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

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.