TITLE: Non-conjugate words with the same trace
QUESTION [10 upvotes]: Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, h in SL_2(Z/pZ).
Question: does the same happen for n>=3?
Could it even be possible that there are words w_1, w_2 that are not conjugate even in G=SL_n(Z/pZ), yet always have the same trace: 
tr(w_1(g_1,g_2,..,g_k)) = tr(w_2(g_1,g_2,...,g_k)) for all g_1,...,g_k in SL_n(Z/pZ)?
This would be extremely helpful.
Actually, I just need a weaker statement:
Wild guess.- Let g, h be elements of SL_n(K), n>2. There is a constant
k (which may depend on n but not on K)
such that there are two elements a, b of the ball
({g,h,g^{-1},h^{-1},e})^k for which
(1) tr(a)=tr(b) and
(2) a is not conjugate to b (for g and h generic).
Does anybody have a clue as to whether this is or isn't true?

REPLY [12 votes]: I don't think you asked the question you intended. Even though this question is old, I'll say a bit:
For any fixed $n$ and $p$, there are a finite number of homomorphisms $F_2 \rightarrow SL_n(Z/pZ)$.
Hence, there are only a finite number of possibilities for the trace of a group element as a function of the representation.  Therefore, there are many duplicates.
I think a better question is to ask whether there are pairs $(g_1, g_2)$ of non-conjugate elements of $F_2$ such that for all primes $p$ and all homomorphisms $F_2 \rightarrow SL_n(Z/pZ)$ the traces are equal. In this form, you can pass to a limit $p \rightarrow \infty$ and conclude if nonconjugate elements can be distinguished by traces for an infinite sequence of $SL_n(Z/pZ)$, they could be distinguished by a  homomorphisms to $SL_n(\mathbb C)$. Conversely, if there are no trace identities in $SL_n(\mathbb C)$, then there are no trace identities true in all $SL_n(Z/pZ)$, by finding $Z/pZ$ quotients of the ring of coefficients.
Martin Kassabov has been interested in the latter question, and in discussions he and I have had, we've both come to the opinion that there are probably no trace identitites for $SL_n(C)$ when $n > 2$, but it's not easy to find a proof.  One possible strategy is to first characterize all trace identities in $SL_2$, and then construct representations in $SL_3$ where they break down.  This is interesting to me in any case because of its meaning in 2 and 3-manifold topology -- the trace identities give collections of distinct elements in $\pi_1$ that are forced to have the equal length in any hyperbolic structure.
A weaker question is whether the characteristic polynomials for representations in $SL_n$ can distinguish conjugacy classes in $F_2$.