In this paper, we study the differences between algebraic and geometric solutions of hyperbolicity equations for ideally triangulated 3-manifolds, and their relations with the variety of representations of the fundamental group of such manifolds into PSL(2, C). We show that the geometric solutions of compatibility equations form an open subset of the algebraic ones, and we prove uniqueness of the geometric solutions of hyperbolic Dehn filling equations. In the last section we study some

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... , doing explicit calculations for three interesting manifolds. Proposition 4. Let M be an ideally triangulated manifold and let z be a choice of moduli. Then, equations C are satisfied if and only if there exist a map D : M → H 3 and a representation h : π 1 (M) → Isom + (H 3 ) such that: