  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 S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show
that if the genus of S is large, then this action is very mixing. In
particular, the action factors through the alternating group of each orbit.
This is analogous to Goldman's theorem that the action of the mapping class
group on the SU(2) character variety is ergodic.
