Virtual Homological Torsion of Closed Hyperbolic 3-manifolds

  title={Virtual Homological Torsion of Closed Hyperbolic 3-manifolds},
  author={Hongbin Sun},
  journal={arXiv: Geometric Topology},
  • Hongbin Sun
  • Published 5 September 2013
  • Mathematics
  • arXiv: Geometric Topology
In this paper, we will use Kahn-Markovic's almost totally geodesic surfaces to construct certain $\pi_1$-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost totally geodesic except along a 1-dimensional subcomplex. Using Agol and Wise's result that fundamental groups of hyperbolic 3-manifolds are LERF and quasi-convex subgroups are virtual retract, we will show that closed hyperbolic 3-manifolds virtually contain any prescribed homological torsion: For… 

Figures from this paper

Virtual domination of 3–manifolds
For any closed oriented hyperbolic 3-manifold M, and any closed oriented 3-manifold N, we will show that M admits a finite cover M ' , such that there exists a degree-2 map f : M ' ! N, i.e. M
Prescribed virtual homological torsion of 3-manifolds
We prove that given any finite abelian group $A$ and any irreducible $3$-manifold $M$ with empty or toroidal boundary which is not a graph manifold there exists a finite cover $M' \to M$ so that $A$
Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic $3$-manifolds
  • Yi Liu
  • Mathematics
    Journal of Differential Geometry
  • 2019
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,
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,
Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston
We will discuss the proof of Waldhausen’s conjecture that compact aspherical 3-manifolds are virtually Haken, as well as Thurston’s conjecture that hyperbolic 3manifolds are virtually fibered. The
Torsion in the homology of finite covers of 3-manifolds
Let $N$ be a prime 3-manifold that is not a closed graph manifold. Building on a result of Hongbin Sun and using a result of Asaf Hadari we show that for every $k\in\Bbb{N}$ there exists a finite
Virtual 1-domination of 3-manifolds
It is shown in this paper that given any closed oriented hyperbolic 3-manifold, every closed oriented 3-manifold is mapped onto by a finite cover of that manifold via a map of degree 1, or in other
Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds
The vision and results of William Thurston (1946–2012) have shaped the theory of 3-dimensional manifolds for the last four decades. The high point was Perelman’s proof of Thurston’s Geometrization
Rank gradient of sequences of subgroups in a direct product
For a sequence $\{U_n\}_{n = 1}^\infty$ of finite index subgroups of a direct product $G = A \times B$ of finitely generated groups, we show that $$\lim_{n \to \infty} \frac{\min\{|X| : \langle X
Virtual homological spectral radii for automorphisms of surfaces
  • Yi Liu
  • Mathematics
    Journal of the American Mathematical Society
  • 2020
In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.


Complex Fenchel-Nielsen coordinates with small imaginary parts
Kahn and Markovic \cite{KahnMark} proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every
Volumes of hyperbolic Haken manifolds, I
In [14] a program was initiated for using the topological theory of 3-manifolds to obtain lower bounds for volumes of hyperbolic 3-manifolds. In [1], by a combination of new geometric ideas with
The virtual Haken conjecture
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are
Approximating L2-Invariants and Homology Growth
In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and
L2-topological invariants of 3-manifolds
SummaryWe give results on theL2-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain
Renormalization and 3-Manifolds Which Fiber over the Circle
Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of
Exponential mixing for the geodesic flow on hyperbolic three-manifolds
We give a short and direct proof of exponential mixing of geodesic flows on compact hyperbolic three-manifolds with respect to the Liouville measure. This complements earlier results of
Immersing almost geodesic surfaces in a closed hyperbolic three manifold
Let M 3 be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into M 3 so that for each, one the corresponding mapping on the universal covering spaces is an
Counting essential surfaces in a closed hyperbolic three-manifold
Let M^3 be a closed hyperbolic three-manifold. We show that the number of genus g surface subgroups of π_1(M^3) grows like g^(2g).