Convex co-compact representations of \(3\)-manifold groups (Q6564511)

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space.\N\NThe paper under review is dedicated to to the study of convex co-compact representations of \(3\)-manifold groups. The authors show that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is either geometric (with \(\mathbb{R}^{3}\), \(\mathbb{H}^{3}\) or \(\mathbb{R} \times \mathbb{H}^{2}\) geometry) or when every component in the geometric decomposition is hyperbolic. Hence in the non-geometric case, the fundamental group is relatively hyperbolic with respect to a collection of rank-two abelian subgroups. In this case the authors prove that the convex co-compact representation, like an Anosov representation, induces an equivariant embedding of the boundary of the group (in his case the Bowditch boundary, see [\textit{B. H. Bowditch}, Int. J. Algebra Comput. 22, No. 3, Article ID 1250016, 66 p. (2012; Zbl 1259.20052)]). However, unlike an Anosov representation, the image is not into a flag manifold but instead into a naturally defined quotient.