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Homological properties of Dunfield-Thurston random 3-manifolds
With a view to understanding the so-called "virtual Haken
conjecture", N. Dunfield and W. Thurston have recently studied a certain
class of random 3-manifolds, and developed tools to grasp some of their
homological properties. We will first summarize their constructions
(from a very down-to-earth point of view!), and then explain how
variants of sieve methods lead to quantitative versions of (some of)
their results, which suggest that a "typical" 3-manifold should have
finite, but quite large, first homology group (with integer
coefficients). Other results due to J. Maher and remaining problems
will also be mentioned.
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