On 4-dimensional convex projective domains invariant by a lattice of $\mathrm{SL}_2 (\mathbb{R})$

This paper is a sequel to the erratum by the authors to a paper by Crampon and Marquis (see arXiv:1202.5442). The main result of the Erratum was relating several notions of geometrical finiteness in round convex projective geometry and we prove here that our series of implications was sharp, by providing counterexamples to the implications that were not established. Our counterexamples are 4-dimensional convex domains $\Omega$ acted on by $\rho (\Gamma)$ where $\Gamma$ is a lattice of $\mathrm{SL}_2 (\mathbb R)$ and $\rho$ is the irreducible representation of $\mathrm{SL}_2 (\mathbb R)$ of dimension $5$. We give a description of all $\rho(\Gamma)$-invariant convex domains, and in particular we construct one which is "close enough" to the convex hull $\mathcal C$ of the limit set of $\rho(\Gamma)$ so that the Hilbert volume $\mathrm{Vol}_{\Omega/\Gamma}(\mathcal C/\Gamma)$ of the convex core is infinite. We include an appendix with a smoothing procedure in the spirit of Cooper, Long and Tillman (arXiv:1511.06206) and Danciger, Gu\'eritaud and Kassel (arXiv:1704.08711).