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I don't think Critch's reply above answers Harald's question; it seems to presume that the map F_2 -> SL_n(Z/pZ) factors through a chosen inclusion of SL_2(Z/pZ), while Harald wants pairs of elements in F_2 (or, more generally, F_k) which have the same trace after applying ANY homomorphism F_k > SL_n(Z/pZ).

EDITED to comment on David's answer below:

"Also, if there is any finite quotient G of F_k, in which the two elements stay nonconjugate, then there is some representation of G in which they have different traces."

Yeah, but not an n-dimensional representation.

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I don't think Critch's reply above answers Harald's question; it seems to presume that the map F_2 -> SL_n(Z/pZ) factors through a chosen inclusion of SL_2(Z/pZ), while Harald wants pairs of elements in F_2 (or, more generally, F_k) which have the same trace after applying ANY homomorphism F_k > SL_n(Z/pZ).