Résumé
In [DP12], Delzant and Py showed that there exist continuous irreducible isometric actions of$\mathrm{PSL}_2(\mathbb{R})$on the infinite-dimensional hyperbolic space$\mathbb{H}^\infty$ . Such continuous irreducible actions do not exist on the hyperbolic spaces$\mathbb{H}^n$when$n>2$and their associated embeddings$\mathbb{H}^2 \to \mathbb{H}^\infty$given by the orbit maps were later called exotic by Monod and Py in [MP14]. In this article, we produce a continuous and irreducible representation of$\mathrm{PSL}_2(\mathbb{R})\to \mathrm{Isom}(\mathbb{H}^\infty)$using the hyperbolic model for convex bodies introduced in [DF22]. This yields a convex cocompact$\mathrm{PSL}_2(\mathbb{R})$ -action on the infinite-dimensional hyperbolic space$\mathbb{H}^\infty$ , of which the compact quotient over the minimal$\mathrm{PSL}_2(\mathbb{R})$ -invariant convex set is homeomorphic to the 2-dimensional oriented Banach--Mazur compactum. Moreover, we study the geometry of one of its orbit maps and compute the Hausdorff dimension of the limit set of this representation.