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

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Abstract

In this paper we are interested in computing representations of the fundamental group of SL 3-manifold into \(\mathrm{PGL}(3,\mathbb {C})\) (in particular in \(\mathrm{PGL}(2,\mathbb {C}), \mathrm{PGL}(3,\mathbb {R})\) and \(\mathrm{PU}(2,1)\)). The representations are obtained by gluing decorated tetrahedra of flags as in Falbel (J Differ Geom 79:69–110, 2008), Bergeron et al. (Tetrahedra of flags, volume and homology of SL(3), 2011). 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.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS UMR 7586 and INRIA EPI-OURAGAN, Université Pierre et Marie Curie, (UPMC Sorbonne Universités), 4, Place Jussieu, 75252 , Paris Cedex 05, France

    E. Falbel, P.-V. Koseleff & F. Rouillier

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  1. E. Falbel
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  2. P.-V. Koseleff
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  3. F. Rouillier
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Correspondence to E. Falbel.

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This work was supported in part by the ANR through the project “Structures Géométriques et Triangulations”.

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Falbel, E., Koseleff, PV. & Rouillier, F. Representations of fundamental groups of 3-manifolds into \(\mathrm{PGL}(3,\mathbb {C})\): exact computations in low complexity. Geom Dedicata 177, 229–255 (2015). https://doi.org/10.1007/s10711-014-9987-x

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  • DOI: https://doi.org/10.1007/s10711-014-9987-x

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