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Let $ M$ be a cusped hyperbolic $ 3$-manifold, e.g. a knot complement.Thurston showed that the space of deformations of its fundamental group in $\mathrm {PGL}(2,\mathbf {C})$ (up to conjugation) is of complex dimension thenumber $ \nu $ of cusps near the hyperbolic representation. It seems natural toask whether some representations remain discrete after deformation. The answeris generically not. A simple reason for it lies inside the cusps: thedegeneracy of the peripheral representation (i.e. representations offundamental groups of the $ \nu $ peripheral tori). They indeed genericallybecome non-discrete, except for a countable set. This last set corresponds tohyperbolic Dehn surgeries on $ M$, for which the peripheral representation isno more faithful.We work here in the framework of $ \mathrm {PGL}(n,\mathbf{C})$. The hyperbolic structure lifts, via the $ n$-dimensional irreduciblerepresentation, to a representation $ \rho \_{\mathrm {geom}}$. We know fromthe work of Menal-Ferrer and Porti that the space of deformations of $ \rho\_{\textrm {geom}}$ has complex dimension $ (n-1)\nu $.We prove here that,unlike the $ \mathrm {PGL}(2)$-case, the generic behaviour becomes thediscreteness (and faithfulness) of the peripheral representation: in aneighbourhood of the geometric representation, the non-discrete peripheralrepresentations are contained in a real analytic subvariety of codimension $\geq 1$.

Deformation of hyperbolic manifolds in $\mathrm {PGL}(n,\mathbf {C})$ and discreteness of the peripheral representations