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).

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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)?
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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?