Say I have $n$ variables, special unitary operators from $SU(k)$, and write a set of $m$ equations that they must satisfy. These have the form $UVW...Z=I$, i.e. each one specifies that some product of the operators has to be equal to the identity. I wish to find out how many solutions there are.
So far, when $n>m$, I believe that the space of solutions is a submanifold of $SU(k)^n$, with dimension $(n-m)(k^2-1)$, which follows from the preimage theorem.
And in the case $m\geq n$, there are countably many solutions. Clearly, all operators being $I$ is a solution, so there are somewhere between 1 and infinite solutions, although I suspect the answer is finite. I've not been able to write down any sets of $m=n$ equations for which there are any solutions other than $I$, but I am unsure what branch of mathematics might contain answers, or if this conjecture is correct. I wondered if noncommutative algebra might have some answers but that seems mostly defined for rings, not groups.
So my question is, is there an area of maths that deals with this?