Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic 3-manifolds

TL;DR

This paper proves that every closed hyperbolic 3-manifold contains immersed quasi-Fuchsian surfaces with odd Euler characteristic, using an enhanced connection principle and good pants method, with applications to homological torsion growth.

Contribution

It introduces an improved connection principle enabling the construction of immersed quasi-Fuchsian surfaces of odd Euler characteristic in all closed hyperbolic 3-manifolds.

Findings

Existence of immersed quasi-Fuchsian surfaces of odd Euler characteristic in all closed hyperbolic 3-manifolds.

Application to sequences of sublattices with exponential homological torsion growth.

Enhanced connection principle for connecting frames in prescribed homology classes.

Abstract

In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two frames with a path of frames in a prescribed relative homology class of the frame bundle. The existence result is applied to show that every uniform lattice of $\mathrm{PSL}(2,\mathbb{C})$ admits an exhausting nested sequence of sublattices with exponential homological torsion growth. However, the constructed sublattices are not normal in general.