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

2025/12/13 by Fillastre, François, Long, Yusen, Xu, David
#22D12 #22E70 #52A39 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2512.12369

Abstract

In [DP12], Delzant and Py showed that there exist continuous irreducible isometric actions of PSL₂(ℝ) on the infinite-dimensional hyperbolic space ℍ^∞. Such continuous irreducible actions do not exist on the hyperbolic spaces ℍⁿ when n>2 and their associated embeddings ℍ² → ℍ^∞ 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 PSL₂(ℝ)→ Isom(ℍ^∞) using the hyperbolic model for convex bodies introduced in [DF22]. This yields a convex cocompact PSL₂(ℝ)-action on the infinite-dimensional hyperbolic space ℍ^∞, of which the compact quotient over the minimal PSL₂(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.

Citations

• Hyperbolic embedding of infinite-dimensional convex bodies
• Complex hyperbolic representations of SL2(R)
• Notes on functions of hyperbolic type
• Hyperbolic geometry of shapes of convex bodies
• Hausdorff dimension and Kleinian groups

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