An explicit exotic representation of a rank-one simple Lie group via convex bodies

In [DP12], Delzant and Py showed that there exist continuous irreducible isometric actions of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ on the infinite-dimensional hyperbolic space ${\mathbb{H}}^{\mathrm{\infty }}$. Such continuous irreducible actions do not exist on the hyperbolic spaces ${\mathbb{H}}^n$ when $n>2$ and their associated embeddings ${\mathbb{H}}^2\to {\mathbb{H}}^{\mathrm{\infty }}$ given by the orbit maps were later called \emph{exotic} by Monod and Py in [MP14]. In this article, we produce a continuous and irreducible representation of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})\to \mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathbb{H}}^{\mathrm{\infty }})$ using the hyperbolic model for convex bodies introduced in [DF22]. This yields a convex cocompact ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$-action on the infinite-dimensional hyperbolic space ${\mathbb{H}}^{\mathrm{\infty }}$, of which the compact quotient over the minimal ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$-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.