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

Author: Antonin Guilloux
Journal: Proc. Amer. Math. Soc. 143 (2015), 2215-2226
MSC (2010): Primary 57M25, 57M60, 53D18
Published electronically: December 9, 2014
MathSciNet review: 3314127
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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$.

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Antonin Guilloux
Affiliation: Institut de Mathématiques de Jussieu, Unité Mixte de Recherche 7586 du CNRS, Université Pierre et Marie Curie, 4, place Jussieu 75252 Paris Cedex 05, France

Received by editor(s): June 26, 2013
Received by editor(s) in revised form: September 16, 2013, and October 1, 2013
Published electronically: December 9, 2014
Additional Notes: This work was partially supported by the French ANR SGT ANR-11-BS01-0018
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.