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 $\geq 1$.