Ergodicity of bending deformations on the Furstenberg boundary

Determine whether the bending deformations $\Gamma_t=\Gamma*_{\pi_1(S)}a^t\Gamma a^{-t}$ in $SO(n,2)$ act ergodically on the Furstenberg boundary $G/P$ for every $t>0$, where $\Gamma<SO(n,1)$ is associated with a hyperbolic manifold having nonempty totally geodesic boundary.

Background

The paper considers a hyperbolic manifold with totally geodesic boundary, doubles it to obtain a lattice, and constructs bending or bulging deformations into SO(n,2)SO(n,2). The main theorem proves ergodicity only for sufficiently small deformations near the inclusion representation.

The authors leave unresolved whether ergodicity persists throughout the entire one-parameter family of bending deformations, rather than merely near the undeformed subgroup.

References

Does $\Gamma_t$ act ergodically on $G/P$ for all $t > 0$?

— On some aspects of discrete groups acting ergodically on the boundary  (2608.27274 - Dey et al., 27 Aug 2026) in Section 5, subsection “Speculations about the critical exponents of compact hyperbolic manifolds with boundary”