## Finite covers of random 3-manifolds (2005)

Citations: | 11 - 0 self |

### BibTeX

@MISC{Dunfield05finitecovers,

author = {Nathan M. Dunfield and William P. Thurston},

title = {Finite covers of random 3-manifolds},

year = {2005}

}

### OpenURL

### Abstract

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface Σ on

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Citation Context ...al. Recent work of the first author and Dylan Thurston shows a similar sharp divergence behavior with respect to fibering over the circle, where here a group “fibers” if is an algebraic mapping torus =-=[DT1]-=-. Surprisingly, the 3-manifolds studied there were much less likely to fiber than similar finitely presented groups. 61.7. Outline of contents. In Section 2, we discuss several different models of ra... |

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Citation Context ...nt technology, it would be difficult to enumerate all subgroups of π1(M) for indices beyond the low 20s. For more about 3-manifolds from the point of view of branched covers of the figure-8 knot, see =-=[Hem1]-=-. 2.12. Random knots based notions. Another notion of random 3-manifold would be to take a Dehn surgery point of view. That is, one could take some notion of a random knot or link in S 3 and do Dehn s... |

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Citation Context ...ining curves is ≤ L. 2.11. Universal link related notions. There are links L in S 3 such that every closed orientable 3-manifold is the cover of S 3 branched over L. One such link is the figure8 knot =-=[HLM]-=-. Let K be this knot and M be its exterior . There are only finitely many branched covers of (S 3 , K) of degree ≤ L, since such a cover corresponds to a finite-index subgroup of π1(M). Thus another n... |