TITLE: Have any publications been made in this area of group theory?
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: From comments it seems you want to know when your group/monoid of functions is finite for all k. These are equivalent and the answer is when G is locally finite of finite exponent. The group you are looking at is the free group of rank k in the variety generated by G. If G has infinite exponent the words $x^n$ are all distinct functions on G. 
If G is of exponent n and is not locally finite then it has an infinite k-generated subgroup so the free group in the variety generated by G is infinite. If G is locally finite, then since varieties of groups are closed under direct limits, it follows G belongs to the variety of groups generated by finite groups of exponent n. This variety has finite free objects on finite generating sets by the solution to the restricted Burnside problem. Thus the free objects in the variety generated by G are finite as well. 
If G is finite things are trivial since there are finitely many k-ary functions on G. In fact it is a classical result of Birkhoff shows the variety generated by a finite universal algebra is locally finite.