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

TL;DR

This paper investigates how deformations of hyperbolic 3-manifolds' fundamental groups in PGL(n,C) affect the discreteness of peripheral representations, showing that generic deformations tend to preserve discreteness unlike in the PGL(2,C) case.

Contribution

It extends the analysis of peripheral representation behavior from PGL(2,C) to PGL(n,C), demonstrating that generic deformations lead to discrete peripheral representations.

Findings

Generic deformations in PGL(n,C) preserve discreteness of peripheral representations.

Non-discrete peripheral representations form a real analytic subvariety of codimension at least 1.

The behavior contrasts with the PGL(2,C) case, where non-discreteness is more prevalent.

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$.