Hurwitz-Radon numbers and proper actions of semisimple Lie groups

TL;DR

This paper investigates proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces, revealing a rigidity phenomenon and connecting the existence of such actions to Hurwitz-Radon numbers.

Contribution

It establishes that only groups isomorphic to Spin(n,1) can act properly on certain boundary symmetric spaces and links these actions to Hurwitz-Radon numbers.

Findings

Proper actions by non-compact semisimple Lie groups are rigidly restricted to Spin(n,1).

Hurwitz-Radon numbers determine the maximal n for Spin(n,1)-proper actions.

Includes pseudo-Riemannian hyperbolic spaces as key examples.

Abstract

We study proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper $SL(2,\mathbb{R})$-actions [J. Differential Geom., 2013], we focus on symmetric spaces lying on the boundary of the existence of proper $SL(2,\mathbb{R})$-actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on these symmetric spaces must be globally isomorphic to $Spin(n,1)$ up to compact factors. Moreover, the Hurwitz-Radon number arises as the largest value of $n$ for the existence of $Spin(n,1)$-proper actions. Our symmetric spaces include the pseudo-Riemannian hyperbolic space $\mathbf{H}_{+}^{N,N-1}$ of signature $(N,N-1)$.