Trace Fields of Subgroups of SU(n,1))

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In this note, we study the field generated by the traces of subgroups of SU(n,1). Under some hypotheses, the trace field of a group Γ⊂ SU(2,1) is equal to the field generated by the coefficients of the matrices in Γ. If the group is the image of a representation of the fundamental group of a triangulated 3-manifold, we can relate the trace field to a geometric invariant. For an arithmetic group of the first type in SU(n,1), up to conjugacy, the trace field and the field of the coefficients are the same.