**Trace fields of subgroups of SU(***n*, 1)

*Juliette Genzmer *

*n*, 1)

**Abstract**

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 three-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.