Deformation of hyperbolic manifolds in $\mathrm {PGL}(n,\mathbf {C})$ and discreteness of the peripheral representations - Archive ouverte HAL Access content directly
Journal Articles Proceedings of the American Mathematical Society Year : 2015

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

Antonin Guilloux

#### 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 $\geq 1$.

#### Domains

Mathematics [math] Geometric Topology [math.GT]

### Dates and versions

hal-01370267 , version 1 (22-09-2016)

### Identifiers

• HAL Id : hal-01370267 , version 1
• ARXIV :
• DOI :

### Cite

Antonin Guilloux. Deformation of hyperbolic manifolds in $\mathrm {PGL}(n,\mathbf {C})$ and discreteness of the peripheral representations. Proceedings of the American Mathematical Society, 2015, 143 (5), pp.2215-2226 ⟨10.1090/S0002-9939-2014-12376-8⟩. ⟨hal-01370267⟩

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