# Finite covering spaces of 3-manifolds

@inproceedings{Lackenby2011FiniteCS, title={Finite covering spaces of 3-manifolds}, author={Marc Lackenby}, year={2011} }

Following Perelman's solution to the Geometrisation Conjecture, a 'generic' closed 3-manifold is known to admit a hyperbolic structure. However, our understand- ing of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyper- bolic 3-manifold has a finite cover that contains a closed embedded orientable �1-injective surface with positive genus. I will give a survey on the progress…

## 21 Citations

Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston

- Mathematics
- 2014

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…

On the treewidth of triangulated 3-manifolds

- MathematicsSoCG
- 2018

It is shown that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth, then the Heegaard genus of M is at most 48(k+1) (resp. pathwidth) k, and it is not shown that there exists an infinite family of closed 3- manifolds not admitting triangulations of bounded pathwidth or trewidth.

Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds

- Mathematics
- 2014

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…

Statistics of Random 3-Manifolds occasionally fibering over the circle

- Mathematics
- 2014

We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the…

Mapping Class Groups, Homology and Finite Covers of Surfaces

- Mathematics
- 2012

Let S be an orientable surface of genus g with n punctures, such that χ(S) = 2 − 2g − n < 0. Let ψ ∈ Mod(S) denote an element in its mapping class group. In this thesis, we study the action of ψ on…

On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

- Mathematics
- 2012

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application…

Heegaard genera in congruence towers of hyperbolic 3-manifolds

- Mathematics
- 2011

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their…

Even triangulations of n–dimensional pseudo-manifolds

- Mathematics
- 2015

This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the…

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