Journal of Differential Geometry

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

Yi Liu

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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.


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

Article information

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

Received: 2 August 2016
First available in Project Euclid: 13 March 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F40: Kleinian groups [See also 20H10] 57M10: Covering spaces

good pants quasi-Fuchsian homological torsion growth


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.

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