Journal of Differential Geometry

Virtual homological torsion of closed hyperbolic 3-manifolds

Hongbin Sun

Abstract

In this paper, we will use Kahn and Markovic’s immersed almost totally geodesic surfaces to construct certain immersed $\pi_1$-injective $2$-complexes in closed hyperbolic $3$-manifolds. Such $2$-complexes are locally almost totally geodesic except along a $1$-dimensional subcomplex. By using Agol’s result that the fundamental groups of closed hyperbolic $3$-manifolds are vitually compact special, and other works on geometric group theory, we will show that any closed hyperbolic $3$-manifold virtually contains any prescribed subgroup in the homological torsion. More precisely, our main result is, for any finite abelian group $A$, and any closed hyperbolic $3$-manifold $M, M$ admits a finite cover $N$, such that $A$ is a direct summand of $\mathit{Tor}(H_1(N; \mathbb{Z}))$.

Article information

Source
J. Differential Geom., Volume 100, Number 3 (2015), 547-583.

Dates
First available in Project Euclid: 28 May 2015

https://projecteuclid.org/euclid.jdg/1432842365

Digital Object Identifier
doi:10.4310/jdg/1432842365

Mathematical Reviews number (MathSciNet)
MR3352799

Zentralblatt MATH identifier
1350.57030

Citation

Sun, Hongbin. Virtual homological torsion of closed hyperbolic 3-manifolds. J. Differential Geom. 100 (2015), no. 3, 547--583. doi:10.4310/jdg/1432842365. https://projecteuclid.org/euclid.jdg/1432842365