Suppose we have a word in $t$ letters, and suppose we have a finite group $G$. We can view the word as a map $G^t \to G$- by plugging into our letters and then doing the multiplication in the group.
I am interested in the existence of an efficiently algorithm to determine if this map is trivial (i.e gives the trivial element).
For example a word with one letter is $a\cdot a.... \cdot a$, (repeated $|G|$ times), will always give the trivial element.
A related question is that it's open if the trivial random tries works (I am also interested in random algorithms of course)- see •Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way? .
Some clarifications -
Assume we have the group multiplication table, i.e if $G$ is of size $n$ and the word of size $w$, I want a $Poly(n,w)$ algorithm.
There is a decent method that works for very specific types of word- if you can find a set of $s$ letters, so that their removal gives you disjoint words no pair of which share a letter, then you can recurse easily (by asking the more general algorithm of determining if a word gives you a specific element)- this gives you $n^s + recursion$ work.
I feel that taking the representations of the group should be helpful.
Obvoiusly this may be NP hard, but I don't know many examples of NP hard problems to try and reduce it to.