Extension to semi-arithmetic hyperbolic three-manifolds

Extend the methods for exponential growth of mean multiplicities and generalized modular embeddings from semi-arithmetic Fuchsian groups to suitable semi-arithmetic lattices in PSL(2,C) acting on hyperbolic three-space.

Background

The paper suggests that the Aurich–Marklof example points toward a higher-dimensional analogue, but its arguments are developed only for surfaces and Fuchsian groups. A rigorous formulation and proof in the PSL(2,C) setting remain unresolved.

References

Can the methods of this paper be extended to semi-arithmetic lattices in $PSL(2,C)=\Isom+(H3)$?

— Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension  (2608.17604 - Zuevsky, 18 Aug 2026) in Section 7, Concluding remarks, subsection “Three-dimensional generalization”