Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

Abstract

In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PGL(3, C) (in particular in PGL(2, C), PGL(3, R) and PU(2, 1)). The representations are obtained by gluing decorated tetrahedra of flags as in [10,2]. We list complete computations (giving 0-dimensional or 1-dimensional solution sets (for unipotent boundary holonomy) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the computer methods used to find them. The methods we use work for non-unipotent boundary holonomy as shown in some examples.This work was supported in part by the ANR through the project "Structures Géométriques et Triangulations".

Deformation Spaces of Discrete Groups of SU(2,1) in Quaternionic Hyperbolic Plane: A Case Study

“…Our knowledge of the character variety χ(Γ W , SU(2, 1)) is far less thorough. Boundary unipotent representations are described in [8], whereas a component of this character variety has been described in [13]. We will consider a representation ρ W inside this component.…”

Complex hyperbolic (3,3,n) triangle groups

“…For the representation where I 1 I 3 I 2 I 3 is parabolic, when n = 4 and 5 we have the following description of the quotient orbifold from the census of Falbel, Koseleff and Rouillier [5]. The case n = 4 combines work of Deraux, Falbel and Wang [3,6].…”

“…Let Γ 2 = I 1 I 2 , I 1 I 3 be the index 2 subgroup of Γ with no complex reflections. Then Γ 2 is conjugate to both ρ 1−1 (π 1 (M 4 )) and ρ 4−1 (π 1 (M 4 )) from [5]. In particular, H 2 C /Γ 2 is a complex hyperbolic orbifold whose boundary is the figure eight knot complement.…”

“…Let Γ 2 = I 1 I 2 , I 1 I 3 be the index 2 subgroup of Γ with no complex reflections. Then Γ 2 is conjugate to both ρ 4−3 (π 1 (M 9 )) and ρ 3−3 (π 1 (M 15 )) from [5].…”

“…Furthermore, if as above Γ 2 = I 1 I 2 , I 1 I 3 is the index two subgroup of representation of the (3, 3, ∞) triangle group for which I 1 I 3 I 2 I 3 is parabolic, then H 2 C /Γ 2 is a complex hyperbolic orbifold whose boundary is the Whitehead link complement. This is one of the representations in the census of Falbel, Koseleff and Rouillier [5].…”