## Journal of Differential Geometry

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

Yi Liu

#### 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.

#### Note

The author was supported by the Recruitment Program of Global Youth Experts of China.

#### Article information

Source
J. Differential Geom., Volume 111, Number 3 (2019), 457-493.

Dates
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.jdg/1552442607

Digital Object Identifier
doi:10.4310/jdg/1552442607

Mathematical Reviews number (MathSciNet)
MR3934597

Zentralblatt MATH identifier
07036513

#### Citation

Liu, Yi. Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic $3$-manifolds. J. Differential Geom. 111 (2019), no. 3, 457--493. doi:10.4310/jdg/1552442607. https://projecteuclid.org/euclid.jdg/1552442607