  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 the
number $ \nu $ of cusps near the hyperbolic representation. It seems natural to
ask whether some representations remain discrete after deformation. The answer
is generically not. A simple reason for it lies inside the cusps: the
degeneracy of the peripheral representation (i.e. representations of
fundamental groups of the $ \nu $ peripheral tori). They indeed generically
become non-discrete, except for a countable set. This last set corresponds to
hyperbolic Dehn surgeries on $ M$, for which the peripheral representation is
no more faithful.We work here in the framework of $ \mathrm {PGL}(n,\mathbf
{C})$. The hyperbolic structure lifts, via the $ n$-dimensional irreducible
representation, to a representation $ \rho \_{\mathrm {geom}}$. We know from
the 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 the
discreteness (and faithfulness) of the peripheral representation: in a
neighbourhood of the geometric representation, the non-discrete peripheral
representations are contained in a real analytic subvariety of codimension $
\geq 1$.
