Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic \(3\)-manifolds.

*(English)*Zbl 1419.57031Every closed hyperbolic \(3\)-manifold contains an immersed quasi-Fuchsian closed subsurface of even Euler characteristic (orientable and non-orientable) by the work of J. Kahn and V. Markovic [Ann. Math. (2) 175, No. 3, 1127–1190 (2012; Zbl 1254.57014)] and H. Sun [J. Differ. Geom. 100, No. 3, 547–583 (2015; Zbl 1350.57030)]. This article addresses the case of odd Euler characteristic, by showing that if \(M\) is a closed hyperbolic \(3\)-manifold, then there exists a connected closed surface \(\Sigma\) of odd Euler characteristic which admits a \(\pi_1\)-injective, quasi-Fuchsian immersion into \(M\). The strategy consists in constructing a suitable oriented panted subsurface \(F\) with boundary, thus obtaining the required surface \(\Sigma\) by attaching a MĂ¶bius band onto \(\partial F\). The tricky part is to obtain \(F\) with the required characteristics. To do so, the author develops an enhanced version of the connection principle in good pants constructions, which allows to connect frames by good paths of frames in any prescribed relative homology class. The author is then able to construct a good curve \(\gamma\) whose canonical lift \(\widetilde{\gamma}\) in the frame bundle SO\((M)\) is null homologous, implying that \(\gamma\) bounds the required good panted subsurface \(F\).

The main result of this paper answers an open question of I. Agol [in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. I: Plenary lectures and ceremonies. Seoul: KM Kyung Moon Sa. 141–170 (2014; Zbl 1379.57001)], and has two interesting consequences, of different flavours. On the one hand, it proves that every closed hyperbolic \(3\)-manifold has an orientable finite cover which admits a degree-one map onto the real projective \(3\)-space. This follows from the fact that an orientable closed \(3\)-manifold containing an embedded closed subsurface of odd Euler characteristic admits a degree-one map onto the real projective \(3\)-space [C. Hayat-Legrand et al., Pac. J. Math. 176, No. 1, 19–32 (1996; Zbl 0877.57007)], by passing to a finite cover. On the other hand, a more algebraic result can be deduced, stating certain exponential torsion growth for uniform lattices of PSL\((2, \mathbb{C})\). More precisely: every uniform lattice \(\Gamma\) of PSL\((2, \mathbb{C})\) contains an exhausting nested sequence of non-normal torsion-free sublattices \(\{\Gamma_i\}_{i\in \mathbb{N}}\) such that \[ \liminf_{n \to \infty} \frac{\log |H_1(\Gamma_n; \mathbb{Z})_{\text{tors}}|}{[\Gamma : \Gamma_n]} > 0. \] Here the sequence being nested means that each subgroup \(\Gamma_n\) contains its successor \(\Gamma_{n+1}\), and being exhausting means that the common intersection of all the subgroups is trivial. The term in the logarithmic function is the cardinality of the torsion subgroup of the first integral homology of the group \(\Gamma_n\).

The main result of this paper answers an open question of I. Agol [in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. I: Plenary lectures and ceremonies. Seoul: KM Kyung Moon Sa. 141–170 (2014; Zbl 1379.57001)], and has two interesting consequences, of different flavours. On the one hand, it proves that every closed hyperbolic \(3\)-manifold has an orientable finite cover which admits a degree-one map onto the real projective \(3\)-space. This follows from the fact that an orientable closed \(3\)-manifold containing an embedded closed subsurface of odd Euler characteristic admits a degree-one map onto the real projective \(3\)-space [C. Hayat-Legrand et al., Pac. J. Math. 176, No. 1, 19–32 (1996; Zbl 0877.57007)], by passing to a finite cover. On the other hand, a more algebraic result can be deduced, stating certain exponential torsion growth for uniform lattices of PSL\((2, \mathbb{C})\). More precisely: every uniform lattice \(\Gamma\) of PSL\((2, \mathbb{C})\) contains an exhausting nested sequence of non-normal torsion-free sublattices \(\{\Gamma_i\}_{i\in \mathbb{N}}\) such that \[ \liminf_{n \to \infty} \frac{\log |H_1(\Gamma_n; \mathbb{Z})_{\text{tors}}|}{[\Gamma : \Gamma_n]} > 0. \] Here the sequence being nested means that each subgroup \(\Gamma_n\) contains its successor \(\Gamma_{n+1}\), and being exhausting means that the common intersection of all the subgroups is trivial. The term in the logarithmic function is the cardinality of the torsion subgroup of the first integral homology of the group \(\Gamma_n\).

Reviewer: Alex Casella (Tallahassee)