Local rigidity of the right-multiplication action on a hyperbolic lattice quotient

Determine whether the right-multiplication action of $SO_0(n-1,1)$ on $\Gamma\backslash SO_0(n,1)$ is locally rigid whenever $n>2$, $\Gamma<SO_0(n,1)$ is a lattice, and the action is uniquely ergodic, equivalently when the associated compact hyperbolic manifold contains no totally geodesic immersed hypersurface.

Background

The paper constructs examples showing that the right-multiplication action of SO0(n−1,1)SO_0(n-1,1) on Γ\SO0(n,1)\Gamma\backslash SO_0(n,1) need not be C0C^0-locally rigid when the compact hyperbolic manifold associated with the uniform lattice Γ\Gamma contains an embedded totally geodesic hypersurface.

The authors then ask whether unique ergodicity, which they identify with the absence of a totally geodesic immersed hypersurface, is sufficient to force local rigidity. This question is presented as potentially characterizing which lattices admit nontrivial deformations into SO(n,2)SO(n,2).

References

Let $n >2$, let $H = SO(n,1)$, $L = SO(n-1,1)$, $\Gamma < H$ be a lattice, and suppose that the right-multiplication action of $L$ is uniquely ergodic (this is equivalent to saying that $\Gamma \backslash X$ does not contain a totally geodesic immersed hypersurface), is this action locally rigid?

— On some aspects of discrete groups acting ergodically on the boundary  (2608.27274 - Dey et al., 27 Aug 2026) in Section 5, subsection “Deformations of actions”