Critical-exponent lower bound for compact hyperbolic three-manifolds with totally geodesic boundary

Prove or disprove that the critical exponent of the fundamental group of every compact hyperbolic three-manifold with totally geodesic boundary satisfies $\delta_{\Gamma}\geq 3/2$.

Background

The proposed lower bound is motivated by the relationship between ergodicity of the bending deformations in SO(n,2)SO(n,2) and the growth indicator function. The authors explain that a positive answer to the preceding ergodicity question could imply a lower bound on the limiting growth rate, and hence on the critical exponent.

They specifically specialize the expected inequality to dimension three, obtaining the threshold $3/2$. They also note that an example involving a noncompact hyperbolic three-manifold has critical exponent approximately $1.3056867280$, so the heuristic may fail outside the compact setting or may require further refinement.

References

Does the critical exponent of a compact hyperbolic three-manifold $\Gamma \backslashH3$ with totally geodesic boundary satisfy $\delta_{\Gamma} \geq 3/2$?

— 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”