Virtual homological torsion of closed hyperbolic 3-manifolds

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 $\mathit{Tor}(H_1(N; \mathbb{Z}))$.