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

Abstract

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$.