$ to be a function which takes $J$ to some product of the elements in $J$. 
So $w(J) = g\_1g\_1g\_2$ for $k \geq 2$ would be an example. 


The structure of the space of all $w$ for a particular group modulo the equivalence relation of functional equality is not trivial.
For instance, over $\mathbb{Z}\_2$, $g\_1g\_1g\_2 = g\_2$ for all $J$ ,and for a finite abelian group the space of $w$ is clearly finite.


I don't know whether this topic has been covered before; It seems simple enough that someone might have done work on it, but I cannot find anything. Does anyone know what this area might be called?



Have any publications been made in this area of group theory?
I believe this is the subject of "word maps". See [this link](http://www.mpim-bonn.mpg.de/node/3421) for a list of relevant authors (there are papers by Shalev and Larsen, e.g.), it is a big area.


