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

Abstract

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 \emph{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.

v2: 37 pages, 5 figures, minor modifications, comments are welcome